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# Thermally driven fission of protocells

Romain Attal  Cité des Sciences et de l’Industrie
30, avenue Corentin-Cariou 75019 Paris, France<EMAIL_ADDRESS>
###### Abstract.

We propose a simple mechanism for the self-replication of protocells. Our main
hypothesis is that the amphiphilic molecules composing the membrane bilayer
are synthesised inside the protocell through globally exothermic chemical
reactions. The slow increase of the inner temperature forces the hottest
molecules to move from the inner leaflet to the outer leaflet of the bilayer.
This asymmetric translocation process makes the outer leaflet grow faster than
the inner leaflet. This differential growth increases the mean curvature and
amplifies any local shrinking of the protocell until it splits in two.

###### Key words and phrases:

protocell, bilayer, translocation, self-replication, thermodynamical
instability.

###### Contents

  1. 1 Protocells and metabolism
  2. 2 Hypotheses of our model
  3. 3 Flows, forces and energy dissipation
    1. 3.1 Main irreversible processes
    2. 3.2 Flows associated to each irreversible process
    3. 3.3 Thermodynamical forces
    4. 3.4 Conductance matrix
    5. 3.5 Entropy production and stability
  4. 4 Membrane geometry and growth equation
    1. 4.1 Conservation of matter and exponential growth
    2. 4.2 Cylindrical growth in steady state
  5. 5 Thermal instability of cylindrical growth
    1. 5.1 The Squeezed Sausage Theorem (SST)
    2. 5.2 Fluctuations, translocation and heat transfer
    3. 5.3 Thermal balance
  6. 6 Translocation between leaflets
    1. 6.1 An effective potential for translocation
    2. 6.2 Computation of $L_{m\theta}$
    3. 6.3 Computation of $L_{mm}$
    4. 6.4 Estimation of $L_{\theta\theta}$
    5. 6.5 Destabilisation
  7. 7 Conclusions and perspectives
  8. A The mean curvature of the membrane
  9. B Solutions of the growth equation
    1. B.1 Case 1 : $\eta\neq 1$ and $\tau>\tau_{+}(\eta)$ or $\tau<\tau_{-}(\eta)$
    2. B.2 Case 2 : $\eta\neq 1$ and $\tau\in\\{\tau_{+}(\eta),\tau_{-}(\eta)\\}$
    3. B.3 Case 3 : $\tau_{-}(\eta)<\tau<\tau_{+}(\eta)$
  10. C Smooth perturbation of cylindrical growth
    1. C.1 Isovolumic variation of the area
    2. C.2 Isovolumic variation of the total mean curvature
  11. D Asymptotic expansion of $F(a)$

##  1\. Protocells and metabolism

The objects modeled in the present article are protocells, the putative
ancestors of modern living cells [23, 34]. In the absence of fossils [38], we
ignore their detailed properties. However, we can sketch a minimalist
functional diagram of protocells (FIG. 1).

FoodWasteMetabolismHeatBiomass

Protocells initiate the fundamental process of life : Food $\to$ Biomass +
Heat + Waste.

The protocell is a vesicle bounded by a bilayer made of amphiphilic molecules.
Nutrient molecules (food) enter by mere diffusion, since they are consumed
inside, where their concentration is lower than outside. Conversely, waste
molecules have a larger concentration inside and therefore diffuse passively
to the outside. The metabolism is a network of unknown chemical reactions
taking place only inside the protocell. The net reaction is supposed to be
exothermic, since living matter is hotter than abiotic matter (under the same
external thermodynamical conditions).

Let us compare this scheme to actual evolved cells. The growth of bacteria in
a nutrient rich medium follows a species dependent periodic process [5, 12].
At regular time intervals, each cell splits to form two daughter cells. This
requires the synchronization of numerous biochemical and mechanical processes
inside the cell, involving cytoskeletal structures positioned at the locus of
the future cut (septum). However, in the history of life, such complex
structures are a high-tech luxury and must have appeared much later than the
ability to split. Protocells must have used a simple splitting mechanism to
ensure their reproduction, before the appearance of genes, RNA, enzymes and
all the complex organelles present today even in the most rudimentary forms of
autonomous life [34].

In this article, we present a simple model for the growth and self-replication
of a protocell, following the laws of irreversible thermodynamics near
equilibrium. Our guide is the rate of entropy production, which is minimal in
a steady state [32, 20]. A key point of our approach is that the heat produced
by the metabolism of the protocell is approximately proportional to its
volume, whereas the heat flow that it can loose is proportional to the area of
its membrane. In a rod-shaped cell (bacillus) growing linearly, these two
quantities are approximately proportional so that each increment of the
membrane area should be sufficient, ideally, to evacuate the heat produced by
the corresponding increment of the cell volume. However, the irreversible
physical and chemical processes produce heat more quickly than the growing
tubular membrane can dissipate to the outside. This increases slowly the inner
temperature and enhances the fluctuations of the shape of the membrane, of the
various concentrations and of the local electric field.

In a growing spherical protocell, the maximal heat flow that the membrane can
expell to the outside without overheating the inside puts an upper limit to
the radius of the protocell. Indeed, the formation of two small protocells
from a big one releases work [33], so that large protocells are mechanically
unstable. However, neither [33] nor [8] provides a path to follow to realise
this deformation.

In our model, we start from a cylindrical shape to simplify the computations.
As the inner temperature increases, the growth of the outer leaflet of the
membrane becomes more probable than the growth of the inner leaflet. If a
random thermal fluctuation lowers slightly the radius of this cylinder, then
its area increases more quickly than during the steady state cylindrical
growth (FIG. 1).

$\downarrow$$\downarrow$$\downarrow$hotcold

Splitting a cylindrical protocell.

This reduction of the radius induces a loss of convexity of the membrane. This
favors the outflow of heat and the ratio area/volume increases slightly,
compared to a convex cylindrical shape.

The plan of the article goes as follows. In Section II, in order to formulate
these ideas mathematically, we state all the physical hypotheses of our model
of protocells. In Section III, we define the various flows of matter and
energy and their associated thermodynamical forces. In the linear
approximation, the rate of entropy production is the scalar product of these
flows and forces and is minimal in a steady state [32]. In Section IV, we
derive a differential equation for the evolution of the area and the integral
of the mean curvature of the membrane, starting from the advancement of the
chemical reaction for the synthesis of the membrane molecules. This linear
equation admits a solution growing exponentially. In Section V, we use
variational calculus [10] to prove that the local reduction of the radius of
the cell increases its length and its area, if its volume is kept constant.
This intuitive property implies that heat is more easily released when the
protocell is squeezed. In Section VI, we propose a molecular mechanism for the
increase of the mean curvature of the membrane associated to this squeezing.
The position of each amphiphilic membrane molecule is reduced to a single
degree of freedom : the distance from the polar head to the middle of the
hydrophobic slice. We use a double well effective potential to describe the
trapping of these molecules in the membrane. Due to the temperature difference
between the inner and outer sides, the membrane molecules go from the inner
leaflet to the outer leaflet more often than in the opposite direction. This
asymmetry forces the membrane to curve and shrink around the middle of the
protocell and initiates its splitting. Our main mathematical result
(Proposition VI.I) states that a stability condition,
$L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$, can not be satisfied at high
temperature, because the squared crossed conductance, $L_{m\theta}^{2}$,
increases more quickly than the product of the diffusion coefficients,
$L_{mm}$ for membrane molecules and $L_{\theta\theta}$ for heat. Hence, the
cylindrical growth process is unstable when the temperature difference is
sufficiently high. We conclude in Section VII with a proposition of an
experimental test for our model. The appendices contain the detailed
computations of our model. The mathematical notions involved are elementary
(linear differential equations and geometry of surfaces).

##  2\. Hypotheses of our model

Let us state more precisely the hypotheses underlying our model :

  1. (1)

Our protocells are made of a membrane of average thickness $2\varepsilon$,
bounding a cytosol of finite volume $\mathcal{V}(t)$.

  2. (2)

The cytosol contains unknown specific molecules (reactants, catalysers,
chromophores, …) which participate to a network of chemical reactions. We
suppose that the concentrations are constant and uniform in the volume
$\mathcal{V}$.

  3. (3)

The protocell starts with a cylindrical shape closed by two hemispherical caps
of fixed radius, $R_{0}$. The total length, $\ell(t)+2R_{0}+2\varepsilon$,
increases with time due to the synthesis of membrane molecules (FIG. 3).

$\ell(t)+2R_{0}+2\varepsilon$$2\varepsilon$${2(R_{0}-\varepsilon)}$${2(R_{0}+\varepsilon)}$$\ell(t)$

Geometry of an idealised cylindrical protocell.

This may seem a rather drastic hypothesis, but the computations could be made
for a generic, approximately spherical shape using an expansion in spherical
harmonics. This would add to the model an unnecessary mathematical complexity
that would hide the main physical phenomena. The use of cylindrical, rotation
invariant shapes allows us to reduce the problem to one dimension. Moreover,
this is a best case scenario for the release of heat in steady state, since
the ratio volume/area can be held constant in a steady growth.

  4. (4)

Due to the surface tension of the membrane, its mean curvature has an upper
bound, $H_{\max}=\mfrac{1}{R_{0}}$. Indeed, due to the attractive forces
between the polar heads of the membrane molecules, and due to their geometry,
they can not form structures arbitrarily small [27].

  5. (5)

Food (nutrients and water) enters the protocell by mere passive diffusion
through the membrane. Waste and heat also diffuse passively but in the
opposite direction. Protocells did not use specialized membrane molecules for
an active transport through the membrane.

  6. (6)

The membrane molecules are synthesised inside the protocell in an unknown
network of chemical reactions. It might use some encapsulated catalyzers or
chromophores trapped in the volume and catching part of the ambient light
[23], but we will make no hypothesis on the details of this network.

  7. (7)

These metabolic reactions generate heat to be evacuated and increase slowly
the internal temperature, $T_{1}$, whereas the external temperature, $T_{0}$,
remains fixed.

  8. (8)

The characteristic time of the variations of $T_{1}(t)$ is much larger than
the characteristic times of chemical reactions and diffusion processes across
the membrane.

  9. (9)

The cytosol is homogenous and contains no organelles, no cytoskeleton, no
enzymes, no RNA/DNA. Just simple chemical reactants uniformly distributed.
(Rashevsky’s model [33] allows for a slight radial variation of concentrations
due to the diffusion of food and waste through the membrane).

  10. (10)

The membrane is a bilayer made of unspecified amphiphilic molecules. We
presume that their hydrophobic tails are long enough (10-12 carbon atoms) to
form a stable bilayer, but not too bulky in order to allow flip-flop (or
translocation) processes between the two leaflets. We do not include sterol
molecules because they are the product of a long biochemical selection process
[27], and a high-tech luxury for protocells.

  11. (11)

The inner leaflet (L1) is at temperature $T_{1}$ whereas the outer leaflet
(L0) is at temperature $T_{0}<T_{1}$. This temperature drop allows the bilayer
to undergo coupled transport phenomena (food and waste diffusion, including
water leaks, heat diffusion, flip-flop, etc.).

  12. (12)

The membrane may contain other molecules, in small concentrations, but we
don’t need them to transport food, waste or any molecule through the membrane.

The validity of these hypotheses will depend on the agreement of their
predictions with the results of future experiments made with real protocells.

##  3\. Flows, forces and energy dissipation

In any living system, some processes release energy whereas other processes
consume energy. Globally, the system takes usable energy from the outside and
rejects unusable energy, in the form of heat and waste, that can be used by
other living systems. In order to describe such a system, we must define the
various flows of matter and energy and the forces causing these flows. Any
gradient of concentration, pressure, temperature, etc. will cause a current of
particles, fluid, heat, etc. These processes are generally irreversible and
dissipate energy to inaccessible degrees of freedom. This dissipation of a
conserved quantity is measured by the entropy function, which increases as
time passes.

The study of irreversible thermodynamical processes near equilibrium [29, 30,
35, 20] is based on the rate of entropy production, represented by a bilinear
function of flows (chemical reaction speed, thermal current, particle current,
electric current, etc.) and forces (chemical affinity, temperature gradient,
concentration gradient, electric tension, etc.). In a first approximation,
flows and forces are related linearly, as in Ohm’s law :

(1) $\displaystyle\text{electric current }=\text{ conductivity }\times\text{
electric field}$

and the power dissipated is a quadratic function of the tension :

(2) power dissipated $\displaystyle=\text{ tension }\times\text{ current}$
$\displaystyle=\text{ conductance }\times\text{ tension}^{2}.$

Similarly, in viscous fluids :

(3) power dissipated $\displaystyle=\text{ friction coefficient }\times\text{
velocity}^{2}.$

We suppose that the protocell metabolism is in a steady state not too far from
equilibrium, so that the various flows, $J_{i}$, and the thermodynamic forces,
$X_{k}$, are linearly related :

(4) $\displaystyle J_{i}=\sum_{k}L_{ik}X_{k}$

and the entropy rate is a quadratic function of $X$ :

(5) $\displaystyle\sigma:=XJ=\sum_{ik}X_{i}L_{ik}X_{k}.$

The coefficients $L_{ik}$ are called phenomenological because their
computation depends on the chosen model of microscopic dynamics (kinetic
theory) and their numerical value has to be compared to a measurement in the
real world to (in)validate this model and the linearity hypothesis. An
important property of the phenomenological coefficients is provided by
Onsager’s relations [29, 30, 20, 35]. Under the hypotheses of microscopic
reversibility and parity of the variables under time reversal (in particular,
in the absence of magnetic coupling and vorticity), the matrix $L$ is
symmetric :

(6) $\displaystyle L_{ik}=L_{ki}.$

This important law has been checked experimentally for various systems near
equilibrium and is satisfied quite accurately in many cases.

###  3.1. Main irreversible processes

To each irreversible physical or chemical process are associated a flow of
matter or energy and a thermodynamical force, just as an electric current and
an electric tension correspond to each branch of an electric network. If we
identify the main processes that take place during the growth of a protocell,
we can compute the global rate of dissipation of energy, or entropy creation.
According to Prigogine’s Theorem [32, 11], this rate reaches a minimum when
the system is in a steady state.

In order to compute this dissipation, we need to define the various
compartiments containing energy. In the sequel of this article, the subscript
$0$ (resp. $1$) will denote the variables outside (resp. inside) the
protocell. The physical and chemical processes are grouped as follows :

$f_{0}\to f_{1}\,$ :

food molecules (nutrients $+$ water) diffuse into the protocell through the
membrane.

$f\to m+c+w\,$ :

food is transformed into membrane, cytosol and waste, inside the protocell.
This is a global process, a superposition of catabolism and anabolism. Taking
into account the stoichiometric coefficients, we can write more precisely :

(7) $\displaystyle\sum_{i}\nu_{f_{i}}f_{i}\ \longrightarrow\
\sum_{j}\nu_{m_{j}}m_{j}+\sum_{k}\nu_{c_{k}}c_{k}+\sum_{l}\nu_{w_{l}}w_{l}$

where $f_{i}$ denotes the food molecules of type $i$, $m_{j}$ the membrane
molecules of type $j$, $c_{k}$ the cytosol molecules of type $k$ and $w_{l}$
the waste molecules of type $l$. If $N_{\alpha}$ is the number of molecules of
type $\alpha$, the advancement of this reaction, $\xi$, is defined by :

(8) $\displaystyle\mathrm{d}\xi:=\frac{\mathrm{d}N_{\alpha}}{\pm\nu_{\alpha}}$

where the stoichiometric coefficients, $\nu_{\alpha}$, are counted positively
for the products and negatively for the reactants. Note that our definition of
$\xi$ involves $N_{\alpha}$ instead of the volumic concentration,
$C_{\alpha}=N_{\alpha}/\mathcal{V}$, because the volume is not fixed.

$w_{1}\to w_{0}\,$ :

waste molecules diffuse out of the protocell through the membrane.

$m_{1}\leftrightarrows m_{0}\,$ :

molecules of the membrane bilayer go from one side to the other. In modern
cells, this process is catalysed by enzymes (flippase for $0\to 1$ and
floppase for $1\to 0$), but in protocells such a complex machinery did not
exist yet [34]. If we suppose that the first membranes were not as thick as
today (most phospholipids in modern and healthy cell walls have hydrophobic
chains made of $\sim$ 16-22 atoms of carbon [27]), the exchange of molecules
between the two leaflets could have been possible in a reasonable time to
allow spontaneous splitting. Medium length lipids (10-14 atoms of carbon)
could be good candidates to make stable, flippable and not too porous
protocells. We isolate the process of translocation
($\restrictwand\xleftrightharpoons{}\restrictwandup$) because the ratio
${N_{m0}}/{N_{m1}}$ of the numbers of membrane molecules on each side is
related to the mean curvature of the bilayer, which is the geometric parameter
monitoring the splitting process.

$q_{1}\to q_{0}\,$ :

electric charges can be transfered from one side of the membrane to the other,
by an ionic bound on the polar head of the membrane molecules. This electric
current builds up an electric tension, $U_{01}$, counteracted by possible
ionic leaks through the membrane. If we suppose that the membrane molecules
are monovalent fatty acids, each one can carry a monocation (H+, Na+, K+, …).
This cotransport process could be the ancestor of the modern sodium-potassium
pump. Anions also can participate to this transmembrane electric current, by
leaking throuh water pores [14].

###  3.2. Flows associated to each irreversible process

The main processes of our model are described by the following flows in the
protocell (see FIG. 3.2) :

$J_{f}\,$ :

the flow of food entering the protocell through its membrane (molecules per
unit time per unit area).

$J_{w}\,$ :

the flow of waste exiting the protocell through its membrane (molecules per
unit time per unit area).

$J_{\theta}\,$ :

the heat flow exiting the protocell by diffusion through its membrane (energy
per unit time per unit area).

$J_{mab}\,$ :

the flow of membrane molecules from $a$ to $b$ (molecules per unit time per
unit area). The possible values of $a$ and $b$ are :

$c\,$ :

the cytosol ;

$1\,$ :

the inner leaflet of the membrane (L1) ;

$0\,$ :

the outer leaflet of the membrane (L0) ;

The net flow of membrane molecules is usually unidirectional,
${\mathbf{C}}\to{\mathbf{L}}_{1}\to{\mathbf{L}}_{0}$, hence $J_{mc1}>0$ and
$J_{m}:=J_{m10}-J_{m01}>0$.

$J_{r}\,$ :

the speed of the synthesis reaction inside the cytosol (molecules per unit
time per unit volume). $\xi$ being the advancement of the reaction $f\to
m+c+w$, defined above, then $J_{r}$ is the time derivative of $\xi$ :

(9) $\displaystyle J_{r}:=\frac{\mathrm{d}\xi}{\mathrm{d}t}.$

$J_{q}\,$ :

some ions can be transported from one side to the other, bounded to the polar
head of the membrane molecules.

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m+c+w$$J_{mc1}$${\it{w}=}$waste$J_{w}$$J_{\theta}$ heat
diffusion$\textbf{E}=\text{environment}$$T_{0}$$\boxed{\textbf{C}=\text{cytosol}}$$T_{1}(t)$

Main flows of energy and matter in our model.

We then have the following linear flow diagram for the synthesis and motion of
membrane molecules :

(10)
$\displaystyle{\mathbf{E}}\xlongrightarrow{J_{f}}{\mathbf{C}}\xlongrightarrow{J_{r}}{\mathbf{C}}\xlongrightarrow{\color[rgb]{.1,.6,1}J_{mc1}}{\mathbf{L}}_{1}\xlongrightarrow{\color[rgb]{.1,.6,1}J_{m10}}{\mathbf{L}}_{0}.$

This picture is however slightly misleading. Indeed, the amphiphilic molecules
being in a liquid phase, their positions fluctuate in each leaflet
(transversal diffusion) and they undergo perpendicular motions (protrusion)
and translocations from one leaflet to the other. The pictures obtained by
molecular dynamics simulations [14, 15, 4, 1] give us a more precise
representation of real world membranes.

###  3.3. Thermodynamical forces

The thermodynamical forces associated to these processes are defined as
follows :

$X_{\theta}\,$ :

the thermal force is the difference of the inverse temperatures inside and
outside the protocell :

(11) $\displaystyle X_{\theta}:=\frac{1}{T_{0}}-\frac{1}{T_{1}}>0.$

$X_{f_{i}}\,$ :

the chemical force driving the food molecules, $f_{i}$, is the difference of
the ratios $-\mu_{f_{i}}/T$ outside and inside the protocell :

(12) $\displaystyle
X_{f_{i}}:=\frac{\mu_{f_{i}0}}{T_{0}}-\frac{\mu_{f_{i}1}}{T_{1}}.$

The influx of food is guided by mere diffusion through the membrane (dedicated
channel and intrinsic proteins did not exist yet in protocells). Since food is
consumed inside the protocell, $[f_{i}]_{1}<[f_{i}]_{0}$. For a spherical
protocell, the profile of the concentration of each molecule (as a function of
the distance to the center) can be computed by solving the diffusion equation
[33]. An important result of this computation is the existence of a
discontinuity in the concentration of each molecule, $f_{i}$, proportional to
the radius, $R$, of the protocell, to the rate of the reaction, $q_{i}$
(concentration/time), and inversely proportional to the permeability, $h_{i}$
(length/time), of the membrane for this molecule :
$[f_{i}]_{1}-[f_{i}]_{0}\propto\mfrac{q_{i}R}{h_{i}}$.

$X_{w_{j}}\,$ :

the force driving the waste molecules to the outside of the protocell is the
difference of chemical potentials divided by the temperature :

(13) $\displaystyle
X_{w_{j}}:=\frac{\mu_{w_{j}0}}{T_{0}}-\frac{\mu_{w_{j}1}}{T_{1}}.$

Note that $X_{w_{j}}$ and $X_{f_{i}}$ must have different signs for waste and
food to go in opposite directions.

$X_{r}\,$ :

the chemical force driving the synthesis reactions (metabolism) is the
chemical reaction affinity, $A_{r}$, of the global process $(f\to m+c+w)$,
divided by the inner temperature of the protocell :

(14) $\displaystyle X_{r}:=\frac{A_{r}}{T_{1}}.$

This affinity is a linear combination of the chemical potentials of the
synthesis equation, weighted by the stoichiometric coefficients, counted
positively for the reactants $(f)$ and negatively for the products $(m,c,w)$ :

(15) $\displaystyle
A_{r}=\sum_{i}\nu_{f_{i}}\mu_{f_{i}}-\sum_{j}\nu_{m_{j}}\mu_{m_{j}}-\sum_{k}\nu_{c_{k}}\mu_{c_{k}}-\sum_{l}\nu_{w_{l}}\mu_{w_{l}}.$

$X_{m^{\prime}}\,$ :

The membrane molecules are synthesised in the cytosol at temperature $T_{1}$.
Their hydrophobic tail enforces the spontaneous organisation of these
molecules into a bilayer. We suppose that the temperature varies only across
the membrane. The driving force of this isothermal process is the affinity of
the reaction $m_{c}\to m_{1}$, divided by the inner temperature, $T_{1}$ :

(16) $\displaystyle
X_{m^{\prime}}=\frac{A_{mc1}}{T_{1}}=\frac{\mu_{mc}-\mu_{m1}}{T_{1}}.$

Here, $\mu_{mc}$ is the chemical potential of the free membrane molecules
inside the cytosol and $\mu_{m1}$ is their chemical potential in the inner
leaflet. The heat released to the inner leaflet during this process is :

(17) $\displaystyle Q_{mc1}=\mu_{mc}-\mu_{m1}=T_{1}X_{m^{\prime}}.$

$X_{m}\,$ :

the membrane molecules are transfered from the inner layer, at temperature
$T_{1}$, to the outer leaflet, at temperature $T_{0}<T_{1}$, releasing the
heat $Q_{m10}$ into the environmental thermostat, at temperature $T_{0}$. The
thermodynamical force of this process is :

(18) $\displaystyle X_{m}=\frac{\mu_{m1}}{T_{1}}-\frac{\mu_{m0}}{T_{0}}.$

$X_{q}\,$ :

the thermodynamical force driving the ions of species $i$, of charge $z_{i}e$,
across the membrane is the difference of electrochemical potentials [2] :

(19) $\displaystyle X_{qi}$ $\displaystyle=\tilde{\mu}_{i1}-\tilde{\mu}_{i0}$
$\displaystyle=\big{(}\mu_{i1}+z_{i}e\psi_{1}\big{)}-\big{(}\mu_{i0}+z_{i}e\psi_{0}\big{)}$
$\displaystyle=\mu_{i1}-\mu_{i0}+z_{i}eU_{10}.$

where $\psi$ denotes the electrostatic potential and
$U_{10}:=\psi_{1}-\psi_{0}$ is the electric tension across the membrane.

Among these forces, only $X_{\theta}$ is a linear function of the small
temperature difference, $\Delta T=T_{1}-T_{0}$. The others have, generically,
a supplementary constant term, of order $0$ in $\Delta T$.

###  3.4. Conductance matrix

The phenomenological coefficients, $L_{ik}$, which couple all the irreversible
processes of our linear model, can be put in a $7\times 7$ matrix :

(20) $\displaystyle L=\begin{pmatrix}L_{\theta\theta}&L_{\theta f}&L_{\theta
w}&L_{\theta m}&L_{\theta m^{\prime}}&L_{\theta q}&L_{\theta r}\\\
L_{f\theta}&L_{ff}&L_{fw}&L_{fm}&L_{fm^{\prime}}&L_{fq}&L_{fr}\\\
L_{w\theta}&L_{wf}&L_{ww}&L_{wm}&L_{wm^{\prime}}&L_{wq}&L_{wr}\\\
L_{m\theta}&L_{mf}&L_{mw}&L_{mm}&L_{mm^{\prime}}&L_{mq}&L_{mr}\\\
L_{m^{\prime}\theta}&L_{m^{\prime}f}&L_{m^{\prime}w}&L_{m^{\prime}m}&L_{m^{\prime}m^{\prime}}&L_{m^{\prime}q}&L_{m^{\prime}r}\\\
L_{q\theta}&L_{qf}&L_{qw}&L_{qm}&L_{qm^{\prime}}&L_{qq}&L_{qr}\\\
L_{r\theta}&L_{rf}&L_{rw}&L_{rm}&L_{rm^{\prime}}&L_{rq}&L_{rr}\\\
\end{pmatrix}.$

In a first approximation, some coefficients can be set equal to zero :

(21) $\displaystyle L\simeq\begin{pmatrix}L_{\theta\theta}&L_{\theta
f}&L_{\theta w}&L_{\theta m}&0&L_{\theta q}&0&\\\
L_{f\theta}&L_{ff}&0&0&0&0&0\\\ L_{w\theta}&0&L_{ww}&0&0&0&0\\\
L_{m\theta}&0&0&L_{mm}&0&L_{mq}&0\\\ 0&0&0&0&L_{m^{\prime}m^{\prime}}&0&0\\\
L_{q\theta}&0&0&L_{qm}&0&L_{qq}&0\\\ 0&0&0&0&0&0&L_{rr}\\\ \end{pmatrix}.$

The diagonal coefficients of $L$ are positive but we let $L_{\bullet
r}=0=L_{r\bullet}$ because the synthesis reactions take place in the cytosol
and are decoupled from the transport processes across the membrane. Similarly,
we let $L_{\bullet m^{\prime}}=0=L_{m^{\prime}\bullet}$, because the transfer
of membrane molecules from the cytosol to the inner leaflet is decoupled from
the other processes. Since the diffusion processes of different molecules
(food, waste, ions or membrane constituents) across the membrane are supposed
to be decoupled, we put $L_{fw}=0=L_{wf}$, $L_{fm}=0=L_{mf}$ and
$L_{wm}=0=L_{mw}$.

$L_{\theta\theta}$ is the thermal diffusion coefficient across the membrane.
$L_{m^{\prime}m^{\prime}}$ is the diffusion coefficient for the transport of
membrane molecules from the cytosol to the inner leaflet of the membrane.
$L_{ff}$, $L_{ww}$, and $L_{mm}$, are the conductance coefficients of food,
waste and membrane molecules through the membrane. We suppose that all these
diagonal coefficients are constant and uniform across the cytosol or the
membrane, because protocells could not rely on local specialised channel
molecules (intrinsic proteins, in evolved cells) to supply their food and
evacuate their waste. We also suppose that food and waste molecules are
electrically neutral and that the electric current is entirely due to the
transport of small ions with the help of the translocation process and water
pores.

The off-diagonal coefficients, $L_{f\theta}=L_{\theta f}$,
$L_{w\theta}=L_{\theta w}$, $L_{m\theta}=L_{\theta m}$ and
$L_{q\theta}=L_{\theta q}$, depend on the heat capacity of the molecules
transported and on the rate constants of this transport. They couple the
transport of matter and the heat flow. For our purpose, the most interesting
off-diagonal coefficient is $L_{\theta m}$. It can be viewed as the ratio of
heat flow, $J_{\theta}$, to the affinity $X_{m}$ when $T_{0}=T_{1}$ and in the
absence of food and waste driving forces :

(22) $\displaystyle L_{\theta
m}=\left(\frac{J_{\theta}}{X_{m}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}.$

In this case, the thermal flow is due only to the asymmetry of the membrane,
induced by its bending. This phenomenon is similar to the Dufour effect [20].
If one can prove experimentally that a bending of the membrane induces a heat
flow through it, this means that $L_{\theta m}\neq 0$, hence, by Onsager’s
reciprocity relations, $L_{m\theta}\neq 0$, i.e. a heat flow modifies the
bending. Indeed, we also have the relation :

(23) $\displaystyle
L_{m\theta}=\left(\frac{J_{m}}{X_{\theta}}\right)_{(X_{m},X_{f},X_{w},X_{m^{\prime}})=0}.$

Hence, $L_{m\theta}$ measures the effect of a slight temperature difference
(between both sides of the membrane) on the induced flow of molecules between
the leaflets, which implies a modification of its mean curvature. This
phenomenon is similar to the thermodiffusion or Soret effect [20]. It is
reciprocal to the previous effect and might be easier to observe and measure.

###  3.5. Entropy production and stability

Just as the power dissipated by Joule effect in an ohmic conductor is

(24) Power dissipated $\displaystyle=\text{Current}\times\text{Voltage}$
$\displaystyle=\text{Conductance}\times\text{Voltage}^{2},$

the rate of dissipation of energy, or entropy creation, in a general chemical
system out of equilibrium is a quadratic function of the thermodynamical
forces acting in the system [32, 20] :

(25) Rate of entropy produced $\displaystyle=\text{Flows}\times\text{Forces}$
$\displaystyle=\text{Forces}\times\text{Conductance
matrix}\times\text{Forces}.$

This relation rests on a linearity hypothesis supposed to be valid only in the
neighbourhood of an equilibrium state. The main difference between the ohmic
conductor and the chemical system is that, in the latter, the conductance is
not a single number but a matrix which, in the general case, couples all the
currents. Taking into account the various thermodynamical forces defined
previously, the rate of entropy production inside the protocell has to the
following expression :

(26) $\displaystyle\sigma(X)$
$\displaystyle=L_{ff}X_{f}^{2}+L_{ww}X_{w}^{2}+L_{mm}X_{m}^{2}+L_{m^{\prime}m^{\prime}}X_{m^{\prime}}^{2}+L_{rr}X_{r}^{2}+L_{\theta\theta}X_{\theta}^{2}+2X_{\theta}(L_{\theta
f}X_{f}+L_{\theta w}X_{w}+L_{\theta m}X_{m})$
$\displaystyle=L_{ff}\left(\frac{\mu_{f0}}{T_{0}}-\frac{\mu_{f1}}{T_{1}}\right)^{2}+L_{ww}\left(\frac{\mu_{w0}}{T_{0}}-\frac{\mu_{w1}}{T_{1}}\right)^{2}+L_{mm}\left(\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}\right)^{2}$
$\displaystyle\qquad+L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)^{2}+L_{m^{\prime}m^{\prime}}\left(\frac{\mu_{mc}-\mu_{m1}}{T_{1}}\right)^{2}+L_{rr}\left(\frac{A_{r}}{T_{1}}\right)^{2}$
$\displaystyle\qquad+2\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\left(L_{\theta
f}\left(\frac{\mu_{f0}}{T_{0}}-\frac{\mu_{f1}}{T_{1}}\right)+L_{\theta
w}\left(\frac{\mu_{w0}}{T_{0}}-\frac{\mu_{w1}}{T_{1}}\right)+L_{\theta
m}\left(\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}\right)\right).$

The stability of this steady state is equivalent to the positivity of the
matrix $L$, which is also the matrix of second order derivatives of $\sigma$
in the coordinate system
$X=(X_{\theta},X_{f},X_{w},X_{m},X_{m^{\prime}},X_{r})$ :

(27) $\displaystyle L_{ik}=\frac{1}{2}\,\frac{\partial^{2}\sigma}{\partial
X_{i}\partial X_{k}}.$

If $P$ is a $n\times n$ matrix with real coefficients, the positivity of $P$,
defined by :

(28) $\displaystyle u^{\text{t}}Pu>0\qquad\forall u\in\mathbb{R}^{n}$

implies the following inequalities :

(29) $\displaystyle P_{ii}>0\quad\forall\,i\qquad\text{and}\qquad
P_{ii}P_{jj}>\left(\frac{P_{ij}+P_{ji}}{2}\right)^{2}\quad\forall\,i,j.$

These conditions are necessary but not sufficient to ensure the positivity of
$P$. In the present case, $L$ being symmetric, we have, in particular :

(30) $\displaystyle L_{ii}$ $\displaystyle>0$ $\displaystyle
L_{\theta\theta}L_{ff}$ $\displaystyle>L_{\theta f}^{2}$ $\displaystyle
L_{\theta\theta}L_{ww}$ $\displaystyle>L_{\theta w}^{2}$ $\displaystyle
L_{\theta\theta}L_{qq}$ $\displaystyle>L_{\theta q}^{2}$ $\displaystyle
L_{mm}L_{qq}$ $\displaystyle>L_{mq}^{2}$ $\displaystyle
L_{\theta\theta}L_{mm}$ $\displaystyle>L_{\theta m}^{2}.$

If one of these inequalities is not satisfied, the growth process is
destabilized. In Section VI, we will prove that the last one can be reversed
as the inner temperature of the protocell increases. In order to prove this
proposition, we must first write down evolution equations for the geometry of
the cell.

##  4\. Membrane geometry and growth equation

Just as the growth of a child depends on his diet, the evolution of the
geometric parameters of a protocell depends on the flow of molecules to its
membrane. This flow is determined by the food intake and by the rate of the
synthesis of these structural molecules. In this section, we establish the
differential equations governing the growth of the volume and area of a
cylindrical protocell by relating them to the flows of matter.

###  4.1. Conservation of matter and exponential growth

The advancement, $\xi$, of the overall synthesis reaction, $f\to m+c+w$, is
the internal clock of the protocell. The corresponding flow of matter,
$J_{r}=\mfrac{\mathrm{d}\xi}{\mathrm{d}t}$, is channeled to all the other
processes in the protocell. In particular, it determines the flux of matter to
the inner leaflet and the growth speed of the membrane. By writing the
equations of conservation of matter, we can then determine the evolution of
the size of the protocell.

Let $a\in\\{c,1,0\\}$ denote the possible position of a membrane molecule :
either in the cytosol $(c)$, or the inner leaflet $(1)$ or the outer leaflet
$(0)$. Let $N_{ma}$ be the number of membrane molecules in each of them. The
time derivatives of these functions are related to the flows defined
previously :

(31) $\displaystyle\frac{\mathrm{d}N_{mc}}{\mathrm{d}t}$
$\displaystyle=-J_{mc1}\mathcal{A}_{1}+J_{rm}\mathcal{V}$
$\displaystyle\frac{\mathrm{d}N_{m1}}{\mathrm{d}t}$
$\displaystyle=J_{mc1}\mathcal{A}_{1}-J_{m10}\mathcal{A}$
$\displaystyle\frac{\mathrm{d}N_{m0}}{\mathrm{d}t}$
$\displaystyle=J_{m10}\mathcal{A}.$

Similarly, the number of food (resp. cytosol and waste) molecules, $N_{f}$
(resp. $N_{c}$ and $N_{w}$), evolves according to the following relations :

(32) $\displaystyle\frac{\mathrm{d}N_{f}}{\mathrm{d}t}$
$\displaystyle=J_{f}\mathcal{A}_{0}-J_{rf}\mathcal{V}$
$\displaystyle\frac{\mathrm{d}N_{c}}{\mathrm{d}t}$
$\displaystyle=J_{rc}\mathcal{V}$
$\displaystyle\frac{\mathrm{d}N_{w}}{\mathrm{d}t}$
$\displaystyle=-J_{w}\mathcal{A}_{1}+J_{rw}\mathcal{V}$

where the flows $J_{r\bullet}$ are defined by :

(33) $\displaystyle J_{rm}$
$\displaystyle:=\nu_{m}\frac{\mathrm{d}\xi}{\mathrm{d}t}$ $\displaystyle
J_{rf}$
$\displaystyle:=\nu_{f}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{f}}{\nu_{m}}\,J_{rm}$
$\displaystyle J_{rc}$
$\displaystyle:=\nu_{c}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{c}}{\nu_{m}}\,J_{rm}$
$\displaystyle J_{rw}$
$\displaystyle:=\nu_{w}\frac{\mathrm{d}\xi}{\mathrm{d}t}\,=\,\frac{\nu_{w}}{\nu_{m}}\,J_{rm}.$

In a steady state, the concentration of membrane molecules in the cytosol is
constant :

(34) $\displaystyle C_{mc}:=\frac{N_{mc}}{\mathcal{V}}=\text{cst.}$

Let $c_{m0}$ and $c_{m1}$ be the average number of membrane molecules per unit
area in each leaflet :

(35) $\displaystyle
c_{m0}:=\frac{N_{m0}}{\mathcal{A}_{0}}\qquad\text{and}\qquad
c_{m1}:=\frac{N_{m1}}{\mathcal{A}_{1}}.$

The conservation equations for $m$ imply the evolution equations of the
geometry of the protocell :

(36) $\displaystyle c_{m0}\frac{\mathrm{d}\mathcal{A}_{0}}{\mathrm{d}t}$
$\displaystyle=J_{m10}\,\frac{\mathcal{A}_{0}+\mathcal{A}_{1}}{2}$
$\displaystyle c_{m1}\frac{\mathrm{d}\mathcal{A}_{1}}{\mathrm{d}t}$
$\displaystyle=J_{mc1}\mathcal{A}_{1}-c_{m0}\frac{\mathrm{d}\mathcal{A}_{0}}{\mathrm{d}t}$
$\displaystyle c_{mc}\frac{\mathrm{d}\mathcal{V}}{\mathrm{d}t}$
$\displaystyle=J_{rm}\mathcal{V}-J_{mc1}\mathcal{A}_{1}.$

Let us introduce the following parameters :

(37) $\displaystyle 2\varepsilon$ $\displaystyle:=\text{average thickness of
the membrane}$ $\displaystyle\eta$
$\displaystyle:=\frac{c_{m1}}{c_{m0}}\quad\text{(layer density ratio $\simeq
1$)}$ $\displaystyle\tau$
$\displaystyle:=\frac{J_{m10}}{J_{mc1}}\quad\text{(transmission rate {through}
the membrane)}$ $\displaystyle t_{1}$
$\displaystyle:=\frac{c_{m1}}{J_{mc1}}\quad\text{(inner leaflet characteristic
time)}$ $\displaystyle\tau_{c}$
$\displaystyle:=\frac{J_{mc1}}{J_{rm}}\quad\text{(transmission rate {to} the
membrane)}$ $\displaystyle t_{c}$
$\displaystyle:=\frac{c_{mc}}{J_{rm}}\quad\text{(cytosol characteristic
time)}.$

The transmission ratio, $\tau$, can be written in terms of thermodynamical
forces :

(38)
$\displaystyle\tau:=\frac{J_{m}}{J_{mc1}}=\frac{L_{mm}X_{m}+L_{m\theta}X_{\theta}+{\ldots}}{L_{m^{\prime}m^{\prime}}X_{m^{\prime}}}.$

Let $\mathcal{U}={\mathcal{V}}/{\varepsilon}$ and
$\dot{X}=t_{1}\mfrac{\mathrm{d}X}{\mathrm{d}t}$. We obtain the following
system of differential equations :

(39) $\displaystyle\dot{\mathcal{A}}_{0}$
$\displaystyle=\frac{\eta\tau}{2}\,(\mathcal{A}_{0}+\mathcal{A}_{1})\,=\,\eta\tau\mathcal{A}$
$\displaystyle\dot{\mathcal{A}}_{1}$
$\displaystyle=-\frac{\tau}{2}\,\mathcal{A}_{0}+\left(1-\frac{\tau}{2}\right)\,\mathcal{A}_{1}\,=\,(1-\tau)\mathcal{A}-\mathcal{B}$
$\displaystyle\dot{\mathcal{U}}$
$\displaystyle=\frac{t_{1}}{t_{c}}\,\mathcal{U}-\frac{c_{m1}}{\varepsilon
c_{mc}}\,\mathcal{A}_{1}.$

In matrix form :

(40) $\displaystyle\dot{X}$
$\displaystyle=\begin{pmatrix}\dot{\mathcal{A}}_{0}\\\
\dot{\mathcal{A}}_{1}\\\ \dot{\mathcal{U}}\\\
\end{pmatrix}=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\
-\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon
c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\begin{pmatrix}\mathcal{A}_{0}\\\
\mathcal{A}_{1}\\\ \mathcal{U}\\\ \end{pmatrix}=MX$ $\displaystyle M$
$\displaystyle:=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\
-\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon
c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\quad\text{and}\quad
X:=\begin{pmatrix}\mathcal{A}_{0}\\\ \mathcal{A}_{1}\\\ \mathcal{U}\\\
\end{pmatrix}.$

This growth equation is solved in Appendix B. The matrix $M$ has a block
diagonal form, hence $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ evolve
independently of $\mathcal{U}$, whereas the equation for $\mathcal{U}$
contains terms linear in $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$. The upper
left $2\times 2$ block is not diagonal, hence $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ are linear combinations of exponential functions of time
(multiplied by an affine function of $t$ in the degenerate, non diagonalisable
case). The rates of growth of these exponential functions are the eigenvalues
of this $2\times 2$ block, plus an exponential of growth rate
$\mfrac{t_{1}}{t_{c}}$ for $\mathcal{U}$.

###  4.2. Cylindrical growth in steady state

When we meet an ordinary differential equation, describing the time evolution
of a dynamical system, a first reflex is to search for constant solutions or
at least steady state solutions, where the speed is constant. In the present
case, we can look for a solution where the length increases steadily whereas
the radius is constant. This corresponds to the observed growth of some
bacterial species in difficult environments [28]. When the sludge content of
wastewater is too high or when the composition is lopsided, a higher
percentage of bacteria adopt a filamentous growth strategy which allows them
to survive in harsher conditions, by catching food more easily.

If the protocell grows like a cylinder of radius $R_{0}$, we have
$\varepsilon\mathcal{A}=R_{0}\mathcal{B}$, hence
$\mfrac{\mathrm{d}\mathcal{A}}{\mathcal{A}}=\mfrac{\mathrm{d}\mathcal{B}}{\mathcal{B}}$
and

(41) $\displaystyle x$
$\displaystyle:=\frac{R_{0}}{\varepsilon}=\frac{\mathcal{A}}{\mathcal{B}}=\frac{\mathrm{d}\mathcal{A}}{\mathrm{d}\mathcal{B}}=\frac{\dot{\mathcal{A}}}{\dot{\mathcal{B}}}$
$\displaystyle=\frac{\big{(}(\eta-1)\tau+1\big{)}\mathcal{A}-\mathcal{B}}{\big{(}(\eta+1)\tau-1\big{)}\mathcal{A}+\mathcal{B}}$
$\displaystyle=\frac{\alpha_{+}\mathcal{A}-\mathcal{B}}{\alpha_{-}\mathcal{A}+\mathcal{B}}$

where $\alpha_{\pm}=(\eta\mp 1)\tau\pm 1$. Therefore, $x$ satisfies the fixed
point equation :

(42) $\displaystyle x=\frac{\alpha_{+}x-1}{\alpha_{-}x+1}\qquad\text{{\it i.e.
}}\qquad\alpha_{-}x^{2}-(\alpha_{+}-1)x+1=0.$

The discriminant of this quadratic equation is

(43) $\displaystyle(\alpha_{+}-1)^{2}-4\alpha_{-}$
$\displaystyle=(\eta-1)^{2}\tau^{2}-4(\eta+1)\tau+4$
$\displaystyle=4\Delta(\eta,\tau)$

(cf. Appendix B) and its roots, $x_{\pm}$, are related to the eigenvalues,
$\lambda_{\pm}$, of the matrix $M$ (Eq. 40) :

(44) $\displaystyle x_{\pm}$
$\displaystyle=\frac{1}{2\alpha_{-}}\left(\alpha_{+}-1\pm\sqrt{(\alpha_{+}-1)^{2}-4\alpha_{-}}\right)$
$\displaystyle=\frac{(\alpha_{+}-1)\pm
2\sqrt{\Delta(\eta,\tau)}}{2\alpha_{-}}$
$\displaystyle=\frac{2\lambda_{\pm}-1}{\alpha_{-}}.$

Consequently, the radius, $R_{0}$, of the cylinder whose length increases in a
steady state is determined by the flows $(J_{mc1},J_{m10},J_{rm})$ and the
concentrations $(C_{mc},c_{m1},c_{m0})$, via the coefficients
$(\varepsilon,\eta,\tau)$ :

(45) $\displaystyle R_{0}$ $\displaystyle=\varepsilon
x_{\pm}=\varepsilon\,\frac{\lambda_{\pm}-1}{2\alpha_{-}}=\frac{\varepsilon\big{(}(\eta+1)\tau\pm
2\sqrt{\Delta}\big{)}}{2\big{(}(\eta+1)\tau-1\big{)}}$
$\displaystyle=\frac{\varepsilon}{2}\,\frac{(\eta+1)\tau\pm\sqrt{(\eta-1)^{2}\tau^{2}-(\eta+1)\tau+1}}{(\eta+1)\tau-1}$

##  5\. Thermal instability of cylindrical growth

As long as the protocell grows by increasing only its length, keeping a
cylindrical shape of fixed radius, $R_{0}$, its volume and its membrane area
grow proportionally, i.e.
$\dot{\mathcal{A}}=\text{cst.}\times\dot{\mathcal{B}}$. If the heat generated
by the metabolic reactions were exactly proportional to the volume increment,
the increase of the area of the membrane would be sufficient to evacuate
steadily the heat generated by the chemical reactions taking place inside the
newly created volume. However, the heat generated by all these irreversible
processes adds up to that coming from the exothermic metabolic reactions and
the inner temperature must therefore increase. This overheating generates
larger fluctuations of all the physical parameters which destabilize the
initial steady state of cylindrical growth. We will see below that the
geometrical parameters $(\mathcal{A},\mathcal{B},\mathcal{V})$ can follow a
path leading to a more efficient release of heat, by reducing the radius
$R_{0}$.

###  5.1. The Squeezed Sausage Theorem (SST)

When we squeeze a sausage, its length increases as well as its area. Indeed,
the stuffing being incompressible, the squeezing is an isovolumic deformation.
The stuffing is pushed longitudinally, away from the squeezed zone, and
increases the length of the sausage, thanks to the elasticity of the gut. The
area of the slice of reduced radius increases consequently to bound the same
volume. Let us prove this mathematically.

A length $\delta x$ of cylinder of radius $R_{0}$ has volume
$\delta\mathcal{V}$ and boundary area $\delta A$ given by :

(46) $\displaystyle\delta\mathcal{V}$ $\displaystyle=\pi R_{0}^{2}\,\delta x$
$\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta x$

Let us suppose that this cylindrical growth is perturbed by a small, local
radius variation, which can be positive (anevrism) or negative (stenosis). We
study here a triangular perturbation and, in the appendix, a smooth
$(\mathcal{C}^{2})$, rotation invariant perturbation of the cylinder. To keep
it simple, we suppose that this perturbation is piecewise linear and
symmetric, with an extremum $\delta R$ at $x=0$, and vanishes outside of the
interval $\left[-\mfrac{\delta x^{\prime}}{2},\mfrac{\delta
x^{\prime}}{2}\right]$. FIG. 5.1 represents the resulting isovolumic
deformation according with the sign of $\delta R$.

$2R_{0}$$\delta x$$\delta\mathcal{V}$$\delta\mathcal{A}$$\delta
x^{\prime}>\delta x$

$2(R_{0}+\delta R)<2R_{0}$

$\delta\mathcal{V}^{\prime}=\delta\mathcal{V}$

$\delta\mathcal{A}^{\prime}>\delta A$

$2(R_{0}+\delta R)>2R_{0}$

$\delta\mathcal{V}^{\prime}=\delta V$

$\delta x^{\prime}<\delta x$$\delta\mathcal{A}^{\prime}<\delta A$

Isovolumic variation of the area of a cylinder under a small triangular
deformation.

In the second and third pictures of FIG. 5.1, the Gaussian curvature is
concentrated on the circular sections at $x=0$ and at $x=\pm\mfrac{\delta
x^{\prime}}{2}$ (dotted lines), where the mean curvature has a finite
discontinuity. The volume and lateral membrane area of this slice of thickness
$\delta x^{\prime}$ (contained between the dotted lines) are therefore :

(47) $\displaystyle\delta\mathcal{V}^{\prime}$
$\displaystyle=\pi\left(R_{0}+\frac{\delta R}{2}\right)^{2}\,\delta
x^{\prime}$ $\displaystyle\delta\mathcal{A}^{\prime}$
$\displaystyle=2\pi\left(R_{0}+\frac{\delta R}{2}\right)\,\delta x^{\prime}.$

The straight slice and the deformed slice have equal volumes
$(\delta\mathcal{V}=\delta\mathcal{V}^{\prime})$ if their thicknesses satisfy
:

(48) $\displaystyle\frac{\delta x^{\prime}}{\delta x}=\left(1+\frac{\delta
R}{2R_{0}}\right)^{-2}\simeq\left(1-\frac{\delta R}{R_{0}}\right).$

Hence the ratio of their areas is

(49)
$\displaystyle\frac{\delta\mathcal{A}^{\prime}}{\delta\mathcal{A}}\simeq\left(1+\frac{\delta
R}{2R_{0}}\right)\left(1-\frac{\delta R}{R_{0}}\right)\simeq 1-\frac{\delta
R}{2R_{0}}.$

The heat flows through these surfaces are, respectively :

(50) $\displaystyle\delta q$
$\displaystyle=L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\,\delta\mathcal{A}$
$\displaystyle\delta q^{\prime}$
$\displaystyle=L_{\theta\theta}\left(\frac{1}{T_{0}}-\frac{1}{T_{1}}\right)\,\delta\mathcal{A}^{\prime}$

hence their ratio is the same as for the areas :

(51) $\displaystyle\frac{\delta q^{\prime}}{\delta
q}=\frac{\delta\mathcal{A}^{\prime}}{\delta\mathcal{A}}=1-\frac{\delta
R}{2R_{0}}.$

When $\delta R<0$, this ratio is larger than $1$. Consequently, the inner
volume being held fixed, a small stenosis of a cylindrical protocell evacuates
heat more efficiently than a small anevrism. This local reduction of the
radius of the protocell increases its mean curvature. For this deformation to
happen, the outer leaflet must grow more rapidly than the inner leaflet.
Therefore, the equilibrium $m_{1}\leftrightarrows m_{0}$ must be shifted
towards $m_{0}$ in order to have $\delta R<0$. This is possible if $T_{1}$
increases slightly and $m_{1}\to m_{0}$ is exothermic. We propose that the
translocation of membrane molecules to the outer leaflet [14, 15, 4, 1] can be
triggered by the increase of the inner temperature, $T_{1}(t)$. The area of
the outer leaflet then increases more quickly than the area of the inner
leaflet, which leads to the bending of the membrane until the total splitting
of the protocell into two daughters.

###  5.2. Fluctuations, translocation and heat transfer

In order to increase $L_{m\theta}$ and destabilise the cylindrical growth, the
transfer coefficient, $\tau$, must also increase. In [14, 15], the authors
present a detailed mechanism for the transfer of membrane molecules between
the leaflets. Due to the fluctuations of ionic densities in the neighbourhood
of the membranes, the local electric field fluctuates strongly enough to push
molecules of water into the membrane, via the field-dipole interaction force
(dielectrophoresis). When it is sufficiently strong, this force can create a
transient water pore that is stable enough to let some membrane molecules dive
into this water pore and join the other side. The increase of the inner
temperature can also enhance these ionic density fluctuations and favor this
translocation process from the hot side to the cold side, since the hottest,
most agitated molecules have a higher probability to dive into the water pore
than the colder molecules. This asymmetric flow of hot molecules to the cold
side enhances the outgoing heat flow and cools down the protocell.

During this process, the shape of the hydrophobic tails is not important, as
long as they remain in the hydrophobic zone, surrounded by siblings. The only
energetic cost is for the hydrophilic head surrounded by these aliphatic
chains, and some clandestine water molecules forming the water pore (not
represented below). The shape of the tail is irrelevant since the energy
depends only on the position of the polar head (FIG. 5.2).

$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$hydrophilic
zonehydrophilic zonehydrophobic zone

Translocation of a membrane molecule from one leaflet to the other.

###  5.3. Thermal balance

Let us make a thermal balance of the whole growth process. After heating its
cold nutrient molecules from $T_{0}$ to $T_{1}$ and processing the isothermal
inner chemical reactions $(J_{r})$, our protocell disposes of its hot waste
(including some water flowing through the water pores) and loses heat by
translocation of membrane molecules from the inside to the outside, and by
diffusion $(J_{\theta})$ without mass transfer. Let $q_{i}$ be the heat
exported out of the protocell by each molecule of type $i$. Cold entering
molecules and hot outgoing molecules both have $q_{i}>0$. Let $\kappa_{i}$ be
the heat capacity of the molecules of type $i$. Let $J_{h}$ be the outgoing
heat flow (energy/(time $\times$ area)). The heat flow exported by the cold
entering food and water molecules is :

(52) $\displaystyle J_{f}q_{f}$ $\displaystyle=J_{f}\kappa_{f}(T_{1}-T_{0}).$

Similarly, the heat flow exported by the outgoing waste and water molecules is
:

(53) $\displaystyle J_{w}q_{w}$ $\displaystyle=J_{w}\kappa_{w}(T_{1}-T_{0}).$

And the heat flow exported by the net translocation of membrane molecules is :

(54) $\displaystyle J_{m}q_{m}$ $\displaystyle=J_{m}\kappa_{m}(T_{1}-T_{0})$

if we suppose that they immediately thermalise from $T_{1}$ to $T_{0}$ once
they reach the outer leaflet. The contact of the hydrophobic tails inside the
membrane allows for a diffusive heat flow :

(55) $\displaystyle J_{\theta}$ $\displaystyle=\sum_{i}L_{\theta k}X_{k}.$

The total heat flow is the sum of these terms :

(56) $\displaystyle J_{h}$
$\displaystyle:=(J_{f}q_{f}+J_{w}q_{w}+J_{m}q_{m})+J_{\theta}$
$\displaystyle=\big{(}J_{f}\kappa_{f}+J_{w}\kappa_{w}\big{)}(T_{1}-T_{0})$
$\displaystyle\quad+(L_{mm}X_{m}+L_{m\theta}X_{\theta}+{\ldots})\kappa_{m}(T_{1}-T_{0})$
$\displaystyle\quad+(L_{\theta\theta}X_{\theta}+L_{\theta m}X_{m}+{\ldots}).$

Since

(57) $\displaystyle
T_{1}-T_{0}=T_{0}T_{1}X_{\theta}=\frac{T_{0}^{2}X_{\theta}}{1-T_{0}X_{\theta}}$

$L_{m\theta}$ appears as a factor of $X_{\theta}^{2}$ in the convective term,
$J_{m}$, whereas $L_{\theta m}$ is a factor of $X_{m}$ in the diffusive term,
$J_{\theta}$. Moreover, $X_{m}$ increases linearly with $X_{\theta}$ :

(58) $\displaystyle X_{m}$
$\displaystyle=\frac{\mu_{m0}}{T_{0}}-\frac{\mu_{m1}}{T_{1}}$
$\displaystyle=\frac{\mu_{m}^{\circ}}{T_{0}}-\frac{\mu_{m}^{\circ}}{T_{1}}+k_{B}\ln\left(\frac{a_{m0}}{a_{m1}}\right)$
$\displaystyle=\mu_{m}^{\circ}X_{\theta}+k_{B}\ln\left(\frac{a_{m0}}{a_{m1}}\right)$

where $\mu^{\circ}$ denotes the standard chemical potential, at temperature
$298$ K and pressure $1$ atm [2]. The $L_{m\theta}$-dependent term in $J_{h}$
becomes :

(59) $\displaystyle
J_{h}=L_{m\theta}\left(\frac{\kappa_{m}T_{0}^{2}X_{\theta}^{2}}{1-T_{0}X_{\theta}}+\mu_{m}^{\circ}X_{\theta}\right)+{\ldots}$

Consequently, as the cytosol heats up, $J_{h}$ increases more quickly by
translocation ($\kappa_{m}$ term) than by diffusion ($\mu_{m}^{\circ}$ term).
Translocation is a particular kind of heat convection and by analogy with the
Rayleigh-Bénard instability [35], we conjecture the existence of a transition
from a diffusive regime to a convective regime, where translocation overtakes
diffusion and expells heat more efficiently.

##  6\. Translocation between leaflets

The energetic barrier, of width $2\varepsilon^{\prime}$ and height $E_{\ast}$,
is difficult to penetrate for the hydrophilic head since this guarantees the
stability of the bilayer under ordinary thermal fluctuations. When the ratio
of concentrations, $\eta=\mfrac{c_{m1}}{c_{m0}}$, becomes too large compared
to unity, the mechanical constraint on the inner leaflet is released by
pushing molecules to the outer leaflet. Conversely, when the outer leaflet is
stretched and the inner leaflet compressed, $\eta$ is slightly greater than
unity (FIG. 6).

$\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup\restrictwandup$$c_{m1}$compressed$c_{m0}<c_{m1}$stretched$\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand\restrictwand$

Mechanical constraints modify the ratio, $\eta$, of leaflet concentrations.

To facilitate this process, some water molecules can leak through the
hydrophobic zone and ease the passage of the hydrophilic head. This leakage of
water lowers the activation energy, $E_{\ast}$, and realizes an aqueous
catalysis of the translocation process [14, 15, 4, 1]. If we suppose that the
density, $n_{p}$, of water pores in the membrane is constant for fixed
temperatures, $T_{0}$ and $T_{1}$, then $J_{m10}$ depends only on this density
and on the net number, $j_{mp}$, of membrane molecules translocated from
$\mathbf{L}_{1}$ to $\mathbf{L}_{0}$ during the lifetime of the pores :

(60) $\displaystyle n_{p}$ $\displaystyle:=\text{ number of water pores per
unit area }$ $\displaystyle j_{mp}$ $\displaystyle:=\text{ net number of
translocations}$ through each water pore $\displaystyle J_{m}$
$\displaystyle=n_{p}j_{mp}.$

This first approximation is based on the hypothesis that the pores have the
same size, the same lifetime and the same number of net translocations during
their short life. However, to be more realistic, we must take into account the
fact that larger pores live longer and leak more (over the same duration) than
smaller short lived pores. We integrate over the interval of possible
lifetimes $(t_{p})$ the density of water pores of lifetime $t_{p}$ created per
unit time $(n_{p}(t_{p}))$ multiplied by the net number $(\nu_{mp}(t_{p}))$ of
molecules each pore of lifetime $t_{p}$ translocates from the inside to the
outside during its existence :

(61) $\displaystyle
J_{m}=\int_{0}^{\infty}\mathrm{d}t_{p}\,n_{p}(t_{p})\nu_{mp}(t_{p}).$

The increase of $X_{\theta}$ enhances at the same time the rate of formation
of pores, hence $n_{p}$, and the net number of translocated molecules, due to
larger thermal fluctuations. Therefore, $J_{m10}$ increases more than linearly
as a function of $X_{\theta}$. Consequently, the crossed conductivity
coefficient, $L_{m\theta}$, increases with $X_{\theta}$. On the other side of
the inequality, $L_{\theta\theta}$ and $L_{mm}$ depend more weakly on the
temperature. Indeed, the heat diffusion coefficient, $L_{\theta\theta}$,
involves the (temperature independant) number of interacting degrees of
freedom between the hydrophobic tails inside the hydrophobic layer, and the
molecular diffusion coefficient :

(62) $\displaystyle
L_{mm}=T_{0}\left(\frac{J_{m}}{\mu_{m1}-\mu_{m0}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$

depends mainly on the ratio of concentrations between the two leaflets, i.e.
on $\eta$. In order to know if the initial inequality,
$L_{m\theta}^{2}<L_{\theta\theta}L_{mm}$, can be reversed, the temperature
dependance of the convective coefficient, $L_{m\theta}$, must be computed and
compared to that of the diffusion coefficients, $L_{\theta\theta}$ and
$L_{mm}$. This necessitates a microscopic model of the interactions of
membrane molecules and water and a precise description of the translocation
process, to go beyond the linear response theory. In the sequel, we adopt a
simple mean field approach where each molecule evolves in the same energetic
landscape as the others.

###  6.1. An effective potential for translocation

The exact shape and position of each membrane molecule is described by dozens
of parameters specifying the position of each atom and the orientation of each
interatomic bond. It would be cumbersome to take them all into account to
describe mathematically the evolution of a single molecule inside the
membrane. However, we can make a simplifying approximation by remarking that
the main energetic cost is in the displacement of the hydrophilic head into
the hydrophobic layer or the protrusion of this head outside of the membrane,
which forces the tail to go into the hydrophilic zone. We can make a mean-
field approximation by considering only the position, $z$, of the hydrophilic
head as a dynamical variable, and defining an adequate effective potential
energy, $U(z)$, that traps the head inside the membrane. In the sequel of this
article, we will use a double well effective potential to compute the net
flow, $J_{m}$, across a plane bilayer subject to a difference of temperatures.
By differentiation, we obtain the coefficients $L_{m\theta}$ and $L_{mm}$ and,
in particular, their dependence on temperature. This model suggests that the
inequality $L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$ can be reversed if the
inner temperature increases sufficiently. Our hypotheses are the following
ones :

  1. (1)

The membrane molecules have length
$\varepsilon=\varepsilon^{\prime}+\varepsilon^{\prime\prime}$, where
$\varepsilon^{\prime\prime}$ is the size of the hydrophilic head and
$\varepsilon^{\prime}$ is the length of the hydrophobic tail.

  2. (2)

The translocation process is described by only one parameter : the position of
the center of mass of the hydrophilic head, varying between $-\varepsilon$ and
$+\varepsilon$.

  3. (3)

On each side of the membrane, the distribution of velocities of the heads
follows a Maxwell-Boltzmann law [35]. The probability of finding a molecule
with velocity $v$ perpendicularly to the membrane is :

(63) $\displaystyle p_{i}(v)=\sqrt{\frac{m}{2\pi
k_{B}T_{i}}}\exp\left(-\frac{mv^{2}}{2k_{B}T_{i}}\right).$

  4. (4)

The translocation requires an energy $E^{\ast}$ and the head of the molecule
evolves in an effective double well potential (FIG. 4).

hydrophilic zonepolar heads\+ water\+ ionshydrophobic zonealiphatic
chainshydrophilic zonepolar heads\+ water\+
ions$z$$-\varepsilon$$-\varepsilon^{\prime}$$T_{1}$$\varepsilon$$\varepsilon^{\prime}$$T_{0}$$E^{\ast}$$\color[rgb]{1,0,0}U(z)$

Potential energy of the hydrophilic head

  5. (5)

The hydrophilic heads trapped in the well
$[-\varepsilon,-\varepsilon^{\prime}]$ have temperature $T_{1}$, whereas those
trapped in the well $[\varepsilon^{\prime},\varepsilon]$ have temperature
$T_{0}$. The thermalisation processes for the motion along the $z$ axis occur
only once the head is trapped in the arrival well. This drastic hypothesis
simplifies the computations and should be refined in a more realistic model.
In reality, the motions of the hydrophobic tails between
$z=-\varepsilon^{\prime}$ and $z=+\varepsilon^{\prime}$ can thermalise the
molecule during the travel across the membrane and this affects the
translocation time.

Only half of the molecules of kinetic energy $E>E_{\ast}$ can escape from a
well to the other side. The time it takes them to go through the barrier is
given by :

(64) $\displaystyle t_{f}$
$\displaystyle=\int_{-\varepsilon^{\prime}}^{+\varepsilon^{\prime}}\,\mathrm{d}z\sqrt{\frac{m}{2(E-E_{\ast})}}$
$\displaystyle=2\varepsilon^{\prime}\sqrt{\frac{m}{2(E-E_{\ast})}}$
$\displaystyle=2\varepsilon^{\prime}\sqrt{\frac{m}{mv^{2}-2E_{\ast}}}.$

The flow of molecules of velocity belonging to the interval
$[v,v+\mathrm{d}v]$, with $v>v_{\ast}:=\sqrt{\mfrac{2E_{\ast}}{m}}$, going
from side $1$ to side $0$, is proportional to the surface density of
molecules, $c_{m1}$, to the Maxwell-Boltzmann weight, $p_{1}(v)\mathrm{d}v$,
of this velocity interval, and to the reciprocal of the translocation time :

(65) $\displaystyle J_{m10}$
$\displaystyle=\int_{v_{\ast}}^{+\infty}\mathrm{d}v\,\frac{c_{m1}p_{1}(v)}{t_{f}}$
$\displaystyle=\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\,\frac{1}{2\varepsilon^{\prime}}\,\sqrt{\frac{2(E-E_{\ast})}{m}}\,\frac{c_{m1}e^{-E/k_{B}T_{1}}}{\sqrt{\frac{2\pi
k_{B}T_{1}}{m}}}$
$\displaystyle=\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{1}}}\,c_{m1}e^{-E/k_{B}T_{1}}.$

The net flow of molecules from leaflet $1$ to leaflet $0$ is :

(66) $\displaystyle J_{m}$ $\displaystyle:=J_{m10}-J_{m01}$
$\displaystyle=\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{1}}}\,c_{m1}e^{-E/k_{B}T_{1}}$
$\displaystyle\quad-\frac{1}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\frac{\mathrm{d}E}{\sqrt{2mE}}\sqrt{\frac{E-E_{\ast}}{k_{B}T_{0}}}\,c_{m0}e^{-E/k_{B}T_{0}}.$

###  6.2. Computation of $L_{m\theta}$

The temperature $T_{0}$ being fixed, we have :

(67) $\displaystyle L_{m\theta}$ $\displaystyle=\frac{\partial J_{m}}{\partial
X_{\theta}}=-\frac{\partial J_{m}}{\partial T_{1}^{-1}}$
$\displaystyle=-\frac{c_{m1}}{2\varepsilon^{\prime}\sqrt{\pi}}\int_{E_{\ast}}^{+\infty}\mathrm{d}E\,\sqrt{\frac{E-E_{\ast}}{2mE}}\,\frac{\partial}{\partial
T_{1}^{-1}}\left(\frac{e^{-E/k_{B}T_{1}}}{\sqrt{k_{B}T_{1}}}\right)$
$\displaystyle=\frac{c_{m1}}{2\varepsilon^{\prime}k_{B}\sqrt{2\pi
mk_{B}T_{1}}}\int_{E_{\ast}}^{+\infty}\mathrm{d}E\,\sqrt{\frac{E-E_{\ast}}{E}}\,\left(E-\frac{k_{B}T_{1}}{2}\right)\,{e^{-E/k_{B}T_{1}}}.$

We set $u_{\ast 1}:=\mfrac{E_{\ast}}{k_{B}T_{1}}$ and change the variable of
integration from $E$ to $s:=\mfrac{E}{E_{\ast}}$ :

(68) $\displaystyle L_{m\theta}$
$\displaystyle=\frac{c_{m1}E_{\ast}\sqrt{k_{B}T_{1}}}{4\varepsilon^{\prime}k_{B}\sqrt{2\pi
m}}\int_{1}^{+\infty}\mathrm{d}s\,\sqrt{1-\frac{1}{s}}\,(2su_{\ast
1}-1)\,{e^{-su_{\ast 1}}}$ $\displaystyle=\alpha_{1}F(u_{\ast 1})$
$\displaystyle\alpha_{1}$
$\displaystyle:=\frac{c_{m1}E_{\ast}\sqrt{k_{B}T_{1}}}{4\varepsilon^{\prime}k_{B}\sqrt{2\pi
m}}$

where the function $F$ is defined by :

(69) $\displaystyle
F(a):=\int_{1}^{+\infty}\mathrm{d}s\,\sqrt{1-\frac{1}{s}}\,(2as-1)e^{-as}.$

We can now compute the relative variations of $L_{m\theta}$ with respect to
relative variations of temperature. Since $L_{m\theta}$ depends on $T_{1}$
through $E_{\ast}$ and $F(u_{\ast 1})$, we have :

(70) $\displaystyle\frac{\partial\ln L_{m\theta}}{\partial\ln T_{1}}$
$\displaystyle=\frac{\partial\ln\alpha_{1}}{\partial\ln
T_{1}}+\frac{\partial\ln F}{\partial\ln T_{1}}$
$\displaystyle=\frac{1}{2}+\frac{\partial\ln E_{\ast}}{\partial\ln
T_{1}}+\frac{\partial\ln u_{\ast 1}}{\partial\ln T_{1}}\,\frac{\partial\ln
F}{\partial\ln u_{\ast 1}}$ $\displaystyle=\frac{1}{2}+\frac{\partial\ln
E_{\ast}}{\partial\ln T_{1}}+\left(\frac{\partial\ln E_{\ast}}{\partial\ln
T_{1}}-1\right)\,\frac{\partial\ln F}{\partial\ln u_{\ast 1}}$
$\displaystyle=\frac{1}{2}+\frac{\partial\ln E_{\ast}}{\partial\ln
T_{1}}\left(1+\frac{\partial\ln F}{\partial\ln u_{\ast
1}}\right)-\frac{\partial\ln F}{\partial\ln u_{\ast 1}}.$

$\mfrac{\partial\ln E_{\ast}}{\partial\ln T_{1}}$ can not be computed in the
present model, because it depends on the microscopic details of the formation
of water pores. However, we know that $E_{\ast}$ diminishes as $T_{1}$
increases, since the water pores become more frequent (and, probably, larger
and more durable) when the ionic density fluctuations increase [14, 15].
Consequently, we have :

(71) $\displaystyle\frac{\partial\ln E_{\ast}}{\partial\ln T_{1}}<0.$

In Appendix C, we prove that $1+\mfrac{\partial\ln F}{\partial\ln u_{\ast 1}}$
is slightly negative at high temperature. Since $\mfrac{\partial\ln
E_{\ast}}{\partial\ln T_{1}}$ is also negative, we obtain the following
estimate :

(72) $\displaystyle\frac{\partial\ln L_{m\theta}}{\partial\ln
T_{1}}\gtrsim\frac{3}{2}\qquad\text{at high temperature.}$

###  6.3. Computation of $L_{mm}$

$L_{mm}$ is obtained by differentiating $J_{m}$ with respect to
$X_{m}=\mfrac{\mu_{m1}-\mu_{m0}}{T_{0}}$ while keeping the other
thermodynamical forces equal to zero :

(73) $\displaystyle L_{mm}$ $\displaystyle=\left(\frac{\partial
J_{m}}{\partial X_{m}}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$
$\displaystyle=T_{0}\left(\frac{\partial
J_{m}}{\partial(\mu_{m1}-\mu_{m0})}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}$
$\displaystyle=\frac{1}{k_{B}}\left(\frac{\partial
J_{m}}{\partial\ln(a_{1}/a_{0})}\right)_{(X_{\theta},X_{f},X_{w},X_{m^{\prime}})=0}.$

In our model, based on the double well effective potential, the activities of
the membrane molecules in each leaflet are equal to their respective
concentrations. A more accurate model, taking into account the attractive
interactions inside each leaflet, is necessary to improve this first
approximation. Replacing $\mfrac{a_{1}}{a_{0}}$ by
$\mfrac{c_{m1}}{c_{m0}}=\eta$, we obtain :

(74) $\displaystyle L_{mm}=\frac{1}{k_{B}}\left(\frac{\partial
J_{m}}{\partial\ln\eta}\right)_{T_{1}=T_{0}}.$

$J_{m}$ is a linear combination of the leaflet concentrations :

(75) $\displaystyle J_{m}$
$\displaystyle=\zeta(T_{1})c_{m1}-\zeta(T_{0})c_{m0}$ $\displaystyle\zeta(T)$
$\displaystyle:=\frac{E_{\ast}e^{-u_{\ast}}}{2\varepsilon^{\prime}\sqrt{2\pi
mk_{B}T}}\int_{0}^{+\infty}\mathrm{d}x\,e^{-u_{\ast}x}\sqrt{\frac{x}{x+1}}$
$\displaystyle u_{\ast}$ $\displaystyle:=\frac{E_{\ast}}{k_{B}T}.$

If the temperatures of both leaflets are equal, then $J_{m}$ is simply
proportional to the difference of their concentrations :

(76)
$\displaystyle\big{(}J_{m}\big{)}_{T_{0}=T_{1}=T}=\zeta(T)(c_{m1}-c_{m0})$

and its derivative with respect to $\ln\eta$, while $c_{m0}$ is held fixed, is
:

(77) $\displaystyle\left(\frac{\partial
J_{m}}{\partial\ln\eta}\right)_{c_{m0}=\text{cst.}}=\zeta(T_{1})c_{m1}=k_{B}L_{mm}.$

Since

(78)
$\displaystyle\int_{0}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}=\frac{1}{a}-\frac{\ln(a)}{2}+\mathcal{O}(1)\qquad(a\to
0^{+})$

the high temperature expansion of $L_{mm}$ gives :

(79) $\displaystyle\left(\frac{\partial\ln L_{mm}}{\partial\ln
T}\right)_{T_{1}=T_{0}=T}=\left(\frac{\partial\ln\zeta}{\partial\ln
T}\right)_{T_{1}=T_{0}=T}=\frac{1}{2}+o(1).$

###  6.4. Estimation of $L_{\theta\theta}$

The heat diffusion coefficient, $L_{\theta\theta}$, depends only on the number
of degrees of freedom that interact in the membrane bilayer. As long as the
structure of the membrane is unchanged, the same hydrophobic tails interact
similarly at any temperature. Therefore, we conjecture that $L_{\theta\theta}$
is independant of the temperature in the liquid disordered phase [27].
Therefore :

(80) $\displaystyle\frac{\partial\ln L_{\theta\theta}}{\partial\ln T}\simeq
0.$

###  6.5. Destabilisation

Putting together the scaling laws for $L_{m\theta}$, $L_{mm}$ and
$L_{\theta\theta}$, we obtain :

(81) $\displaystyle\frac{\partial}{\partial\ln
T_{1}}\left(\frac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}\right)=3-\frac{1}{2}-0=\frac{5}{2}$

The main mathematical proposition of the present article is the following.

######  Proposition 6.1.

Since $\mfrac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}$ grows as
$T_{1}^{5/2}$, the stability condition,
$L_{m\theta}^{2}<L_{mm}L_{\theta\theta}$, can not be satisfied at high
temperature.

The exact value of $T_{1}$ for which this transition occurs can not be
computed in our simple model, but the only characteristic temperature being
$\mfrac{E_{\ast}}{k_{B}}$, the critical temperature must be of this order of
magnitude.

This destabilisation of the steady growth regime is comparable with the onset
of heat convection in a fluid subject to a strong temperature gradient. In
fine, the self-replication of protocells could be interpreted as a convective
phenomenon inside their membrane, triggered by their metabolic activity.

##  7\. Conclusions and perspectives

We have proposed a toy model of protocell growth, fission and reproduction.
The scenario thus described can be viewed as the ancestor of mitosis. The main
force driving this irreversible process is the temperature difference between
the inside and the outside of the protocell, due to the inner chemical
activity. We propose that the increase of the inner temperature, due to a
rudimentary inner metabolism, enhances the transfer of membrane molecules from
the inner leaflet to the outer leaflet, as described in silico by models of
molecular dynamics [14, 15]. Due to this transfer of molecules, coupled to a
heat transfer, the difference of their areas and the total mean curvature of
the median surface increase. The cylindrical growth becomes unstable and any
slight local reduction of the radius of the initial cylinder increases until
the protocell is cut into two daughter protocells, each one containing
reactants and catalysers to continue the growth and fission process. The cut
occurs near the hottest zone, around the middle. This model is based on the
idea [23] that the early forms of life were simple vesicles containing a
particular network of chemical reactions, precursor of modern cellular
metabolism :

Protolife = Cellularity + Inner Metabolism.

With a large supply of reactants in the so-called prebiotic soup [31, 16, 23],
and with an optimal salinity and pH, these ingredients are sufficient to
induce an exponential growth of prebiomass and make possible the exploration
of a large number of chemical reactions in these miniature chemical factories.
The possibility to sythesize complex molecules (sterols, RNA, DNA, proteins,
etc.) comes later, once these factories self-replicate and thrive.

In order to test our model experimentally, we have to manipulate vesicles that
can be heated from within in a controled way. Let us imagine, in a solution
maintained at temperature $T_{0}$, vesicles containing molecules of type $A$
able to absorb visible radiation, with which the surrounding molecules do not
interact. Let us suppose that $A$ re-emits radiation in the near infrared. The
heat thus generated inside the vesicle creates a controled temperature
difference, $T_{1}-T_{0}>0$, between both sides of the membrane. If
$L_{m\theta}$ is large enough, we should observe a bending of the membrane of
the vesicles due to the transfer of the hottest molecules from the inner
leaflet to the outer leaflet.

Another experimental test of our model can be made by observing eukaryotic
cells, where the mitochondria are the main source of heat. It seems possible
to measure their temperature variations using fluorescent molecules [3].
Although the very notion of temperature at this scale and far from a
thermodynamical equilibrium is not clear, the measurement of the temperature
variations inside the cell during its life cycle could be correlated with the
onset of mitosis and with the shape of mitochondrial network [21].

Our model is obviously oversimplified since the polar heads of membrane
molecules are treated as an ideal gas in a box. In particular, we haven’t
taken into account the interaction between these molecules and the surrounding
solution. This calls for the development of a better model to treat the effect
of these interactions on the temperature dependence of the conductance
coefficients. The scaling law of the ratio
$\mfrac{L_{m\theta}^{2}}{L_{mm}L_{\theta\theta}}$ at temperatures higher than
$\mfrac{E_{\ast}}{k_{B}}$ is the key argument that explains the splitting of
the protocell. Future investigations and experiments will decide of the
plausibility of this proposition.

Acknowledgments : We thank Jorgelindo Da Veiga Moreira (Université de
Montréal), Marc Henry (Université de Strasbourg), Olivier Lafitte (Institut
Galilée, Université Paris XIII), Kirone Mallick (Institut de Physique
Théorique, CEA, Saclay), Laurent Schwartz (AP-HP) and Jean-Yves Trosset
(SupBiotech, Villefuif) for their advice and helpful discussions.

##  Appendix A The mean curvature of the membrane

Let $\Sigma_{t}$ be a family of surfaces, indexed by a time parameter
$t\in[t_{0},+\infty[$. We suppose that each $\Sigma_{t}$ is a smooth,
orientable and closed (compact, without boundary) hence diffeomorphic to the
standard $2$-sphere. At each point $P\in\Sigma_{t}$, the Taylor expansion of
the distance from $Q\in\Sigma_{t}$ to the tangent plane, $T_{P}\Sigma_{t}$,
defines a quadratic form whose eigenvalues (homogenous to the inverse of a
length) do not depend on the coordinate system in the neighbourhood of $P$. We
denote them $R_{-}$ and $R_{+}$. The mean curvature of $\Sigma_{t}$ at $P$ is
the arithmetic mean of the principal curvatures :

(82) $\displaystyle
H:=\frac{1}{2}\left(\frac{1}{R_{+}}+\frac{1}{R_{-}}\right)$

and the gaussian curvature is their product :

(83) $\displaystyle K:=\frac{1}{R_{+}R_{-}}.$

In the case of a cylinder, $R_{+}=+\infty$ and $R_{-}=R_{0}=$ its radius,
hence $H_{\text{cyl.}}(P)=\mfrac{1}{2R_{0}}$ and $K(P)=0$ at every point
$P\in\Sigma_{t}$ (except on the end hemispheres).

Let $\Sigma_{t0}$ and $\Sigma_{t1}$ be the surfaces obtained by shifting
$\Sigma_{t}$ in the normal direction, over an infinitesimal distance
$\varepsilon$ on both sides of $\Sigma_{t}$. Let $\mathcal{A}_{0}(t)$ (resp.
$\mathcal{A}_{1}(t)$) be the average area of the outer (resp. inner) layer of
the membrane, measured at the hydrophilic heads, and $\mathcal{A}=\mfrac
12(\mathcal{A}_{0}+\mathcal{A}_{1})$ the average area of the median surface,
where the hydrophobic tails join. The difference of their areas,
$\mathcal{A}_{1}-\mathcal{A}_{0}$, is given by the first term of Weyl’s Tube
Formula [13] :

(84)
$\displaystyle\mathcal{A}_{0}-\mathcal{A}_{1}=4\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A+\mathcal{O}(\varepsilon^{2}).$

Let $\mathcal{B}$ be the infinitesimal variation of area along the outer
normal :

(85) $\displaystyle\mathcal{B}:=2\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A.$

$\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ can also be written as functions of
$\mathcal{A}$ and $\mathcal{B}$ :

(86)
$\displaystyle\mathcal{A}_{0}=\mathcal{A}+\mathcal{B}\qquad\text{and}\qquad\mathcal{A}_{1}=\mathcal{A}-\mathcal{B}.$

Our dynamical variables are the area of the median surface,
$\mathcal{A}(t)=\int_{\Sigma_{t}}\mathrm{d}A$, the volume of the cytosol,
$\mathcal{V}(t)$, and the variation of area,
$\mathcal{B}(t)=2\varepsilon\int_{\Sigma_{t}}H\,\mathrm{d}A$. In the next
section, we will establish their evolution equations as a consequence of the
balance equations for the number of membrane molecules.

Remark : In the case of a cylinder of radius $R_{0}$, we have
$H=\mfrac{1}{2R_{0}}$ and

(87) $\displaystyle\mathcal{A}_{0}-\mathcal{A}_{1}=2\mathcal{B}=4\varepsilon
H\mathcal{A}=\frac{2\varepsilon\mathcal{A}}{R_{0}}.$

Since
$2\varepsilon\,\mfrac{\mathcal{A}_{0}+\mathcal{A}_{1}}{2}=2\varepsilon\mathcal{A}$
is also the volume, $v$, of this normal thickening of $\Sigma_{t}$, we have :

(88) $\displaystyle v=\frac{2\mathcal{B}}{H}=4\mathcal{B}R_{0}.$

##  Appendix B Solutions of the growth equation

In this appendix, we solve the growth equation, using basic linear algebra and
standard results about linear differential equations [19]. The matrix form of
the growth equation is

(89) $\displaystyle\dot{X}=\begin{pmatrix}\dot{\mathcal{A}}_{0}\\\
\dot{\mathcal{A}}_{1}\\\ \dot{\mathcal{U}}\\\
\end{pmatrix}=\begin{pmatrix}\frac{\eta\tau}{2}&\frac{\eta\tau}{2}&0\\\
-\frac{\tau}{2}&\frac{2-\tau}{2}&0\\\ 0&-\frac{c_{m1}}{\varepsilon
c_{mc}}&\frac{t_{1}}{t_{c}}\\\ \end{pmatrix}\begin{pmatrix}\mathcal{A}_{0}\\\
\mathcal{A}_{1}\\\ \mathcal{U}\\\ \end{pmatrix}=MX.$

As long as no flow vanishes, the determinant of $M$ is non-zero :

(90) $\displaystyle\det(M)=\frac{\eta\tau
t_{1}}{2t_{c}}=\frac{c_{m1}^{2}J_{m10}J_{rm}}{2c_{m0}c_{mc}J_{mc1}^{2}}$

and the protocell grows exponentially :

(91) $\displaystyle X(t)=e^{\frac{t}{t_{1}}M}X(0).$

In general, the two leaflets of the membrane grow at different speeds. Indeed,
the characteristic polynomial of $M$ is :

(92) $\displaystyle\det(M-\lambda\,\mathrm{Id})$
$\displaystyle=\begin{vmatrix}\frac{\eta\tau}{2}-\lambda&\frac{\eta\tau}{2}&0\\\
-\frac{\tau}{2}&\frac{2-\tau}{2}-\lambda&0\\\ 0&-\frac{c_{m1}}{\varepsilon
c_{mc}}&\frac{t_{1}}{t_{c}}-\lambda\\\ \end{vmatrix}$
$\displaystyle=\left(\lambda^{2}-\lambda\left(1+\frac{(\eta-1)\tau}{2}\right)+\frac{\eta\tau}{2}\right)\left(\frac{t_{1}}{t_{c}}-\lambda\right).$

Its roots are $\mfrac{t_{1}}{t_{c}}$ and the two roots,
$\lambda_{\pm}(\eta,\tau)$, of the polynomial
$\lambda^{2}-\lambda\big{(}1+(\eta-1)\mfrac{\tau}{2})\big{)}+\mfrac{\eta\tau}{2}$
:

(93) $\displaystyle\lambda_{\pm}(\eta,\tau)$
$\displaystyle:=\frac{1}{2}\left(1+\frac{(\eta-1)\tau}{2}\pm\sqrt{\Delta(\eta,\tau)}\right)$
$\displaystyle\Delta(\eta,\tau)$
$\displaystyle:=\frac{1}{4}(\eta-1)^{2}\tau^{2}-(\eta+1)\tau+1.$

$\bullet$ If $\eta=1$, i.e. if both leaflets have the same density, then
$\Delta$ is an affine function of $\tau$ :

(94)
$\displaystyle\Delta(1,\tau)=1-2\tau\qquad\text{and}\qquad\lambda_{\pm}(1,\tau)=\frac{1\pm\sqrt{1-2\tau}}{2}.$

If, moreover, $\tau=\mfrac 12$, i.e. the inner leaflet transmits half of the
incoming membrane molecules to the outer leaflet, then

(95)
$\displaystyle\Delta\left(1,\frac{1}{2}\right)=0\qquad\text{and}\qquad\lambda_{\pm}\left(1,\frac{1}{2}\right)=\frac{1}{2}$

and both leaflets grow at the same speed.

$\bullet$ If $\eta\neq 1$, then $\Delta(\eta,\tau)$ is a quadratic function of
$\tau$, bounded from below, of discriminant

(96) $\displaystyle\delta=(\eta+1)^{2}-(\eta-1)^{2}=4\eta>0$

and has distinct roots :

(97)
$\displaystyle\tau_{\pm}(\eta)=2\,\frac{\eta+1\pm\sqrt{4\eta}}{(\eta-1)^{2}}=2\left(\frac{\sqrt{\eta}\pm
1}{\eta-1}\right)^{2}=\frac{2}{(\sqrt{\eta}\mp 1)^{2}}.$

Physically, $\eta\simeq 1$ and $\tau\simeq\mfrac 12$. If $\eta=1+h$, with
$0<h\ll 1$ then $\tau_{+}\simeq\mfrac{8}{h^{2}}\gg 1>\tau_{-}$ and

(98)
$\displaystyle\tau_{-}\simeq\frac{2}{\left(2+\frac{h}{2}\right)^{2}}\simeq\frac{1}{2}-\frac{h}{4}.$

Consequently, $\tau$ stays $<\tau_{-}$ (FIG. B).

$\tau_{-}$$\mfrac 12$physicalregionnon-physical
region$\Delta>0$$\Delta<0$$\Delta>0$$\tau_{+}\gg\tau_{-}$

Exponential growth necessitates to keep $\tau<\tau_{-}$.

Mathematically, we have three possibilities :

$\displaystyle(1)$
$\displaystyle\tau<\tau_{-}(\eta)\quad\text{or}\quad\tau>\tau_{+}(\eta)\ \to\
\Delta(\eta,\tau)>0$
$\displaystyle\quad\lambda_{+}(\eta,\tau)\neq\lambda_{-}(\eta,\tau)\quad\text{(real
numbers) ;}$ $\displaystyle(2)$
$\displaystyle\tau=\tau_{-}(\eta)\quad\text{or}\quad\tau=\tau_{+}(\eta)\ \to\
\Delta(\eta,\tau_{\pm})=0$
$\displaystyle\quad\lambda_{+}\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\lambda_{-}\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\frac{\sqrt{\eta}}{\sqrt{\eta}\pm
1};$ $\displaystyle(3)$ $\displaystyle\tau_{-}(\eta)<\tau<\tau_{+}(\eta)\ \to\
\Delta(\eta,\tau)<0$
$\displaystyle\quad\lambda_{+}(\eta,\tau)=\overline{\lambda}_{-}(\eta,\tau)\quad\text{(complex
numbers).}$

###  B.1. Case 1 : $\eta\neq 1$ and $\tau>\tau_{+}(\eta)$ or
$\tau<\tau_{-}(\eta)$

In these intervals, $N$ is diagonalisable and a basis of eigenvectors of $N$
is given by :

(99) $\displaystyle\mathcal{B}_{\pm}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}\\\
\frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}-\frac{1}{2}\end{pmatrix}$
$\displaystyle=\frac{\mathcal{A}_{0}-\mathcal{A}_{1}}{2}+\frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}\,\mathcal{A}_{1}$
$\displaystyle=\mathcal{B}+\frac{\lambda_{\pm}(\eta,\tau)}{\eta\tau}\,\mathcal{A}_{1}$

i.e. $\mathcal{B}_{+}$ and $\mathcal{B}_{-}$ grow exponentially, with a rate
of growth $\lambda_{\pm}/t_{1}$, respectively :

(100)
$\displaystyle\mathcal{B}_{\pm}(t)=\mathcal{B}_{\pm}(0)\,\exp\left(\frac{\lambda_{\pm}(\eta,\tau)}{t_{1}}\,t\right).$

The area of the inner leaflet is :

(101) $\displaystyle\mathcal{A}_{1}(t)$
$\displaystyle=\frac{\mathcal{B}_{+}(t)-\mathcal{B}_{-}(t)}{\frac{\lambda_{+}}{\eta\tau}-\frac{\lambda_{-}}{\eta\tau}}$
$\displaystyle=\frac{\eta\tau}{\sqrt{\Delta}}\left(\mathcal{B}_{+}(0)e^{t\lambda_{+}/t_{1}}-\mathcal{B}_{-}(0)e^{t\lambda_{-}/t_{1}}\right).$

The area of the outer leaflet is :

(102) $\displaystyle\mathcal{A}_{0}(t)$
$\displaystyle=\frac{(2\lambda_{+}-\eta\tau)\mathcal{B}_{-}(t)-(2\lambda_{-}-\eta\tau)\mathcal{B}_{+}(t)}{\lambda_{+}-\lambda_{-}}$
$\displaystyle=\frac{2\lambda_{+}-\eta\tau}{\sqrt{\Delta}}\mathcal{B}_{-}(0)e^{t\lambda_{-}/t_{1}}-\frac{2\lambda_{-}-\eta\tau}{\sqrt{\Delta}}\mathcal{B}_{+}(0)e^{t\lambda_{+}/t_{1}}.$

And $\mathcal{U}(t)$ is obtained from $\mathcal{A}_{1}(t)$ :

(103) $\displaystyle
e^{tt_{1}/t_{c}}\,\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathcal{U}(t)e^{-tt_{1}/t_{c}}\right)$
$\displaystyle=-\frac{c_{m1}}{\varepsilon c_{mc}}\,\mathcal{A}_{1}(t)$ (104)
$\displaystyle\mathcal{U}(t)\,e^{-tt_{1}/t_{c}}$
$\displaystyle=-\frac{\eta\tau c_{m1}\mathcal{B}_{+}(0)}{\varepsilon
c_{mc}\sqrt{\Delta}\left(\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}\right)}\,e^{t\,\left({\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)}$
$\displaystyle+\frac{\eta\tau c_{m1}\mathcal{B}_{-}(0)}{\varepsilon
c_{mc}\sqrt{\Delta}\left(\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}\right)}\,e^{t\,\left({\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)}+\text{cst.}$
$\displaystyle\mathcal{U}(t)$ $\displaystyle=\frac{\eta\tau
c_{m1}}{\varepsilon
c_{mc}\sqrt{\Delta}}\left(\frac{e^{t\lambda_{-}/t_{1}}}{\frac{\lambda_{-}}{t_{1}}-\frac{t_{1}}{t_{c}}}-\frac{e^{t\lambda_{+}/t_{1}}}{\frac{\lambda_{+}}{t_{1}}-\frac{t_{1}}{t_{c}}}\right)$
$\displaystyle\ +\text{cst.}\,e^{tt_{1}/t_{c}}$

where the integration constant is determined by $\mathcal{U}(0)$.

###  B.2. Case 2 : $\eta\neq 1$ and
$\tau\in\\{\tau_{+}(\eta),\tau_{-}(\eta)\\}$

In this singular case, the upper-left $2\times 2$ submatrix is not
diagonalisable but conjugate to a lower triangular matrix of Jordan form :

(105) $\displaystyle N:=\frac{1}{2}\begin{pmatrix}\eta\tau&\eta\tau\\\
-\tau&2-\tau\\\ \end{pmatrix}=T\begin{pmatrix}\Lambda_{\pm}(\eta)&0\\\
1&\Lambda_{\pm}(\eta)\\\ \end{pmatrix}T^{-1}$

where $\Lambda_{\pm}(\eta)$ is the single eigenvalue of $N$ when $\tau$ is
fixed equal to $\tau_{+}(\eta)$ or $\tau_{-}(\eta)$ :

(106) $\displaystyle\Lambda_{\pm}(\eta)$
$\displaystyle:=\lambda\big{(}\eta,\tau_{\pm}(\eta)\big{)}=\frac{2+(\eta-1)\tau_{\pm}(\eta)}{4}$
$\displaystyle=\frac{2+\frac{2(\eta-1)}{(\sqrt{\eta}\mp
1)^{2}}}{4}=\frac{\eta\pm\sqrt{\eta}}{\eta-1}=\frac{\sqrt{\eta}}{\sqrt{\eta}\pm
1}.$

An easy computation gives us :

(107)
$\displaystyle\frac{2\Lambda_{+}}{\eta\tau}-1=\frac{1-3\sqrt{\eta}}{\eta+\sqrt{\eta}}\quad\text{and}\quad\frac{2\Lambda_{-}}{\eta\tau}-1=\frac{1+3\sqrt{\eta}}{\eta-\sqrt{\eta}}.$

$N$ has a unique proper line, generated by the vector

(108) $\displaystyle\mathcal{B}_{\ast}^{\pm}$
$\displaystyle:=\begin{pmatrix}\frac{1}{2}\\\
\frac{\Lambda_{\pm}}{\eta\tau_{\pm}}-\frac{1}{2}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1\\\
\frac{1\mp 3\sqrt{\eta}}{\eta\pm\sqrt{\eta}}\end{pmatrix}$
$\displaystyle=\frac{\mathcal{A}_{0}}{2}+\left(\frac{1\mp
3\sqrt{\eta}}{\eta\pm\sqrt{\eta}}\right)\frac{\mathcal{A}_{1}}{2}$

where the lower $\ast$ means that $\lambda_{+}=\lambda_{-}$, whereas the upper
$\pm$ depends on the choice between $\tau=\tau_{+}(\eta)$ and
$\tau=\tau_{-}(\eta)$. Since

(109)
$\displaystyle\mathcal{B}_{\ast}^{\pm}(t)=\mathcal{B}_{\ast}^{\pm}(0)\,e^{t\Lambda_{\pm}/t_{1}}$

we obtain :

(110) $\displaystyle\mathcal{A}_{0}(t)+\left(\frac{3\eta\mp
1}{\eta\pm\sqrt{\eta}}\right)\,\mathcal{A}_{1}(t)=2\,\mathcal{B}_{\ast}^{\pm}(0)\,e^{t\Lambda_{\pm}/t_{1}}.$

Since $\mathcal{B}_{\ast}^{\pm}=T{0\choose 1}$, the vector
$\mathcal{B}_{\ast}^{\pm}$ is the right column of $T$. The left column of $T$
is the vector $\mathcal{B}_{\bullet}^{\pm}={x\choose y}$ which satisfies the
equation
$(N-\Lambda_{\pm})\mathcal{B}_{\bullet}^{\pm}=\mathcal{B}_{\ast}^{\pm}$, or in
extenso :

(111)
$\displaystyle\left(\frac{\eta\tau}{2}-\Lambda_{\pm}\right)x+\left(\frac{\eta\tau}{2}\right)y$
$\displaystyle=\frac{1}{2}$
$\displaystyle-\frac{\tau}{2}\,x+\frac{2-\tau-2\Lambda_{\pm}}{2}\,y$
$\displaystyle=\frac{\Lambda_{\pm}}{\eta\tau}-\frac{1}{2}.$

Taking $x=0$ and $y=\mfrac{1}{\eta\tau}$ gives a solution :

(112) $\displaystyle\big{(}N-\Lambda_{\pm}\big{)}\mathcal{B}_{\bullet}^{\pm}$
$\displaystyle=\begin{pmatrix}\frac{\eta\tau}{2}-\Lambda_{\pm}&\frac{\eta\tau}{2}\\\
-\frac{\tau}{2}&\frac{2-\tau}{2}-\Lambda_{\pm}\end{pmatrix}\begin{pmatrix}0\\\
\frac{1}{\eta\tau}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\frac{1}{2}\\\
\frac{2-\tau-2\Lambda_{\pm}}{\eta\tau}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\\\
\frac{2\Lambda_{\pm}-\eta\tau}{\eta\tau}\end{pmatrix}=\mathcal{B}_{\ast}^{\pm}.$

The matrix $T$ and its inverse, $T^{-1}$, are therefore :

(113) $\displaystyle T=\begin{pmatrix}0&1\\\
\frac{1}{\eta\tau}&\frac{2\Lambda-\eta\tau}{2\eta\tau}\end{pmatrix}\qquad\text{and}\qquad
T^{-1}=\begin{pmatrix}\frac{\eta\tau-2\Lambda}{2}&\eta\tau\\\
1&0\end{pmatrix}.$

Since
$\mathcal{B}_{\bullet}^{\pm}=\mfrac{\mathcal{A}_{1}}{\eta\tau_{\pm}(\eta)}$,
we have
$\mathcal{B}_{\bullet}^{\pm}(t)=\mathcal{B}_{\ast}(0)\,\mfrac{t}{t_{1}}\,e^{t\Lambda_{\pm}/t_{1}}$
and :

(114)
$\displaystyle\mathcal{A}_{1}(t)=\eta\tau_{\pm}(\eta)\,\mathcal{B}_{\ast}^{\pm}(0)\,\frac{t}{t_{1}}\,e^{t\Lambda_{\pm}/t_{1}}$

###  B.3. Case 3 : $\tau_{-}(\eta)<\tau<\tau_{+}(\eta)$

In this interval, $\Delta<0$ and $M$ has two distinct complex conjugated
eigenvalues, $\lambda$ and $\overline{\lambda}$, functions of $\eta$ and
$\tau$. Let $\alpha,\beta\in\mathbb{R}$ be the real and imaginary parts of
$\lambda$ :

(115) $\displaystyle\alpha$ $\displaystyle:=\frac{2+(\eta-1)\tau}{4}\,>0$
$\displaystyle\beta$ $\displaystyle:=\frac{\sqrt{-\Delta}}{2}\,>0$
$\displaystyle\lambda(\eta,\tau)$
$\displaystyle=\alpha+\mathbf{i}\,\beta\qquad(\mathbf{i}\,^{2}=-1).$

Let $V$ (resp. $\overline{V}$) be a complex eigenvector of $N$, of eigenvalue
$\lambda$ (resp. $\overline{\lambda}$), for instance :

(116) $\displaystyle
V:=\mathcal{B}+\frac{\lambda}{\eta\tau}\,\mathcal{A}_{1}\qquad\text{and}\qquad\overline{V}:=\mathcal{B}+\frac{\overline{\lambda}}{\eta\tau}\,\mathcal{A}_{1}$

then the real and imaginary parts of $V$, defined by $V^{\prime}:=\mfrac
12(V+\overline{V})$ and
$V^{\prime\prime}:=\mfrac{1}{2\mathbf{i}\,}(V-\overline{V})$, form a basis of
$\mathbb{R}^{2}$ on which $N$ acts as an orthogonal matrix [19] :

(117) $\displaystyle\frac{1}{2}\begin{pmatrix}\eta\tau&\eta\tau\\\
-\tau&2-\tau\\\ \end{pmatrix}$ $\displaystyle=U\begin{pmatrix}\alpha&\beta\\\
-\beta&\alpha\\\ \end{pmatrix}U^{-1}$ $\displaystyle
V^{\prime}=U\begin{pmatrix}1\\\ 0\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\\\
\frac{\alpha}{\eta\tau}-\frac{1}{2}\end{pmatrix}$ $\displaystyle\quad
V^{\prime\prime}=U\begin{pmatrix}0\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\
\frac{\beta}{\eta\tau}\end{pmatrix}$

i.e. the matrix $U$ has $V^{\prime}$ and $V^{\prime\prime}$ as columns :

(118) $\displaystyle U=\begin{pmatrix}\frac{1}{2}&0\\\
\frac{\alpha}{\eta\tau}-\frac{1}{2}&\frac{\beta}{\eta\tau}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}&0\\\
\frac{2-(\eta+1)\tau}{4\eta\tau}&\frac{\sqrt{-\Delta}}{2\eta\tau}\end{pmatrix}.$

Let $s=\mfrac{t}{t_{1}}$. Since our evolution operator, the exponential of
$sN$, is :

(119) $\displaystyle e^{sN}=e^{\alpha s}U\begin{pmatrix}\cos(\beta
s)&\sin(\beta s)\\\ -\sin(\beta s)&\cos(\beta s)\end{pmatrix}U^{-1}$

we have :

(120) $\displaystyle V^{\prime}(s)$
$\displaystyle=e^{sN}V^{\prime}(0)=e^{\alpha s}U\begin{pmatrix}\cos(\beta
s)&\sin(\beta s)\\\ -\sin(\beta s)&\cos(\beta
s)\end{pmatrix}\begin{pmatrix}\frac{1}{2}\\\
\frac{\alpha}{\eta\tau}-\frac{1}{2}\end{pmatrix}$
$\displaystyle=\frac{e^{\alpha s}}{2\eta\tau}\begin{pmatrix}\frac{1}{2}&0\\\
\frac{\alpha}{\eta\tau}-\frac{1}{2}&\frac{\beta}{\eta\tau}\end{pmatrix}\begin{pmatrix}\eta\tau\cos(\beta
s)+(2\alpha-\eta\tau)\sin(\beta s)\\\ -\eta\tau\sin(\beta
s)+(2\alpha-\eta\tau)\cos(\beta s)\end{pmatrix}$
$\displaystyle=\frac{e^{\alpha
s}}{2\eta\tau}\begin{pmatrix}\frac{\eta\tau}{2}\cos(\beta
s)+\frac{2\alpha-\eta\tau}{2}\sin(\beta s)\\\
\frac{(2\alpha-\eta\tau)(2\beta+\eta\tau)}{2\eta\tau}\cos(\beta
s)+\left(\frac{(2\alpha-\eta\tau)^{2}}{2\eta\tau}-\beta\right)\sin(\beta
s)\end{pmatrix}.$

Similarly, we have the expression of $V^{\prime\prime}(s)$ :

(121) $\displaystyle V^{\prime\prime}(s)$
$\displaystyle=e^{sN}V^{\prime\prime}(0)$ $\displaystyle=\frac{\beta e^{\alpha
s}}{\eta\tau}\begin{pmatrix}\frac{1}{2}\sin(\beta s)\\\
\left(\frac{2\alpha-\eta\tau}{2\eta\tau}\right)\sin(\beta
s)+\frac{\beta}{\eta\tau}\cos(\beta s)\end{pmatrix}.$

Finally, $\mathcal{A}_{1}$ and $\mathcal{B}$ are obtained from $V^{\prime}$
and $V^{\prime\prime}$ by the linear relations :

(122) $\displaystyle\mathcal{B}(s)$ $\displaystyle=\frac{\beta
V^{\prime}(s)-\alpha V^{\prime\prime}(s)}{\beta-\alpha}$
$\displaystyle\mathcal{A}_{1}(s)$
$\displaystyle=\frac{\eta\tau}{\alpha-\beta}\big{(}V^{\prime}(s)-V^{\prime\prime}(s)\big{)}.$

##  Appendix C Smooth perturbation of cylindrical growth

In this appendix, we compute the variation of the area and of the total mean
curvature of a surface of revolution under a small variation of its generating
curve. We will work in an orthonormal system of coordinates $(x,y,z)$. Let us
suppose now that $\Sigma$ is a revolution surface whose generating curve,
rotated around the axis $\\{y=0=z\\}$, is given by :

(123) $\displaystyle\sqrt{y^{2}+z^{2}}=R(x)=R_{0}+\delta R(x)$

with $|\delta R(x)|\ll R_{0}$. The function $\delta R$ represents an
infinitesimal normal perturbation around the cylindrical shape. The variable
$x$ satisfies $0\leq x\leq\ell$ and the deformed cylinder is glued smoothly
with two hemispherical caps of radius $R_{0}$. In other words, we suppose that

(124) $\displaystyle\delta R(0)=\delta R(\ell)$ $\displaystyle=0$
$\displaystyle\delta R^{\prime}(0)=\delta R^{\prime}(\ell)$ $\displaystyle=0.$

Let us compute the variations of area, $\delta\mathcal{A}$, of length,
$\delta\ell$, and of total mean curvature, $\delta\mathcal{H}$, for a fixed
volume.

###  C.1. Isovolumic variation of the area

$\mathcal{A}$ is a functional of the length, $\ell$, the radius, $R$, and its
derivative, $R^{\prime}$ :

(125)
$\displaystyle\mathcal{A}(\ell,R,R^{\prime})=\int_{0}^{\ell}\mathrm{d}x\,2\pi
R\sqrt{1+R^{\prime 2}}.$

Its variation under infinitesimal changes of $\ell$ and $R$ is :

(126) $\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta\ell$
$\displaystyle+2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta R\left(\sqrt{1+R^{\prime
2}}-\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{RR^{\prime}}{\sqrt{1+R^{\prime
2}}}\right)\right).$

Since

(127)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{RR^{\prime}}{\sqrt{1+R^{\prime
2}}}\right)$ $\displaystyle=\frac{RR^{\prime\prime}+R^{\prime
2}}{\sqrt{1+R^{\prime
2}}}-R^{\prime}R^{\prime\prime}\frac{RR^{\prime}}{\big{(}1+R^{\prime
2}\big{)}^{3/2}}$ $\displaystyle=(1+R^{\prime
2})^{-3/2}\left(\big{(}RR^{\prime\prime}+R^{\prime 2}\big{)}\big{(}1+R^{\prime
2}\big{)}-RR^{\prime 2}R^{\prime\prime}\right)$ $\displaystyle=(1+R^{\prime
2})^{-3/2}\big{(}RR^{\prime\prime}+R^{\prime 2}+R^{\prime 4}\big{)}$

we obtain

(128) $\displaystyle\delta\mathcal{A}$ $\displaystyle=2\pi R_{0}\,\delta\ell$
$\displaystyle+2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta R\big{(}1+R^{\prime
2}\big{)}^{-3/2}\big{(}1-RR^{\prime\prime}-R^{\prime 4}\big{)}.$

Similarly, the volume, $\mathcal{V}$, is a functional of $\ell$ and $R$ :

(129) $\displaystyle\mathcal{V}(\ell,R)=\frac{4\pi
R_{0}^{3}}{3}+\int_{0}^{\ell}\mathrm{d}x\,\pi R^{2}$

and its variation under infinitesimal changes of $\ell$ and $R$ is :

(130) $\displaystyle\delta\mathcal{V}=\pi
R_{0}^{2}\,\delta\ell+2\pi\int_{0}^{\ell}\mathrm{d}x\,R\,\delta R.$

If $\mathcal{V}$ is held constant, then $\delta\mathcal{V}=0$ and :

(131)
$\displaystyle\big{(}\delta\ell\big{)}_{\mathcal{V}=\text{cst.}}=-\frac{2}{R_{0}^{2}}\int_{0}^{\ell}\mathrm{d}x\,R\,\delta
R.$

Inserting this expression of $\delta\ell$ into that of $\delta\mathcal{A}$, we
obtain the isovolumic variation of area :

(132) $\displaystyle(\delta\mathcal{A})_{\mathcal{V}=\text{cst.}}$
$\displaystyle=2\pi\int_{0}^{\ell}\mathrm{d}x\,\delta
R\left(\big{(}1+R^{\prime 2}\big{)}^{-3/2}\big{(}1-RR^{\prime\prime}-R^{\prime
4}\big{)}-\frac{2R}{R_{0}}\right).$

######  Theorem C.1.

The isovolumic variational derivatives of the length and of the area of a
(nearly cylindrical) closed revolution surface are negative :

(133) $\displaystyle\left(\frac{\delta\ell}{\delta
R}\right)_{\mathcal{V}=\text{cst.}}<0\qquad\text{and}\qquad\left(\frac{\delta\mathcal{A}}{\delta
R}\right)_{\mathcal{V}=\text{cst.}}<0.$

In other words, since the stuffing is incompressible whereas the gut is
elastic, the length and the area of a squeezed sausage increase. We call this
simple statement the Squeezed Sausage Theorem (SST).

###  C.2. Isovolumic variation of the total mean curvature

The circles $\\{x=\text{cst.}\\}$ and the meridians, obtained by rotating the
generating curve of equation $z^{2}=R^{2}(x)$, form an orthogonal system of
geodesics [7], and the mean curvature of $\Sigma$ is given by :

(134) $\displaystyle H=\frac{1}{2}\left(\frac{1}{R\sqrt{1+R^{\prime
2}}}+\frac{R^{\prime\prime}}{\big{(}1+R^{\prime 2}\big{)}^{3/2}}\right).$

The lateral area of a slice of width $\mathrm{d}x$, perpendicular to the axis
of the surface, is :

(135) $\displaystyle\mathrm{d}A=2\pi R\sqrt{1+R^{\prime 2}}\,\mathrm{d}x$

and the total mean curvature is :

(136) $\displaystyle\mathcal{H}$ $\displaystyle:=\int_{\Sigma}H\,\mathrm{d}A$
$\displaystyle=\int_{\text{caps}}H\,\mathrm{d}A+\int_{0}^{\ell}2\pi
R\sqrt{1+R^{\prime 2}}\,\mathrm{d}x$ $\displaystyle=4\pi
R_{0}^{2}\cdot\frac{1}{R_{0}}+2\pi\int_{0}^{\ell}\frac{1}{2}\left(1+\frac{R\,R^{\prime\prime}}{1+R^{\prime
2}}\right)\mathrm{d}x$ $\displaystyle=4\pi
R_{0}+\pi\ell+\pi\int_{0}^{\ell}\frac{R\,R^{\prime\prime}}{1+R^{\prime
2}}\,\mathrm{d}x.$

Since $\mathcal{H}$ is a functional of $\ell$, $R$, $R^{\prime}$ and
$R^{\prime\prime}$, its variation under a change $\delta R$ of the radius of
gyration and a change of length $\delta\ell$, is obtained after a double
integration by parts [10] :

(137) $\displaystyle\delta\mathcal{H}$
$\displaystyle=\pi\delta\ell+\pi\int_{0}^{\ell}\mathrm{d}x\
\frac{R^{\prime\prime}}{1+R^{\prime
2}}+\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{2RR^{\prime}R^{\prime\prime}}{\big{(}1+R^{\prime
2}\big{)}^{2}}\right)$ $\displaystyle\hskip
85.35826pt+\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\left(\frac{R}{1+R^{\prime
2}}\right)\,\delta R.$

Instead of computing each term of the integrand, let us make the approximation
$R^{\prime 2}\ll 1$, valid when the initial cylinder is only slightly
deformed. The expression of $\delta\mathcal{H}$ then simplifies to

(138) $\displaystyle\delta\mathcal{H}$
$\displaystyle\simeq\pi\delta\ell+\pi\delta\int_{0}^{\ell}\mathrm{d}x\,RR^{\prime\prime}$
$\displaystyle\simeq\pi\delta\ell+2\pi\int_{0}^{\ell}\mathrm{d}x\,R^{\prime\prime}\,\delta
R.$

Using the expression of
$\delta\ell=-\mfrac{2}{R_{0}^{2}}\int_{0}^{\ell}\mathrm{d}x\,R\,\delta R$ when
$\mathcal{V}$ is held constant, we obtain :

(139)
$\displaystyle\big{(}\delta\mathcal{H}\big{)}_{\mathcal{V}=\text{cst.}}\simeq
2\pi\int_{0}^{\ell}\mathrm{d}x\left(R^{\prime\prime}-\frac{R}{R_{0}^{2}}\right)\,\delta
R.$

As long as $R_{0}^{2}|R^{\prime\prime}|\ll R$, the isovolumic variational
derivative of $\mathcal{H}$ with respect to $R$ is negative :

(140) $\displaystyle\left(\frac{\delta\mathcal{H}}{\delta
R}\right)_{\mathcal{V}=\text{cst.}}<0\qquad\text{if}\quad R^{\prime 2}\ll
1\quad\text{and}\quad R_{0}^{2}|R^{\prime\prime}|\ll R.$

When $\delta R$ approaches $-R_{0}$ and the protocell is ready to split, the
two radii of curvature are small compared to $R_{0}$ but have opposite sign,
hence the Gaussian curvature around the septum is large and negative. After
the cut, when the two caps are formed, the mean curvature and the Gaussian
curvature are positive again.

##  Appendix D Asymptotic expansion of $F(a)$

The change of variable $t=\sqrt{a(s-1)}$ in the integral defining $F$ gives us
:

(141) $\displaystyle F(a)$
$\displaystyle=\frac{e^{-a}}{a}\int_{0}^{+\infty}\mathrm{d}t\,f(a,t)$
$\displaystyle f(a,t)$
$\displaystyle:=2t^{2}\,e^{-t^{2}}\left(2\sqrt{t^{2}+a}-\frac{1}{\sqrt{t^{2}+a}}\right).$

Let

(142) $\displaystyle G(a):=\int_{0}^{+\infty}\mathrm{d}t\,f(a,t)=ae^{a}F(a).$

The function $f(0,\cdot)$ is integrable over the half line $[0,+\infty[$ and

(143) $\displaystyle G(0)$
$\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,f(0,t)$
$\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,2t\,e^{-t^{2}}(2t^{2}-1)$
$\displaystyle=\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}(2u-1)=1.$

Let us compute the asymptotic expansion of $G(a)$ when $a\to 0^{+}$ :

(144) $\displaystyle{\,}G(a)-G(0)$
$\displaystyle=\int_{0}^{+\infty}\mathrm{d}t\,\big{(}f(a,t)-f(0,t)\big{)}$
$\displaystyle=2\int_{0}^{+\infty}\mathrm{d}t\,t^{2}e^{-t^{2}}\left(2\big{(}\sqrt{t^{2}+a}-t\big{)}-\left(\frac{1}{\sqrt{t^{2}+a}}-\frac{1}{t}\right)\right)$
$\displaystyle=\int_{0}^{+\infty}2t\,\mathrm{d}t\,e^{-t^{2}}\big{(}\sqrt{t^{2}+a}-t\big{)}\left(2t+\frac{1}{\sqrt{t^{2}+a}}\right)$
$\displaystyle=\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\big{(}\sqrt{u+a}-\sqrt{u}\big{)}\left(2\sqrt{u}+\frac{1}{\sqrt{u+a}}\right)$
$\displaystyle=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}+\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}(1-2u)-\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{\frac{u}{u+a}}$
$\displaystyle=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}-1-\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{\frac{u}{u+a}}.$

Hence :

(145) $\displaystyle G(a)=\varphi(a)-\varphi^{\prime}(a)$

where

(146) $\displaystyle\varphi(a)$
$\displaystyle:=2\int_{0}^{+\infty}\mathrm{d}u\,e^{-u}\sqrt{u(u+a)}$
$\displaystyle=2a^{2}\int_{0}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{x(x+1)}.$

$\varphi(a)$ being the Laplace transform of the function $x\mapsto
2a^{2}\sqrt{x(x+1)}$, its expansion as $0^{+}$ is given by integrating the
expansion of $\sqrt{x(x+1)}$ at $+\infty$ term by term :

(147) $\displaystyle\sqrt{x(x+1)}$
$\displaystyle=x+\frac{1}{2}-\frac{1}{8x}+\mathcal{O}(x^{-2})$
$\displaystyle\varphi(a)$
$\displaystyle=2a^{2}\left(\frac{1}{a^{2}}+\frac{1}{2a}-\frac{1}{8}\int_{1}^{+\infty}\mathrm{d}x\,\frac{e^{-ax}}{x}+\mathcal{O}(1)\right)$
$\displaystyle=2+a-\frac{a^{2}}{4}\ln(a)+\mathcal{O}(a^{2}).$

Similarly, for $\varphi^{\prime}(a)$ we have :

(148) $\displaystyle\sqrt{\frac{x}{x+1}}$
$\displaystyle=1-\frac{1}{2x}+\frac{3}{8x^{2}}+\mathcal{O}(x^{-3})\qquad(x\to+\infty)$
$\displaystyle\varphi^{\prime}(a)$
$\displaystyle=a\int_{0}^{1}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}$
$\displaystyle+a\int_{1}^{+\infty}\mathrm{d}x\,e^{-ax}\sqrt{\frac{x}{x+1}}$
$\displaystyle=a\int_{0}^{1}+a\left(\frac{e^{-a}}{a}-\frac{1}{2}\int_{1}^{+\infty}\mathrm{d}x\,\frac{e^{-ax}}{x}+\mathcal{O}(1)\right)$
$\displaystyle=1-\frac{a\ln(a)}{2}+\mathcal{O}(a).$

Consequently :

(149) $\displaystyle G(a)$ $\displaystyle=1+\frac{a\ln(a)}{2}+\mathcal{O}(a)$

and

(150) $\displaystyle F(a)$
$\displaystyle=\frac{e^{-a}}{a}+\frac{e^{-a}\ln(a)}{2}+\mathcal{O}(a)$
$\displaystyle=\frac{1}{a}+\frac{1}{2}\ln(a)+o(1).$

The asymptotic expansion of $\frac{\partial\ln F}{\partial\ln a}$ is therefore
:

(151) $\displaystyle\frac{\partial\ln F}{\partial\ln
a}=-1+\frac{a\ln(a)}{2}+\mathcal{O}(a).$

In particular, since $a\ln(a)<0$ for $0<a<1$, we have

(152) $\displaystyle\frac{\partial\ln F}{\partial\ln a}<-1\qquad(a\to 0^{+}).$

## References

  * [1] J. S. Allhusen & J. C. Conboy : The Ins and Outs of Lipid Flip-Flop (Acc. Chem. Res., 2017, 50, 1, 58-65).
  * [2] P. W. Atkins & J. de Paula : Physical chemistry (Oxford University Press, 11th edition, 2017).
  * [3] D. Chrétien et al. : Mitochondria are physiologically maintained at close to 50 $\circ$C (PLoS Biol 16(1): e2003992. https://doi.org/ 10.1371/journal.pbio.2003992 ; January 25, 2018)
  * [4] F.-X. Contreras, L. Sánchez-Magraner, A. Alonso, F. M. Goñi : Transbilayer (flip-flop) lipid motion and lipid scrambling in membranes (FEBS Letters, 584 (2010) 1779–1786).
  * [5] S. Cooper : Bacterial growth and division (Academic Press, 1991).
  * [6] J. da Veiga Moreira, S. Peres, J.-M. Steyaert, E. Bigan, L. Paulevé, M.-L. Nogueira & L. Schwartz : Cell cycle progression is regulated by intertwined redox oscillators. (Theoretical Biology and Medical Modelling, 12(1), 1-14, 2015).
  * [7] M. Do Carmo : Differential Geometry of Curves and Surfaces (Prentice hall, 1976).
  * [8] J. England : Statistical physics of self-replication (J. Chem. Phys., 139 , 121923 (2013)).
  * [9] A. Erdélyi : Asymptotic expansions (Dover Publications, 1956).
  * [10] I. M. Gel’fand & S. V. Fomin : Calculus of variations (Prentice-Hall, 1963).
  * [11] P. Glansdorff & I. Prigogine : Thermodynamic theory of structure, stability and fluctuations (Wiley Interscience, New York, 1971).
  * [12] G. Gottschalk : Bacterial metabolism (2nd edition, Springer-Verlag, 1986).
  * [13] A. Gray : Tubes (2nd edition, Birkhäuser, 2004).
  * [14] A. A. Gurtovenko & I. Vattulainen : Ion Leakage through Transient Water Pores in Protein-free Lipid Membranes Driven by Transmembrane Ionic Charge Imbalance (Biophys. J. , 92, March 2007, 1878-1890).
  * [15] A. A. Gurtovenko & I. Vattulainen : Molecular Mechanism for Lipid Flip-Flops (J. Phys. Chem. B 2007, 111, 13554-13559).
  * [16] J. B. S. Haldane : The origin of Life (1929).
  * [17] M. Henry : Thermodynamics of life (Unpublished notes).
  * [18] T. L. Hill : Free energy transduction and biochemical cycle kinetics (Dover Publications, 2005).
  * [19] M. W. Hirsch, S. Smale & R. L. Devaney : Differential Equations, Dynamical Systems and an Introduction to Chaos (Academic Press, 3rd edition, 2013).
  * [20] A. Katchalsky & P. F. Curran : Nonequilibrium Thermodynamics in Biophysics (Harvard University Press, 1965).
  * [21] K. Mitra, C. Wunder, B. Roysam, G. Lin & J. Lippincott-Schwartz : A hyperfused mitochondrial state achieved at G1-S regulates cyclin E buildup and entry into S phase (PNAS, 2009, 106, 29, 11960-11965).
  * [22] H. J. Morowitz : Energy Flow in Biology (Academic Press, 1968).
  * [23] H. J. Morowitz : Beginnings of Cellular Life (Yale University Press, 1992).
  * [24] H. J. Morowitz & E. Smith : The origin and nature of life on Earth (Cambridge University Press, 2016).
  * [25] H. J. Morowitz & E. Smith : Universality in intermediate metabolism (PNAS, vol. 101, n∘ 36, 2004, 13168-13173).
  * [26] H. J. Morowitz & E. Smith : Energy flow and the organization of life (2007).
  * [27] O. Mouritsen : Life as a matter of fat (2nd edition, Springer-Verlag, 2016).
  * [28] P. H. Nielsen, C. Kragelund, R. J. Seviour & J. Lund Nielsen : Identity and ecophysiology of filamentous bacteria in activated sludge (FEMS Microbiol. Rev. 33 (2009) 969-998).
  * [29] L. Onsager : Reciprocal relations in irreversible processes. I (Phys. Rev., 37, pp. 405-426, 1931).
  * [30] L. Onsager : Reciprocal relations in irreversible processes. II (Phys. Rev., 38, pp. 2265-2279, 1931).
  * [31] I. Oparin : The Origin of Life on Earth (Oliver and Boyd, Edinburgh, 1957).
  * [32] I. Prigogine : Introduction to thermodynamics of irreversible processes (John Wiley and Sons, 1962).
  * [33] N. Rashevsky : Mathematical biophysics, vol. 1 (Dover Publications, 3rd edition, 1960).
  * [34] S. Rasmussen et al., editors : Protocells. Bridging Nonliving and Living Matter (MIT Press, 2009).
  * [35] L. Reichl : A Modern Course in Statistical Physics (Arnold, 1980).
  * [36] M. Salazar-Roa & M. Malumbres : Fueling the cell division cycle (Trends Cell Biol. 2017 Jan. ; 27(1):69-81).
  * [37] E. Schrödinger : What is life ? (Cambridge University Press, 1945).
  * [38] S. M. Stanley : Exploring Earth and Life through Time (W. H. Freeman, 1993).