# Dynamics of unfolded protein aggregation

Utkarsh Upadhyay, Chandrima Barua, Shivani Devi, Jay Prakash Kumar and R.K.
Brojen Singh<EMAIL_ADDRESS>(Corresponding author) School of Computational
& Integrative Sciences, Jawaharlal Nehru University, New Delhi 110019, India.

###### Abstract

Unfolded protein aggregation in cellular system is a problem causing various
types of diseases depending on which type unfolded proteins aggregate. This
phenomenon of aggregation may take place during production, storage, shipment
or delivery in the cellular medium. In the present work, we studied a
simplified and extended version of unfolded protein aggregation model by Lumry
and Eyring [J. Phys. Chem. 58:110–120 (1954).] using stochastic approach. We
solved analytically the Master equation of the model for the probability
distribution $P(x,t)$ of the unfolded protein population and the solution was
found to be time dependent complex binomial distribution. In the large
population limit $P(x,t)\sim\Lambda(x,t)\times Pois(x,t)$. Further, the
distribution became Normal distribution at large population and mean of the
distribution limit: $P(x,t)\sim\Lambda(x,t)\times N(\langle qx\rangle,\langle
qx\rangle)$. The fluctuations inherent in the dynamics measured by Fano factor
can have sub-Poisson, Poisson and super-Poisson at different situations.

Keywords: Unfolded protein; Protein aggregation; Master equation; Generating
function; Fano factor.

Introduction
Aggregation of unfolded proteins could be a nuisance factor and lethal in many
cellular dynamics causing various pathological states Fink ; Albert . This
aggregation may lead to few important factors which might enhance potential
immunogenicity Singh , may cause conformational diseases (aggregation of
mutant proteins) Gow  and neurodegenerative diseases (aggregation of prions in
brain) Stefani . The aggregation of unfolded proteins or non-native
aggregation is a process of clustering together of monomers to stable
complexes. Individual monomer could be composed of a single folded chain or
multiple protein chains that are disulfide bonded to one another such as
multimeric complex Roberts . Few key features which favor protein aggregates
are: the process is irreversible i.e. do not easily dissociate, and the
process retains certain fraction of their original structureJoubert . Active
research in this area has grown up fast because of few reasons, first,
unfolded protein aggregation is risk to develop immune system, second, in
becoming potential key drug target for possible therapeutic intervention
Jiskoot , and these aggregated unfolded proteins can be possible key drug
target for Martinez . However, general theory of protein aggregation and
control mechanism are still open question.

Unfolded protein aggregation may happen during production, storage, shipment
or delivery to the patient. The process is subjected to various fluctuations
(temperature, light, shaking, surfaces, pH adjustments, etc.), and these
fluctuations favor protein aggregation in the cellular environment Chi . There
are many examples of protein aggregation, silicone oil droplets Tyagi  and
freezing can induce aggregation Bhatnagar . Further, this aggregation process
is quite similar to the protein folding mechanism and the whenever,
hydrophobic groups of a protein are exposed to the solvent, kinetic
competition arises between folding and aggregation Mahler . First step of the
aggregation process could be to allow the the unfolded or partially folded
protein in the process to choose a state either to move to native state or
formation of dimer together with another unfolded molecule (aggregation) Zett
. This aggregation kinetics, structure and formation could be dependent on
various factors which leads to the cause various diseases, for example,
aggregation of amyloid-$\beta$ is believed to be cause of a number of brain
diseases, including Alzheimer’s disease and Huntington’s disease Dobson .
Further, the aggregation was found to be linked with induction of allergic
responses, type 1 hypersensitivity responses, such as urticaria, anaphylaxis
etc Mazzeo . Sometimes such aggregation of proteins may lead to protein
modifications from its own normal situation which cause neuronal dysfunction
and neurotoxicity directed to widespread neurodegeneration Tarawneh .

Another aspect of unfolded protein aggregation is that it may occur throughout
the lifetime of a protein causing modification in size, shape, morphology,
chemical modifications, degradation and folding mechanism Moussa . Such
proteins in aggregated state generally have different biological functions as
proteins at native states do causing toxic effects in the cellular systems
Mahler . Since slight aggregation levels for certain period of time may cause
clinically unacceptable situation, such aggregation should be avoided Braun .
One possible strategy to prevent aggregation is to add molecules that hinders
aggregation. For example, molecules that slow down aggregation urea,
guanidinium chloride, amino acids, sugars, polyols, polymers, surfactants, and
antibodies Cleland ; Cleland1 . One reason causing this could be the decrease
in the binding free energy of the unfolded protein and transition states
allowing to increase in the aggregation activation energy slowing down
aggregation rate. Sugars (sucrose and other molecules) that are preferentially
excluded from the protein-solvent interface generally favors proteins to be in
native state against aggregation Lee ; Arakawa  leading to increase in free
energy barrier of unfolding states Kendrick . However, even though we know
unfolded protein aggregation could be a potential target for prevention/cure
of various types diseases, choice of molecules which can disrupt such
aggregation state is a field of active research now.

We present a extended model of unfolded protein aggregation model by Lumry and
Eyring Lumry  within stochastic formalism to understand dynamics and
distribution of unfolded proteins. W solved exactly the master equation of the
system for probability distribution of the unfolded proteins. Then we
estimated noise fluctuation in the dynamics and analyzed the role of
fluctuations in regulating the protein aggregation mechanism. We concluded few
findings based on the results we obtained.

Theory of protein aggregation model and solution
Protein aggregation is problem in cellular process as the biological activity
of aggregated proteins is not the same as that of native protein, and cause
acute toxic in the cellular environment. Hence, one has to device strategies
to prevent protein aggregation and is a subject of active research in
pharmaceutic companies and biotechnological research. The number of proteins
per human cell is roughly $\sim 10^{10}$ Milo  which are subjected to various
forms of stress and a small fraction of marginally stable folded proteins in
such stress and crowded environment cause error in folding leading to protein
misfolding Dobson . This process of misfolding is irreversible to folded state
and cause the generation of unfolded protein aggregation Vendruscolo . If $N$,
$U$ and $A$ denote natured, denatured and aggregated proteins in the simple
model of protein aggregation, the model can be described by the following set
of reactions which is the extension of the model due to Lumry ,

$\displaystyle N\stackrel{{\scriptstyle
k_{1}}}{{\rightarrow}}U;~{}U\stackrel{{\scriptstyle
k_{2}}}{{\rightarrow}}N;~{}U\stackrel{{\scriptstyle
k_{3}}}{{\rightarrow}}A;~{}A\stackrel{{\scriptstyle k_{4}}}{{\rightarrow}}U$
(1)

where, $k_{1},k_{2},k_{3},k_{4}$ are rate constants. Using mass action
chemical kinetics, this set of reactions (1) can be translated into the
following set of ordinary differential equations,

$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle-k_{1}N+k_{2}U$
$\displaystyle\frac{dU}{dt}$ $\displaystyle=$ $\displaystyle
k_{1}N-k_{2}U-k_{3}U+k_{4}A$ (2) $\displaystyle\frac{dA}{dt}$ $\displaystyle=$
$\displaystyle k_{3}U-k_{4}A$

Summing over these three differential equations we have,
$\frac{dN}{dt}+\frac{dU}{dt}+\frac{dA}{dt}=0$ which implies $N+U+A=constant$.
It has been reported that at room temperature the rate of folding protein is
comparatively much higher than the rate of misfolding protein Mayor . Hence,
let us consider the dynamics of unfolded protein as few of large population of
$N$ which move to unfolded protein aggregation state such that
$U+A=constant-N\sim constant$. It has been reported that misfolded protein
aggregation could be stable and long life span Roberts  and hence, one can
take $A$ as constant or $A=constant-U$ as the mean length of the aggregated
unfolded protein during exponential growth is nearly constant Kunes .

.

Figure 1: Schematic diagram illustrating the basic mechanism of the possible
transitions of protein states in unfolded protein aggregation model.
$\\{k_{i}\\}$, $i=1,2,3,4$ are the rates at which the transitions of states
take place.

The role of protein aggregation can be studied if we obtain the distribution
of unfolded protein $U$ from which protein aggregation takes place during the
course of interaction. Let us take the population of $U$ at any instant of
time ’$t$’ be $x$. The probability distribution of $x$ at time $t$ $P(x,t)$
can be calculated from the Master equation built from the reaction set (1).
The Master equation of the reactions set in (1) can be constructed as follows
McQuarrie ; Gillespie . First, the stochastic rate constants are calculated
using $c_{i}=V^{1-\nu_{i}}k_{i}$ Gillespie , where, $\nu_{i}$ is the ith state
change or stoichiometric parameter and since the reactions are all
unimolecular reactions $\nu_{i}=1$ such that $c_{i}=k_{i}$. Then transition
probabilities of each reaction are calculated by assuming the process is
Markovian in forward arrow of time and taking all possible molecular
interaction during time interval $[t,t+\Delta t]$ unto first order
approximation, $\omega_{i}=f_{i}(x,N,A)\Delta t+O\left(\Delta^{2}\right)$,
where, $f_{i}$ is the function obtained from the notion of molecular
interaction McQuarrie ; Gillespie . The three variable model can be reduced to
simple one dimension $f(x,t)$ by assuming $x\langle\langle N$ ($x$ is moving
in the sea of $N$), and $x\rangle A$ (protein aggregation is fast decay). Then
one can construct the Master equation by incorporating the above mentioned
processes as given below,

$\displaystyle\frac{\partial P(x,t)}{\partial
t}=(k_{1}N+k_{4}A)P(x-1,t)+(k_{2}+k_{3})(x+1)P(x+1,t)+(k_{1}N+k_{2}x+k_{3}x+k_{4}A)P(x,t)$
(3)

This Master equation (3) can be solved using generating function technique
defined by the transformation of $P(x,t)$ from $x$-space to $s$-space given
by, $G(s,t)=\displaystyle\sum_{x}s^{x}P(x,t)$ provided initial condition
$G(s,0)=s^{M}$ (initially there were $M$ population of $x$ at $t=0$) and
normalization condition. Now multiplying equation (3) by $s^{x}$ and doing the
summation over $x$ in both sides of the equation and after some algebra, we
have the following differential equation in $G(s,t)$,

$\displaystyle\frac{\partial G}{\partial t}+(k_{2}+k_{3})(s-1)\frac{\partial
G}{\partial s}=(s-1)(k_{1}N+k_{4}A)G$ (4)

This partial differential equation in $G(s,t)$ eqrefgf can be be used using
Lagrange-Charpit characteristic method Delgado . This can be done by writing
the partial differential equation (4) as in the following form,

$\displaystyle\frac{\partial G}{(s-1)(k_{1}N+k_{4}A)G}=\frac{\partial
s}{(s-1)(k_{3}+k_{2})}=\frac{\partial t}{1}$ (5)

By integrating both sides of the last equation in (5) connecting $\partial s$
and $\partial t$, $(k_{3}+k_{2})\displaystyle\int\partial t=\int\frac{\partial
s}{s-1}+C_{1}$, we arrived at the relation between $s$ and $t$ as,
$\displaystyle\lambda_{1}=(s-1)e^{-(k_{2}+k_{3})t}$, where,
$\displaystyle\lambda_{1}=e^{C_{1}}$ is a constant in terms of integrating
constant $C_{1}$. Next, from the equation connecting $\partial G$ and
$\partial s$ in equation (5), by integrating both sides and rearranging the
terms we get, $\displaystyle\int\frac{\partial
G}{G}=\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\int\partial s+C_{2}$ and solution is
$\displaystyle G=\lambda_{2}e^{\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}s}$, where,
$\lambda_{2}$ is a constant in terms of integrating constant $C_{2}$ by
$\lambda_{2}=e^{C_{2}}$. Now, combining the two solutions, we get the complete
solution of $G$ in equation (4) as follows,

$\displaystyle
G(s,t)=f\left[(s-1)e^{-(k_{2}+k_{3})t}\right]e^{\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}s}$
(6)

Now, we have to find out the functional form of $f$ in the equation (6) by
using the initial condition, $G(s,0)=s^{M}$ which is given by,

$\displaystyle f(s)=(s+1)^{M}e^{-\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}(s+1)}$ (7)

Putting this functional form $f(s)$ in equation (7) to equation (6) and
rearranging the terms, we get the $G(s,t)$,

$\displaystyle
G(s,t)=\left[(s-1)e^{-(k_{2}+k_{3})t}+1\right]^{M}e^{-\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}(1-s)\left[1-e^{-(k_{2}+k_{3})t}\right]}$
(8)

Next, we rearrange the terms in the first factor in equation (8) and doing the
binomial expansion, we have,
$\displaystyle\left[(s-1)e^{-(k_{2}+k_{3})t}+1\right]^{M}=\left[se^{-(k_{2}+k_{3})t}+\left(1-e^{-(k_{2}+k_{3})t}\right)\right]^{M}=\sum_{i}\binom{M}{i}\left[se^{-(k_{2}+k_{3})t}\right]^{i}\left[1-e^{-(k_{2}+k_{3})t}\right]^{M-i}$.
From the second factor of the generating function in equation (8), expanding
the exponential term having $s$, we have, $\displaystyle
e^{\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}s\left[1-e^{-(k_{2}+k_{3})t}\right]}=\sum_{j}\frac{1}{j!}\left[\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}s\left(1-e^{-(k_{2}+k_{3})t}\right)\right]^{j}$.
Now combining the two expressions in the equation eqrefgff and rearranging the
terms, we have,

$\displaystyle G(s,t)$ $\displaystyle=$ $\displaystyle
e^{-\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\left[1-e^{-(k_{2}+k_{3})t}\right]}\sum_{i}\sum_{j}\binom{M}{i}\frac{s^{i+j}}{j!}e^{-i(k_{2}+k_{3})t}\left[1-e^{-(k_{2}+k_{3})t}\right]^{M-i}$
(9)
$\displaystyle\times\left[\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}(1-e^{-(k_{2}+k_{3})t})\right]^{j}$

Putting $i+j=x$ in the above expression and comparing with the definition of
Generating function $G(s,t)=\sum_{x}s^{x}P(x,t)$, we finally obtain the
probability density function as the following,

$\displaystyle
P(x,t)=e^{\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\left[1-e^{-(k_{2}+k_{3})t}\right]}\sum_{i}\binom{M}{i}\left(\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\right)^{x-i}\frac{\left[1-e^{-{(k_{2}+k_{3})t}}\right]^{M+x-2i}}{(x-i)!}e^{-i{(k_{2}+k_{3})t}}$
(10)

From this probability distribution (10), it clearly indicates that as
$(k_{2}+k_{3})\rangle\rangle(k_{1}N+k_{4}A)$ and $(k_{2}+k_{3})\rightarrow
large$ limits $e^{-(k_{2}+k_{3})t}\rightarrow 0$, so that $P(x,t)\rightarrow
0$. The situation in this limit indicates that all the available unfolded
proteins are converted to folded proteins ($N$) and aggregated unfolded
proteins ($A$) as evident from equations (1) and (10). A simple possible
condition to control aggregation of unfolded proteins is to find a situation
where, $k_{2}\rangle\rangle
k_{3};~{}(k_{2}+k_{3})\rangle\rangle(k_{1}N+k_{4}A)$.

Now we calculated the observables $\langle x\rangle$ and $\langle
x^{2}\rangle$ which are expectation values of unfolded protein population $x$
using equation (8). Putting $\alpha=\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}$ and
$\beta=k_{2}+k_{3}$, we have,

$\displaystyle\langle x\rangle$ $\displaystyle=$
$\displaystyle\left.\frac{\partial G(s,t)}{\partial s}\right|_{s=1}$ (11)
$\displaystyle=$ $\displaystyle M\left[(s-1)e^{-\beta
t}+1\right]^{M-1}e^{-\beta t}e^{\alpha(1-s)(1-e^{-\beta t})}$
$\displaystyle+\left.\left[(s-1)e^{-\beta
t}+1\right]^{M}\left[-\alpha(1-e^{-\beta
t})\right]e^{\alpha(1-s)\left[1-e^{-\beta t}\right]}\right|_{s=1}$
$\displaystyle=$ $\displaystyle\left[M-\alpha(e^{\beta t}-1)\right]e^{-\beta
t}$

Further, the second moment of the unfolded protein population $x$ can be
obtained as follows,

$\displaystyle\langle x^{2}\rangle$ $\displaystyle=$
$\displaystyle\left.\frac{\partial^{2}G}{\partial
s^{2}}\right|_{s=1}+\left.\frac{\partial G}{\partial s}\right|_{s=1}$ (12)
$\displaystyle=$ $\displaystyle M(M-1)\left[(s-1)^{e^{-\beta
t}}+1\right]^{M-2}e^{-2\beta t}e^{\alpha(1-s)(1-e^{-\beta t})}$
$\displaystyle+M\left[(s-1)^{e^{-\beta t}}+1\right]^{M-1}e^{-\beta
t}(-\alpha\left[1-e^{-\beta t})\right]e^{\alpha(1-s)(1-e^{-\beta t})}$
$\displaystyle+M\left[(s-1)^{e^{-\beta t}}+1\right]^{M-1}e^{-\beta
t}\left[-\alpha(1-e^{-\beta t})\right]e^{\alpha(1-s)(1-e^{-\beta t})}$
$\displaystyle+\left[(s-1)^{e^{-\beta
t}}+1\right]^{M}\left[-\alpha(1-e^{-\beta
t})\right]^{2}e^{\alpha(1-s)(1-e^{-\beta t})}$ $\displaystyle+\langle
x\rangle$ $\displaystyle=$ $\displaystyle M(M-1)e^{-2\beta t}-2Me^{-\beta
t}\alpha(1-e^{-\beta t})+\left[\alpha(1-e^{-\beta t})\right]^{2}$

Now, from the equations (11) and (12), we arrived at the variance of the
population $x$,

$\displaystyle\sigma_{x}^{2}$ $\displaystyle=$ $\displaystyle\langle
x^{2}\rangle-\langle x\rangle^{2}$ (13) $\displaystyle=$
$\displaystyle\left[M-\alpha(e^{\beta t}-1)\right]e^{-\beta t}-Me^{-2\beta t}$

Proposition 1: Long time behavior of the population of unfolded protein is
given by:

  * •

Asymptotic limit of the mean $|\langle x\rangle|$ at long time becomes a
constant: $\displaystyle\lim_{t\rightarrow\infty}|\langle
x\rangle|=\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}$.

  * •

Long time behavior of the $\langle x^{2}\rangle$ is given by:
$\displaystyle\lim_{t\rightarrow\infty}\langle
x^{2}\rangle=\left[\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\right]^{2}$.

Proof: From the equation (11) taking limit $t\rightarrow\infty$, we have,

$\displaystyle\lim_{t\rightarrow\infty}|\langle x\rangle|$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow\infty}|\left[M-\alpha(e^{\beta
t}-1)\right]e^{-\beta t}|$ (14) $\displaystyle=$
$\displaystyle\alpha=\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}$

This means that the mean population of the unfolded proteins at the long time
limit becomes constant which depends on the rate constants and initial
populations of $N$ and $A$. Similarly, by taking the limit
$t\rightarrow\infty$ of the equation eqrefmoment we easily get
$\displaystyle\lim_{t\rightarrow\infty}\langle
x^{2}\rangle=\left[\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\right]^{2}$.

Theorem 1: In the limit $M,x\rightarrow\infty$, the probability distribution
function P(x,t) follows the time dependent Poisson distribution given by,

$\displaystyle P(x,t)=F(x,t)Pois(\langle qx\rangle)$ (15)

where,
$F(x,t)=\displaystyle\Gamma(t)e^{-\frac{M\alpha}{M+\alpha}}e^{\frac{M-\alpha}{M+\alpha}x}\frac{M-\alpha}{M+\alpha}\frac{a}{b}q^{-x}$,
$q=\frac{M}{M+\alpha}$.

Proof: Let us put,
$\displaystyle\Gamma(t)=e^{\frac{k_{1}N+k_{4}A}{k_{2}+k_{3}}\left[1-e^{-(k_{2}+k_{3})t}\right]}=e^{\alpha\left[1-e^{-\beta
t}\right]}$, $p=e^{-(k_{2}+k_{3})t}=e^{-\beta t}$ and rearranging the terms
the equation (10) can be written as,

$\displaystyle
P(x,t)=\Gamma(t)\sum_{i}\binom{M}{i}p^{i}(1-p)^{M-i}\frac{\alpha^{x-i}(1-p)^{x-i}}{(x-i)!}$
(16)

Now in the limit, $M\rightarrow\infty$, we have,
$\displaystyle\lim_{M\rightarrow\infty}\binom{M}{i}p^{i}(1-p)^{M-i}\rightarrow\frac{\lambda^{i}e^{-\lambda}}{i!},~{}\lambda=Mp$.
Substituting this expression to equation eqrefpfa and rearranging the term, we
have,
$P(x,t)=\displaystyle\frac{\Gamma(t)e^{-\lambda}}{x!}\sum_{i}\left[M^{i}\alpha^{x-i}\right]\frac{x!}{(x-i)!i!}p^{i}(1-p)^{x-i}$.
In the limit $x\rightarrow\infty$, we have,

$\displaystyle P(x,t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(t)e^{-\lambda}}{x!}\sum_{i}\left[M^{i}\alpha^{x-i}\right]\frac{\gamma^{i}e^{-\gamma}}{i!},~{}~{}~{}~{}~{}\gamma=xp$
(17) $\displaystyle=$
$\displaystyle\frac{\Gamma(t)\alpha^{x}e^{-(\lambda+\gamma)}}{x!}\sum_{i}\frac{\left[\frac{M\gamma}{\alpha}\right]^{i}}{i!}$
$\displaystyle\approx$
$\displaystyle\frac{\Gamma(t)\alpha^{x}e^{-(M+x)p}}{x!}e^{\frac{Mxp}{\alpha}}$
$\displaystyle=$ $\displaystyle\Gamma(t)e^{-Mp}\frac{\left[\alpha
e^{-p+\frac{Mp}{\alpha}}\right]^{x}}{x!}$

From equation (11), we can write first factor in equation eqrefpxt,
$p=\displaystyle\frac{\langle x\rangle+\alpha}{M+\alpha}$. The second factor
in the equation (17) can be written as,
$e^{-Mp}=e^{-\frac{M\alpha}{M+\alpha}}e^{-\frac{M}{M+\alpha}\langle
x\rangle}$. Further,
$e^{-p+\frac{M}{\alpha}p}=e^{\frac{M-\alpha}{M+\alpha}x}\left[e^{\frac{1}{\alpha}\frac{M-\alpha}{M+\alpha}\langle
x\rangle}\right]^{x}$. Since, $\alpha<1$ and $M$ is large, we know that
$\frac{1}{\alpha}\frac{M-\alpha}{M+\alpha}\langle x\rangle>1$. If we take
$z=\frac{1}{\alpha}$, then in this function
$\frac{1}{z}e^{z\frac{M-\alpha}{M+\alpha}\langle x\rangle}$, for small values
$z$ $1/z$ dominates whereas for large $z$, the exponential function dominates.
Now to have approximate solution which have both contributions from both $1/z$
and exponential part is to consider linear approximation in the functions. For
this let us take $\frac{1}{v}=e^{z\frac{M-\alpha}{M+\alpha}\langle
x\rangle}<1$, so that, $-z\displaystyle\frac{M-\alpha}{M+\alpha}\langle
x\rangle=ln(v)=\sum_{i=0}^{\infty}(-1)^{i}\frac{(q-1)^{i+1}}{i+1}$. Then we
consider only the linear terms in the expansion which can be written in the
form $av+b\approx-z\displaystyle\frac{M-\alpha}{M+\alpha}\langle x\rangle$,
where, $a$ and $b$ are constants, $a,b\in mathcal{R}$. Rewriting this
expression again in the form,
$[1+\frac{a}{b}v]^{-1}=-\displaystyle\frac{b}{z\frac{M-\alpha}{M+\alpha}\langle
x\rangle}$ and collecting only linear terms we have at large $z$ limit,
$v=\displaystyle e^{-z\frac{M-\alpha}{M+\alpha}\langle
x\rangle}\sim\frac{b}{a}$. Now, from the third factor, for
$z\rightarrow\infty$ limit, we have,
$\displaystyle\lim_{z\rightarrow\infty}\frac{e^{z\frac{M-\alpha}{M+\alpha}\langle
x\rangle}}{z}=\lim_{z\rightarrow\infty}\frac{e^{z\frac{M-\alpha}{M+\alpha}\langle
x\rangle}\frac{M-\alpha}{M+\alpha}\langle
x\rangle}{1}~{}(L^{\prime}Hospital^{\prime}s~{}rule)\sim\frac{M-\alpha}{M+\alpha}\langle
x\rangle\times\frac{a}{b}$. Putting all these results and rearranging the
terms, we have,

$\displaystyle P(x,t)$ $\displaystyle\approx$ $\displaystyle
F(x,t)e^{-q\langle x\rangle}\frac{\langle x\rangle^{x}}{x!}$ (18)
$\displaystyle\approx$ $\displaystyle F(x,t)\left[\frac{e^{-\langle
qx\rangle}\langle qx\rangle^{x}}{x!}\right]$ $\displaystyle\approx$
$\displaystyle F(x,t)Pois(\langle qx\rangle)$

where,
$F(x,t)=\displaystyle\Gamma(t)e^{-\frac{M\alpha}{M+\alpha}}e^{\frac{M-\alpha}{M+\alpha}x}\frac{M-\alpha}{M+\alpha}\frac{a}{b}q^{-x}$,
$q=\frac{M}{M+\alpha}$.

Theorem 2: The asymptotic limiting value of $P(x,t)$ at $M,x,\langle
x\rangle\rightarrow\infty$ is a normal distribution given by,

$\displaystyle P(x,t)\approx F(x,t)\times N(\langle qx\rangle,\langle
qx\rangle)$ (19)

Proof: Let us take the Poisson distribution in equation (18)
$Pois(qx)=\displaystyle\frac{e^{-\langle qx\rangle}\langle
qx\rangle^{x}}{x!}$. In $x\rightarrow large$ limit, $x!$ can be approximated
by Stirling’s formula, $x!\approx x^{x}e^{-x}\sqrt{2\pi x}$. Now, one can
write, $\displaystyle lnPois(\langle qx\rangle)=xln(\langle qx\rangle)-\langle
qx\rangle-xln(x)+x-ln(\sqrt{2\pi x})$. Then $\varepsilon-$expansion can be
done by expanding $x$ around $\langle qx\rangle$ by, $x=\langle
qx\rangle+\varepsilon$, where, $\varepsilon$ is small arbitrary parameter such
that $\displaystyle\frac{\varepsilon}{\langle qx\rangle}\langle\langle 1$.
Within this approximation and after some algebra, one can reach,
$\displaystyle lnPois(\langle qx\rangle)=-\frac{\varepsilon^{2}}{2\langle
qx\rangle}-ln\sqrt{2\pi\langle qx\rangle}$. Then we have,

$\displaystyle Pois(\langle qx\rangle)$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{2\pi\langle qx\rangle}}e^{-\frac{(x-\langle
qx\rangle)^{2}}{\sqrt{2\langle qx\rangle}}}$ (20) $\displaystyle\approx$
$\displaystyle N(\langle qx\rangle,\langle qx\rangle)$

Hence, equation eqrefpois becomes,

$\displaystyle P(x,t)\approx F(x,t)\times N(\langle qx\rangle,\langle
qx\rangle)$ (21)

Proposition 2: The Fano factor of the distribution of $x$ is given by,

$\displaystyle F_{f}=1-\frac{M}{\langle x\rangle}e^{-\beta t}$ (22)

The processes become:

  * •

Statistically independent in the limit,
$\displaystyle\lim_{t\rightarrow\infty}F_{f}\rightarrow 1$.

  * •

Sub-Poissonian process for the case $\displaystyle\frac{M}{\langle
x\rangle}e^{-\beta t}>0$ and finite.

  * •

Noise enhanced process if $\displaystyle\frac{M}{\langle x\rangle}e^{-\beta
t}<0$.

Proof: From equations (11) and (13), and using the formula for calculating
Fano factor, $F_{f}=\displaystyle\frac{\sigma_{x}}{\langle x\rangle}$, which
can estimate noise associated with the dynamics Fano ; Chanu . It is given by,

$\displaystyle F_{f}=1-\frac{M}{\langle x\rangle}e^{-\beta t}$ (23)

Taking limit $t\rightarrow\infty$ in the equation (23), we have,
$F_{f}=1-\displaystyle\lim_{t\rightarrow\infty}\frac{M}{\langle
x\rangle}e^{-\beta t}\rightarrow 1.$ This is the case of Poisson process,
where, the processes are statistically independent of each other.

For finite values of $\displaystyle\frac{M}{\langle x\rangle}e^{-\beta t}$, we
have $F_{f}<1$. This is the case of sub-Poissonian process, where, noise does
not have significant role in regulating the dynamics.

The role of noise in regulating the system’s dynamics become visible if the
process is noise enhanced or super-Poissonian process. It can be observed if
$\frac{M}{\langle x\rangle}e^{-\beta t}<1$. It is possible only when $\beta
t<0$ such that $e^{-\beta t}\sim 1-\beta t<0$ satisfying the condition
$t>\frac{1}{\beta}$.

Now let us consider the situation of controlling aggregation of unfolded
protein, where, $k_{2}\gg k_{3}$, the condition at which the rate of
converting unfolded to native state of the proteins is much larger than the
rate of aggregation of unfolded protein. This condition allows us,
$\beta=k_{2}\displaystyle\left(1+\frac{k_{3}}{k_{2}}\right)\sim k_{2}$ such
that, $\displaystyle e^{-\beta t}\sim e^{-k_{2}t}\sim(1-k_{2})t$. This case
could be the situation at which unfolded protein is approximately controlled.
Then, the Fano factor in equation (23) becomes,

$\displaystyle F_{f}\sim 1+\frac{M}{\langle x\rangle}(k_{2}-1)t$ (24)

In this situation, the process becomes 1. Poissonian if $F_{f}=1$; 2. sub-
Poissonian if $k_{2}<1$ and 3. Noise enhanced process if $k_{2}>1$. Hence, the
dynamics as well as distribution of the unfolded protein aggregation is quite
different depending on $k_{2}$ and it can able to drive the dynamics to
different dynamical states where processes are drastically different.

Conclusion
Unfolded protein aggregation is nuisance to the living beings which could be
one of the origin of various diseases, namely, conformational diseases Gow ,
neurodegenerative diseases Stefani  etc. This aggregation mechanism is quite
complicated process but can be simplified into a simple model we presented
which is an extension of the model by Limry and Erying Lumry . We considered
the model as a Markov process and can be represented by a Masted equation
which can be solved using generating function technique by taking certain
initial condition. The probability distribution $P(x,t)$ we obtained is quite
complicated function which depends on various parameters involved in the
model, initial conditions and concentrations of the native protein and
aggregated protein. In the long time or aging behavior of the unfolded
protein, which can be obtained approximately from the mean value unfolded
protein at long time limit, is found to be nearly constant. Since population
of such unfolded protein should be minimized to prevent from their aggregation
possible condition should be $[k_{2}+k_{3}]\gg[k_{1}N+k_{4}A]$. This means
that total rate of conversion of native state protein and formation of
aggregate from unfolded protein should be comparatively larger than the sum of
converted unfolded protein populations from native state and aggregated state.
Further, conversion from unfolded state to native state should be favored
$k_{2}\gg k_{3}$.

The behavior of the probability distribution $P(x,t)$ at large population of
the unfolded limit show modified Poisson distribution with a factor $F(x,t)$
which is mainly dominated by exponential function at large population limit.
Further, if we take large limit of mean population of the unfolded protein,
the distribution $P(x,t)$ becomes modified normal distribution of same mean
and standard deviation. This indicates that the distribution of the
interacting unfolded protein is deeply rooted to the universally accepted
poisson and normal distributions at various situations. However, the
distributions are significantly deviated from the distribution of non-
reactively interacting molecules which follow Maxwell-Boltzmann distribution.
One reason of the origin of this deviation could be birth and death process
involved in the molecular interaction process.

The fluctuations inherent to the dynamics of the unfolded protein has
significant role in the regulation of the unfolded protein dynamics and has
different roles at different situations. This fluctuation parameter can be
estimated using standard Fano factor from the mean and variance of the protein
dynamics. At large time limit or aging limit of the dynamics indicates that
the processes are statistically independent of each other which is the case of
completely memory lost dynamics and are not correlated. For finite values of
the parameters, $F<1$ indicating possibility of sub-poissinian process where
the involved fluctuations try to stabilize the system and hence the system
tends to stay at equilibrium state. Noise enhancement process, which generally
try to drive the system at non-equilibrium state, could be possible for
certain range of time otherwise the dynamics is dominated by sub-poissonian
process. Unfolded protein aggregation causes various types of diseases and
could be a potential target to prevent from the diseases. The question is how
to select aggregation destabilizing molecule for a specific disease. These are
open questions which are needed to be investigated systematically.

Acknowledgements
We would like to thank Jawaharlal Nehru University for providing us facilities
to carry out this work.

Author Contributions:
RKBS conceptualized and supervised the work. UU, CB, SD, JPK and RKBS did the
analytical work. UU, CB, SD, JPK and RKBS wrote the manuscript. All the
authors read and approved the manuscript.

Additional Information
Competing financial interests: The authors declare no competing interests.

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