# Morphology and Interaction between Lipid Domains

Tristan S. Ursell1, William S. Klug2 and Rob Phillips1111Address
correspondence to<EMAIL_ADDRESS>
1Department of Applied Physics, California Institute of Technology, Pasadena,
CA 91125
2Department of Mechanical and Aerospace Engineering, Program in Biomedical
Engineering,
and California NanoSystems Institute, University of California Los Angeles,
Los Angeles, CA 90095

Cellular membranes are a heterogeneous mix of lipids, proteins and small
molecules. Special groupings of saturated lipids and cholesterol form a
liquid-ordered phase, known as ‘lipid rafts,’ serving as platforms for
signaling, trafficking and material transport throughout the secretory
pathway. Questions remain as to how the cell maintains heterogeneity of a
fluid membrane with multiple phases, through time, on a length-scale
consistent with the fact that no large-scale phase separation is observed. We
have utilized a combination of mechanical modeling and in vitro experiments to
show that membrane morphology can be a key player in maintaining this
heterogeneity and organizing such domains in the membrane. We demonstrate that
lipid domains can adopt a flat or dimpled morphology, where the latter
facilitates a repulsive interaction that slows coalescence and tends to
organize domains. These forces, that depend on domain morphology, play an
important role in regulating lipid domain size and in the lateral organization
of lipids in the membrane.

The plasma and organelle membranes of cells are composed of a host of
different lipids, lipophilic molecules and membrane proteins [1]. Together,
they form a heterogeneous layer capable of regulating the flow of materials
and signals into and out of the cell. Lipid structure and sterol content play
a key role in membrane organization, where steric interactions and
energetically costly mismatch in the hydrophobic structure of lipid tails
result in lateral phase-separation. Saturated lipids and cholesterol are
sequestered into liquid-ordered ($L_{\mbox{\footnotesize o}}$) domains, often
known as ‘lipid rafts’, from an unsaturated liquid-disordered
($L_{\mbox{\footnotesize d}}$) phase [2, 3, 4]. Domains composed of saturated
sphingolipids and cholesterol, with sizes in the range of $\sim
50-500\,\mbox{nm}$, have been implicated in a range of biological processes
from lateral protein organization and virus uptake to signaling and plasma-
membrane tension regulation [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
How the cell maintains the lateral heterogeneity of lipids over time, and what
physical mechanism might be responsible for the spatial organization of these
domains, challenges classical theories of phase-separation and ‘domain
ripening’ (such as Cahn-Hilliard kinetics [18]). The maintenance of lateral
heterogeneity is thought to arise from a combination of lipid recycling and
energetic barriers to domain coalescence [19, 20, 21] (potentially provided by
transmembrane proteins [22]), resulting in a stable distribution of domain
sizes. The precise origin of this energy barrier and the nature of its
dependence on membrane elastic properties remains unclear.

The simplest physical model that describes the evolution of lipid domain size
and position predicts that domains diffuse and coalesce, such that the number
of domains constantly decreases, while the average domain size constantly
increases [18]. Indeed, models of two-dimensional phase separation have been
studied in detail for many physical systems [23, 24, 25, 26], and where the
phase boundary is unfavorable and characterized by an energy per unit length
[27], the domain size grows continuously ($\propto t^{1/3}$) [18, 28, 29].
However, membranes can adopt three-dimensional morphologies that affect the
kinetics of phase separation [30, 31, 32, 33, 34]. In those cases where
morphology is considered as part of the phase separation model, novel
coalescence kinetics emerge [31]. Experimentally, model membranes have shown
that nearly complete phase separation on the surface of a cell-sized vesicle
can be reached in as little as one minute [2]. This seems inconsistent with
the fact that on the cell surface, much smaller domains persist on that same
time-scale [21] and no large-scale phase separation is observed. With these
facts in mind, our central questions are: how can membranes that have phase-
separated maintain their lateral heterogeneity on long time scales and short
length scales? Are there membrane-mediated (i.e. elastic) forces that inhibit
coalescence and spatially organize domains?

We begin to answer these questions by examining the energetics of the membrane
using a linear elastic model. A phase-separated membrane is endowed with
bending stiffness, membrane tension, an energetic cost at the phase boundary,
and domains of a particular size. Membrane bending and tension establish a
natural length-scale over which a morphological instability develops that
switches domains from a flat to ‘dimpled’ shape, similar to classical Euler
buckling [35] (see Figure 1). The dimpling instability is size-selective and
‘turns on’ a membrane-mediated interaction that inhibits domain coalescence.
This transition is a precursor to budding, and is distinct from transitions
that require spontaneous curvature. While variations in membrane composition
may change specific parameter values, the mechanical effects we describe are
generic. Thus, these systems exhibit shape-dependent coarsening kinetics, that
are relevant for a broad class of two-dimensional binary fluid systems. The
interaction between domains is a mechanical effect, and we use a model
treating dimpled domains as curved rigid inclusions to distill the main
principles governing this interaction. The confluence of membrane properties
required for this morphological change and its attendant forces lies squarely
in the biological regime. Experimentally, we use a model mixture of lipids and
cholesterol to show that such an interaction exists between dimpled domains
and is well approximated by a simple model. Combined with lipid recycling
[19], we offer elastic interactions as a mechanism for the maintenance of
lipid lateral-heterogeneity and organization of domains in cellular membranes.

Figure 1: Three dimensional rendering of a dimpled lipid domain in
dimensionless coordinates. For a domain (shown in red), a competition between
bending, membrane tension, and phase boundary line tension results in a
morphological transition from a flat to a dimpled state as depicted above. The
dimple costs bending energy but relieves line tension by reducing the phase
boundary length (shown as a white line around the domain). This morphology
facilitates interactions between domains that significantly alter the kinetics
of coalescence and lateral lipid organization. The projected domain radius is
$\rho_{o}=r_{o}/\lambda_{2}$.

The first section of the paper outlines the energetic contributions to the
mechanical model, and predicts the conditions under which domain dimpling
occurs. The second section outlines how dimpled domains facilitate an elastic
interaction and compares the model interaction to our measurements made in
phase-separated giant unilamellar vesicles.

The Elastic Model and Morphological Transitions
The energetics of a lipid domain are dominated by a competition — on one hand
the applied membrane tension and bending stiffness both energetically favor a
flat domain; on the other hand the phase boundary line tension prefers any
domain morphology (in 3D) that reduces the boundary length. We use a continuum
mechanical model that couples these effects, relating the energetics of
membrane deformation to domain morphology. As we will show, this competition
results in a morphological transition from a flat to dimpled domain shape,
where two dimpled domains are then capable of interacting elastically.

Lipid domains in a liquid state naturally adopt a circular shape to minimize
the phase boundary length [2], allowing us to formulate our continuum
mechanical model in polar coordinates. We employ a Monge representation, where
the membrane mid-plane is described by a height function $h({\bf r})$ in the
limit of small membrane deformations (i.e. $|\nabla h|<1$). With this height
function, we characterize how membrane tension, bending, spontaneous curvature
and line tension all contribute to domain energetics.

Changes in membrane height alter the projected area of the membrane and hence
do work against the applied membrane tension, resulting in an increase in
energy written as

${G_{\mbox{\tiny tens}}=\pi\tau\left(\int_{0}^{r_{o}}(\nabla h_{1})^{2}r{\rm
d}r+\int_{r_{o}}^{\infty}(\nabla h_{2})^{2}r{\rm d}r\right),}$ (1)

where $\tau$ is the constant membrane tension, $r_{o}$ is the projected radius
of the domain, $h_{1}$ is the height function of the domain and $h_{2}$ is the
height function of the surrounding membrane [36, 37]. Membrane curvature is
penalized by the bending stiffness with a bending energy written as [39, 36]

${G_{\mbox{\tiny
bend}}=\pi\kappa_{b}^{\mbox{\tiny(2)}}\left(\sigma\int_{0}^{r_{o}}\left(\nabla^{2}h_{1}\right)^{2}r{\rm
d}r+\int_{r_{o}}^{\infty}\left(\nabla^{2}h_{2}\right)^{2}r{\rm d}r\right).}$
(2)

Our model allows the domain and surrounding membrane to have differing
stiffnesses, $\kappa_{b}^{\mbox{\tiny(1)}}$ and $\kappa_{b}^{\mbox{\tiny(2)}}$
respectively, characterized by the parameter
$\sigma=\kappa_{b}^{\mbox{\tiny(1)}}/\kappa_{b}^{\mbox{\tiny(2)}}$, and from
this point on we drop the superscript on $\kappa_{b}^{\mbox{\tiny(2)}}$.
Recent experiments suggest that the bending moduli of a cholesterol-rich
domain and the surrounding membrane are roughly equal [4, 38], and hence for
simplicity, we assume the bending moduli of the two regions are equal (i.e.
$\sigma=1$), unless otherwise noted. In addition to bending stiffness, the
domain may exhibit a preferred ‘spontaneous’ curvature due to lipid asymmetry
or protein binding [40, 34]. The contribution of domain spontaneous curvature
can be simplified to a boundary integral, which couples to the overall
curvature field by

${G_{\mbox{\tiny
spont}}=-2\pi\sigma\kappa_{b}c_{o}\int_{0}^{r_{o}}\left(\nabla^{2}h_{1}\right)r{\rm
d}r=-2\pi\sigma\kappa_{b}c_{o}r_{o}\epsilon,}$ (3)

where $c_{o}$ is the spontaneous curvature of the domain and $\epsilon$ is the
membrane slope at the phase boundary as shown in Figure 1. Further, we assume
the saddle-splay curvature moduli are equal in the two regions, yielding no
dependence on Gaussian curvature. In principle, this contribution could be
accounted for with a boundary term, explored in detail in the supplementary
information (SI). The phase boundary line tension is applied to the projected
circumference of the domain, as shown in Figure 1, by $G_{\mbox{\tiny
line}}=2\pi r_{o}\gamma$ where $\gamma$ is the energy per unit length at the
phase boundary.

Finally, a constraint must be imposed that relates the actual domain area,
$\mathcal{A}$, to the projected domain radius $r_{o}$. The energetic cost to
change the area per lipid molecule is high ($\sim
50-100\,k_{B}T/\mbox{nm}^{2}$ where $k_{B}=1.38\times 10^{-23}\,J/\mbox{K}$
and $T=300\,\mbox{K}$ [41]), hence we assume the domain area is conserved
during any morphological change (see SI for details). We impose this
constraint using a Lagrange multiplier, $\tau_{o}$, with units of tension by

${G_{\mbox{\tiny area}}=\tau_{o}\left(\pi\int_{0}^{r_{o}}(\nabla
h_{1})^{2}r{\rm d}r+\pi r_{o}^{2}-\mathcal{A}\right).}$ (4)

This results in an effective membrane tension within the domain
$\tau_{1}=\tau+\tau_{o}$, which must be negative to induce dimpling. Examining
the interplay between bending and membrane tension, we see that two natural
length scales emerge - within the domain we define
$\lambda_{1}=\sqrt{\sigma\kappa_{b}/\tau_{1}}$ and outside the domain we
define $\lambda_{2}=\sqrt{\kappa_{b}/\tau}$. These length scales allow us to
define the relevant dimensionless parameters in this system.

The total free energy of an elastic domain and its surrounding membrane is
then the sum of these five terms, $G=G_{\mbox{\tiny tens}}+G_{\mbox{\tiny
bend}}+G_{\mbox{\tiny spont}}+G_{\mbox{\tiny line}}+G_{\mbox{\tiny area}}$.
Details on all the terms in the free energy can be found in the SI. With this
free energy in hand, we examine how the morphology of a circular domain
evolves as we tune domain size and the elastic properties of the membrane.

The height field and radius can be rescaled by the elastic decay lengths such
that the Euler-Lagrange equation for the domain can be written in the
parameter-free form $\nabla^{2}(\nabla^{2}+\beta^{2})\eta_{1}=0$, while the
equation for the surrounding membrane is $\nabla^{2}(\nabla^{2}-1)\eta_{2}=0$,
where the dimensionless variables are defined by $\lambda_{2}\eta_{i}=h_{i}$,
$\lambda_{2}\rho=r$, $\lambda_{2}\rho_{o}=r_{o}$ and
$\beta=i\lambda_{2}/\lambda_{1}$. Using the same dimensionless notation, the
energy from line tension and spontaneous curvature can be written as
$G_{\mbox{\tiny line}}=2\pi\kappa_{b}\rho_{o}\chi$ and $G_{\mbox{\tiny
spont}}=-2\pi\sigma\kappa_{b}\epsilon\rho_{o}\upsilon_{o}$, with
$\upsilon_{o}=\lambda_{2}c_{o}$ and $\chi=\gamma\lambda_{2}/\kappa_{b}$. The
dimensionless line tension, $\chi$, is simply a rescaled version of the line
tension $\gamma$ and is one of two key parameters that characterize the
morphological transition; the dimensionless domain area,
$\alpha=\mathcal{A}/\lambda_{2}^{2}$, is the second key parameter.

The admissible solutions for $\eta_{1}(\rho)$ and $\eta_{2}(\rho)$ are zeroth
order Bessel functions $J_{0}(\beta\rho)$ and $K_{0}(\rho)$, respectively,
with the boundary conditions $|\nabla\eta_{1}(0)|=|\nabla\eta_{2}(\infty)|=0$
and $|\nabla\eta_{1}(\rho_{o})|=|\nabla\eta_{2}(\rho_{o})|=\epsilon$. The
boundary slope, $\epsilon$, is the parameter that indicates the morphology of
the domain; $\epsilon=0$ indicates a flat domain, while $0<|\epsilon|\lesssim
1$ indicates a dimpled domain. The five contributions to membrane deformation
energy yield a relatively simple expression for the total free energy, given
by

$\displaystyle G$ $\displaystyle=$
$\displaystyle\pi\kappa_{b}\rho_{o}\left[\epsilon^{2}\left(\sigma\beta\frac{J_{0}(\beta\rho_{o})}{J_{1}(\beta\rho_{o})}+\frac{K_{0}(\rho_{o})}{K_{1}(\rho_{o})}\right)+2(\chi-\epsilon\sigma\upsilon_{o})\right]$
$\displaystyle-\kappa_{b}(\sigma\beta^{2}+1)(\pi\rho_{o}^{2}-\alpha).$

Mechanical equilibrium is enforced by rendering the energy stationary with
respect to unknown parameters $\epsilon$, $\rho_{o}$, and $\beta$,

${\frac{\partial G}{\partial\epsilon}=0,\quad\frac{\partial
G}{\partial\rho_{o}}=0,\quad\frac{\partial G}{\partial\beta}=0.}$ (6)

These equilibrium equations physically correspond to torque balance at the
phase boundary, lateral force balance at the phase boundary and domain area
conservation, respectively.

Analysis of the equilibrium equations reveals a second-order transition at a
critical line-tension, $\chi_{c}$, as shown in Figure 2. For $\chi$ less than
this critical value, only the flat, trivial solution with $\epsilon=0$ exists.
At $\chi_{c}$ a non-trivial solution describing buckled or dimpled
morphologies emerges. For zero spontaneous curvature, the bifurcation is
defined by a transcendental characteristic equation

${\sigma\beta\frac{J_{0}(\beta\rho_{o})}{J_{1}(\beta\rho_{o})}+\frac{K_{0}(\rho_{o})}{K_{1}(\rho_{o})}=0,}$
(7)

with $\beta=\sqrt{(\chi_{c}/\rho_{o}-1)/\sigma}$ and
$\rho_{o}=\sqrt{\alpha/\pi}$. For a given dimensionless domain area, $\alpha$,
this defines the critical line tension required to dimple the domain. In
Figure 2a(inset), this relation is used to generate a morphological phase
diagram that shows where in the space of dimensionless domain area and line
tension we find the discontinuous transition (i.e. bifurcation) from a flat
domain, to a dimpled domain. Near the morphological transition the boundary
slope grows as $|\epsilon|\propto\sqrt{\chi/\chi_{c}-1}$, indicating that a
dimple rapidly deviates from the flat state. The transition is symmetric, in
that both possible dimple curvatures have the same energy, and hence the
domain is equally likely to dimple upwards or downwards. In the experimentally
relevant limit of small dimensionless domain area, the complexity of eqn. 7 is
reduced to

${\chi_{c}\sqrt{\alpha}=\frac{\gamma_{c}}{\kappa_{b}}\sqrt{\mathcal{A}}\simeq
8\sigma\sqrt{\pi}.}$ (8)

This leads to the conclusion that the dominant parameter governing domain
dimpling at zero spontaneous curvature is $\chi\sqrt{\alpha}$. For a small
domain, the dimpling transition is directly regulated by domain area, the
bending modulus, and line tension, but only weakly depends on applied membrane
tension. Intuitively, domains dimple when line tension or domain size increase
(subject to small $\alpha$), as shown in Figure 2a(inset). Likewise, a
decrease in bending stiffness, due, for instance, to changes in membrane
sterol content [42, 43], can also induce dimpling. The effects of applied
membrane tension are weak because the change in projected area upon dimpling
does not lead to a significant energy cost relative to the cost of bending and
line tension.

If membrane elastic properties are fixed (i.e. fixed $\kappa_{b}$, $\tau$ and
$\gamma$), the dimpling-induced interactions ‘turn on’ only after a critical
domain size is achieved. This scenario is encountered when two domains, too
small to dimple on their own, diffusively coalesce into a larger domain
capable of dimpling and hence interacting. Indeed, such a size-selective
coalescence mechanism was observed recently in model membrane vesicles [44].
This constitutes a distinct class of coarsening dynamics, where classical
diffusion-limited kinetics are obeyed until the domain size distribution has
matured past the critical size for dimpling - then domain coalescence is a
relatively slow, interaction-limited process.

Figure 2: Bifurcation diagram for dimpling transition at constant area
($\alpha=\pi/4$, $\kappa_{b}=25\,k_{B}T$, $\lambda_{2}=500\,\mbox{nm}$,
$\sigma=1$). Constant line tension and increasing area produces a
qualitatively similar graph. a) At zero spontaneous curvature
($\upsilon_{o}=0\rightarrow\mbox{black}$) the bifurcation is symmetric, the
upper and lower branches are at the same energy, and $\epsilon=0$ becomes
unstable above the critical point (horizontal black dashed line). With finite
spontaneous curvature ($\upsilon_{o}=2$,
$c_{o}=(250\,\mbox{nm})^{-1}\rightarrow\mbox{blue}$) the lower energy branch
(upper) has non-zero $\epsilon$ for all line tensions, asymptoting to the
$\upsilon_{o}=0$ branch. At a line tension slightly higher than $\chi_{c}$ for
the $\upsilon_{o}=0$ case, a bifurcation yields a higher energy dimple with
the opposite curvature as $\upsilon_{o}$ (indicated by the second vertical
dashed line). Inset: Equilibrium phase diagrams for $\sigma=0.5$(red),
$\sigma=1$(green), and $\sigma=2$(blue) (the dashed lines are the
approximation of eqn. 8) showing flat (F) and dimpled (D) domains. b) Energy
difference between the flat and dimpled state, normalized by $\kappa_{b}$, for
domains with and without spontaneous curvature
($\upsilon_{o}=0\rightarrow\mbox{black}$;
$\upsilon_{o}=2\rightarrow\mbox{blue}$).

For the model domain considered in Figure 2, with area $\alpha=\pi/4$
($r_{o}\simeq 250\,\mbox{nm}$), the critical dimensionless line tension is
$\chi_{c}\simeq 13$, corresponding to a critical line tension of
$\gamma_{c}\simeq 0.65\,k_{B}T/\mbox{nm}$. This value compares well with both
theoretical estimates of the line tension [27, 45], and the higher side of
experimentally measured values [4, 46, 47].

Spontaneous curvature does not affect the Euler-Lagrange equations, and hence
will not effect the class of equilibrium membrane shapes. However, domains
with zero and nonzero spontaneous curvature exhibit qualitatively different
behavior. Biological membranes can be asymmetric with respect to leaflet
composition [42, 5, 48], endowing a domain with potentially large spontaneous
curvature. The energetic contribution from spontaneous curvature takes the
form of an additional line tension depending linearly on the slope taken by
the domain boundary, $\epsilon$. This breaks the symmetry of the membrane,
giving an energetic preference to a dimple with the same curvature as the
spontaneous curvature, and eliminating the trivial $\epsilon=0$ solution even
at small line-tensions. As line tension increases, a bifurcation produces a
second, stable, higher-energy dimple of the opposite curvature as
$\upsilon_{o}$. The more energetically stable branch of this transition
corresponds to a dimpled state for all values of line tension and non-zero
values of domain area, as demonstrated in Figure 2a. This predicts that as
soon as a domain with finite spontaneous curvature forms, it dimples,
regardless of size, and begins to experience interactions with any nearby
dimpled domains. It is reasonable to expect that domains with similar
composition will have similar spontaneous curvature, and hence form dimples
whose curvature has the same sign. As we will show, dimples whose curvature
has the same sign tend to interact repulsively. Such a mechanism of
coalescence inhibition was observed recently in simulation [34].

This indicates that control of spontaneous curvature via domain composition or
protein binding can regulate dimpling and hence domain interaction [49, 48].
Indeed, recent experimental [50] and theoretical [51] work shows that protein
binding and lipid asymmetry, respectively, lead to precisely these kinds of
dimpled domains.

Figure 3: Theoretical and experimental dimpled domain shapes. Domains are
shown in red, surrounding membrane in blue. a) Minimum energy dimples with and
without spontaneous curvature (see legend, $\alpha=\pi/4$, $\sigma=1$). b)
Epi-fluorescence cross-section of a dimple on the surface of a GUV; the red
and blue lines are a guide to the eye. c) 1D model of interaction - dimples
maintain shape, but tilt ($\phi$) as a function of separation distance ($d$).
Dimples with the same sign of curvature repel, while dimples with opposite
sign attract. d) Epi-fluorescence cross-section of two dimpled domains
interacting on the surface of a GUV. Scale bars are $3\,\mu\mbox{m}$.

Calculated shapes of dimpled domains induced by line tension and spontaneous
curvature are shown in Figure 3a, alongside dimpled domains observed on giant
unilamellar vesicles, shown in Figure 3b and d.

Elastic Interactions of Dimpled Domains
Given two domains that have met the criteria for dimpling, the deformation in
the membrane surrounding the domains mediates an elastic interaction when they
are within a few elastic decay lengths ($\lambda_{2}$) of each other. This
equips us to begin addressing how short length-scale and long time-scale
membrane heterogeneity might be achieved. As previously stated, free diffusion
sets the maximum rate at which a quenched membrane can evolve into a fully
phase-separated membrane [18], where this evolution can happen in as little as
a minute on the surface of a cell-sized vesicle [2]. On the other hand,
recycling and hence homogenization of cellular membrane is a process that
takes place on the time-scale of an hour or more [52]. Our measurements of
domain interactions (detailed below and other data shown in SI) estimate the
coalescence barrier between dimpled domains at $\sim 5-10\,k_{B}T$. Hence,
given the diffusion-limited rate of phase separation, interactions slow this
process by approximately $e^{-5}\simeq 0.007$ to $e^{-10}\simeq 0.00005$. This
makes the time-scale of lipid heterogeneity comparable to the time-scale of
membrane recycling and even eukaryotic cell division.

The physical origin of domain interaction is explained by a simple model based
on the assumption that the dimpled domain shape is constant during
interaction, but the domains are free to tilt by an angle $\phi$, as shown in
Figure 3c. This assumption was, in part, inspired by experimental observations
of domain shapes on the surface of giant unilamellar vesicles, as shown, for
example, in Figure 3d. The interaction energy is roughly an order of magnitude
less than the free energy associated with the morphological transition itself
(see Figure 2b), thus interaction does not perturb the domain shape
significantly. Only allowing domains to rotate simplifies the interaction
between two domains to a change in the boundary conditions in the three
regions of interest, shown in blue in Figure 3c. Applying the small gradient
approximation, the boundary slope is given by $|\epsilon-\phi|$ in the outer
regions and by $|\epsilon+\phi|$ in the inner region. With the single domain
boundary slope, $\epsilon$, set by the energy minimization of the previous
section (i.e. eqn. 6), the pairwise energy is minimized at every domain
spacing, $d$, by $\partial G/\partial\phi=0$ to find the domain tilt angle
that minimizes the deformation energy (see SI for details). This results in
two qualitatively distinct scenarios: two domains whose curvatures have the
same sign repel each other, while two domains whose curvatures have the
opposite sign attract each other. Scaling arguments can be used to show that
the strength of interaction between two dimpled domains increases roughly
linearly with their area, so long as they are both larger than some critical
area (see SI for details). Mathematically, the assumption of rigidly rotating
dimpled domains on a membrane is identical to a previous 2D model of bending-
mediated interactions between intramembrane proteins represented by rigid
conical inclusions [53].

Independent of the effects of spontaneous curvature, slight osmolar imbalances
and constriction due to the lipid phase boundaries create small pressure
gradients across the membrane that tend to orient all dimples in a cell or
vesicle in the same direction, resulting in net repulsive interactions between
all domains. Transitions between ‘upward’ and ‘downward’ dimples are
infrequent, due to a large energy barrier. In the simplest case, where the
domains are the same size, the tilt angle $\phi$ monotonically increases as
two domains get closer, $\phi(d)\simeq-\epsilon e^{-d}$. Likewise, the
interaction energy, $V_{\mbox{\tiny int}}(d)\simeq
2\pi\kappa_{b}\epsilon^{2}\rho_{o}^{2}e^{-d}$, increases monotonically with
decreasing separation. For direct comparison, we fit both the 1D model
outlined here and the 2D inclusion model [53] to the data of Fig.4, showing
that they are experimentally indistinguishable, though with a slightly
different elastic decay length.

To quantitatively compare our interaction model with experiment, we examined
the thermal motion of small domains on the surface of giant unilamellar
vesicles, as described in ‘Materials and Methods.’ Membrane tension was
regulated by balancing the internal and external osmolarity, giving us coarse
control over the elastic decay length $\lambda_{2}$. Through time, the
distance between every domain pair was measured and the net results were used
to construct a histogram. The potential of mean force as a function of
distance between domains is shown in Figure 4b. We selected vesicles that had
a low density of approximately equal-sized domains, and thus generally the
interactions were described by a repulsive pairwise potential. Though areal
density of domains and generic data quality varied in our experiments (see
SI), all data sets exhibit the repulsive core of the elastic interaction.
Multi-body interactions occur, though infrequently; their effect can be seen
as a small variation in the baseline of Figure 4b, which is not captured by
the pairwise interaction model. At high membrane tension, when we would not
expect dimpled domains, we qualitatively verified that domains coalesce in a
rapid manner as compared to our low tension experiments (data not shown).
Other recent experiments have also observed repulsive interactions between
domains on low membrane tension vesicles and the lack thereof on taut vesicles
[44].

Figure 4: Measuring domain interactions on the surface of a vesicle. a) Three
images of dilute interacting domains on the surface of the same vesicle (scale
bar is $10\,\mu\mbox{m}$). b) The repulsive interaction potential of domains
on the surface of the same vesicle as (a). The energy is measured in $k_{B}T$
and distance is domain center-to-center. The blue dashed line is a fit to the
1D interaction model in this paper, $V_{\mbox{\tiny
int}}(r)=a_{1}e^{-r/\lambda_{2}^{\mbox{\tiny(1D)}}}+a_{2}$, with elastic decay
length $\lambda_{2}^{\mbox{\tiny(1D)}}=240\,\mbox{nm}$. The orange dashed line
is a fit to the model, $V_{\mbox{\tiny
int}}(r)=2\pi\kappa_{b}\left[(a_{1}a_{2})^{2}K_{0}(r/\lambda_{2}^{\mbox{\tiny(2D)}})+a_{2}^{2}a_{3}^{4}K_{2}^{2}(r/\lambda_{2}^{\mbox{\tiny(2D)}})\right]+a_{4}$,
with elastic decay length $\lambda_{2}^{\mbox{\tiny(2D)}}=270\,\mbox{nm}$,
based on the theory of Weikl et al [53]. Both elastic decay lengths indicate a
membrane tension of $\sim 4\times 10^{-4}\,k_{B}T/\mbox{nm}^{2}$. Errors bars
are shown in green on the $x$-axis.

Our measurement of the pairwise potential allows us to estimate elastic
properties of the membrane. The elastic decay length was fit to the 1D and 2D
interactions models described above, and found to be
$\lambda_{2}^{\mbox{\tiny(1D)}}\simeq 240\,\mbox{nm}$ and
$\lambda_{2}^{\mbox{\tiny(2D)}}\simeq 270\,\mbox{nm}$, respectively. Taken
with a nominal bending modulus of $25\,k_{B}T$, we estimate the membrane
tension to be $\sim 4\times 10^{-4}\,k_{B}T/\mbox{nm}^{2}$. From the images,
we measure the size of the domains at $r_{o}\simeq 350-400\,\mbox{nm}$, and
hence $\rho_{o}\simeq 1.5$. We estimate the line tension, $\gamma$, using eqn.
8, based on the fact that the domains are dimpled, and find a lower bound of
$\gamma\simeq 0.49\,k_{B}T/\mbox{nm}$. This is in good agreement with
theoretical estimates and values determined from AFM measurements [47], though
somewhat higher than the value of $\gamma\simeq 0.22\,k_{B}T/\mbox{nm}$
measured via shape analysis of fully phase separated vesicles [4] and
$\gamma\simeq 0.40\,k_{B}T/\mbox{nm}$ from micropipette aspiration experiments
[46]. Finally, viewing the repulsive core of the interaction as an effective
activation barrier to coalescence, a simple Arrhenius argument suggests a
decrease in coalescence kinetics by two to three orders of magnitude. Indeed,
such a slowing of coalescence was recently observed in a similar model
membrane system [44].

Discussion
Comparing biologically relevant domain sizes ($\sim 50-500\,\mbox{nm}$) with
the elastic decay length ($\lambda_{2}$), we expect physiologically relevant
domains to be small (i.e. small $\alpha$), as presumed in eqn. 8. Estimating
the elastic decay length requires knowledge of the membrane tension and
bending stiffness. We note that in vitro experiments of osmotically balanced
single and multicomponent vesicles, and measurements of the plasma membrane of
unstressed cells suggest membrane tensions of
$10^{-4}-10^{-2}\,k_{B}T/\mbox{nm}^{2}$ [4, 41, 54, 55]. The typical bending
modulus of a phosphocholine bilayer is $\sim 10-50\,k_{B}T$, depending on the
exact lipid and cholesterol content [41, 56, 57]. Choosing a nominal membrane
tension of $10^{-4}\,k_{B}T/\mbox{nm}^{2}$ and nominal bending modulus of
$25\,k_{B}T$ [41, 4] corresponds to an elastic decay length of
$\lambda_{2}\simeq 500\,\mbox{nm}$, suggesting that for lipid domains on the
order of $50-500\,\mbox{nm}$, small $\alpha$ is an appropriate approximation.

Our experiments on the surface of GUVs have three potentially confounding
effects, all due to the spherical curvature of the vesicle. First, the surface
metric is not entirely flat with respect to the image plane. Thus,
measurements of distance are underestimated the farther they are made from the
projected vesicle center. This problem is ameliorated by concentrating on
domains which are at the bottom (or top) of the vesicle where the surface is
nearly flat and demanding that our tracking software exclude domains that are
out of focus; see SI for a more detailed explanation. The second potential
complication is that we use a flat 2D coordinate system for our theoretical
analysis, however domains reside on a curved surface. Given that the domain
deformation, and hence energy density, decays exponentially with
$\lambda_{2}$, as long as $\lambda_{2}$ is small with respect to the vesicle
radius, the energetics that govern morphology converge on an essentially flat
surface metric. The final complication is that the circular area of focus
creates a fictitious confining potential for the domains, such that the
effective measured potential of mean force is the sum of the elastic pairwise
potential and a fictitious potential, $V_{\mbox{\tiny eff}}=V_{\mbox{\tiny
int}}+V_{\mbox{\tiny fict}}$. The fictitious potential is removed by
simulating non-interacting particles in a circle the same size as the radius
of focus (see SI for details).

The constant tension ensemble used in our theoretical analysis has a range of
validity, determined by the excess area available on a thermally fluctuating
membrane with conserved volume and total surface area $\mathcal{A}_{o}$ (i.e.
a vesicle). In the limit where the morphological transitions use only a small
portion ($\Delta\mathcal{A}$) of this excess area, defined by
$k_{B}T/8\pi\kappa_{b}\gg\Delta\mathcal{A}/\mathcal{A}_{o}$, the tension is
constant. Outside this regime the tension rises exponentially with reduction
in excess area, tending to stabilize dimples from fully budding (see SI for
details).

In addition to the elastic mechanism of interaction, described herein, there
may be other organizing forces at work in a phase-separated membrane: those of
elastic [27], entropic [58, 59] and electrostatic origin [60], however their
putative length-scale, on the order of ten nanometers or less, is not
accessible to the spatial and temporal resolution of our experiments, and not
consistent with our measurement of an interaction length-scale of hundreds of
nanometers.

Conclusion
We have shown that lipid domains are subject to a morphological dimpling
transition that depends on the bilayer elastic properties and domain size.
Dimpling allows two domains in proximity to repulsively interact due to the
deformation in the surrounding membrane. Our model makes some key predictions:
at zero spontaneous curvature the domain size distribution reaches a critical
point where coalescence is arrested by repulsive interactions; domains with
finite spontaneous curvature are always subject to interaction and hence
should always coalesce at a rate slower than the diffusion-limited rate.
Additionally, the strength of elastic interactions is augmented by increasing
line tension or domain area, with an approximately linear scaling. The domain
size and bilayer elastic parameters necessary to induce the dimpled morphology
are consistent with physiological conditions. Further, careful regulation of
membrane cholesterol in cells may be related to the membrane mechanical
properties necessary for morphological transitions. Combined with lipid
recycling, our work offers a mechanism working against diffusion-driven
coalescence, to maintain fine-scale lateral heterogeneity of lipids over time.

We proposed a simple 1D model of an elastic interaction that mediates dimpled-
domain repulsion, and then used a standard ternary membrane system to verify
the existence of dimpled domains and their subsequent repulsive interaction.
Finally, it follows that the morphologies and elastic forces which organize
lipid domains might play an important role in the binding and lateral
organization of proteins in the membrane.

Materials and Methods
Giant unilamellar vesicles (GUVs) were prepared from a mixture of DOPC
(1,2-Dioleoyl-sn-Glycero-3-Phosphocholine), DPPC (1,2-Dipalmitoyl-sn-
Glycero-3-Phosphocholine) and cholesterol (Avanti Polar Lipids)
(25:55:20/molar) that exhibits liquid-liquid phase coexistence [2].
Fluorescence contrast between the two lipid phases is provided by the
rhodamine head-group labeled lipids: DOPE (1,2-Dioleoyl-sn-
Glycero-3-Phosphoethanolamine-N- (Lissamine Rhodamine B Sulfonyl)) or DPPE
(1,2-Dipalmitoyl-sn-Glycero-3-Phosphoethanolamine-N- (Lissamine Rhodamine B
Sulfonyl)), at a molar fraction of $\sim 0.005$. The leaflet compositions are
presumed symmetric and hence $\upsilon_{o}=0$.

GUVs were formed via electroformation [2, 61]. Briefly, $3-4\,\mu\mbox{g}$ of
lipid in chloroform were deposited on an indium-tin oxide coated slide and
dessicated for $\sim 2\,\mbox{hrs}$ to remove excess solvent. The film was
then hydrated with a $100\,\mbox{mM}$ sucrose solution and heated to $\sim
50\,\mbox{C}$ to be above the miscibility transition temperature. An
alternating electric field was applied; $10\,\mbox{Hz}$ for 120 minutes,
$2\,\mbox{Hz}$ for 50 minutes, at $\sim 500\,\mbox{Volts/m}$ over $\sim
2\,\mbox{mm}$. Low membrane tensions were achieved by careful osmolar
balancing with sucrose ($\sim 100\,\mbox{mM}$) inside the vesicles, and
glucose ($\sim 100-108\,\mbox{mM}$) outside.

Domains were induced by a temperature quench (see SI) and imaged using
standard TRITC epi-fluorescence microscopy at 80x magnification with a cooled
(-30 C) CCD camera (Roper Scientific, $6.7\times 6.7\,\mu\mbox{m}^{2}$ per
pixel, 20 MHz digitization). Images were taken from the top or bottom of a GUV
where the surface metric is approximately flat (see SI). Data sets contained
$\sim 500-1500$ frames collected at 10-20 Hz with a varying number of domains
(usually $5-10$). The frame rate was chosen to minimize exposure-time blurring
of the domains, while allowing sufficiently large diffusive domain motion.
Software was written to track the position of each well-resolved domain and
calculate the radial distribution function. The raw radial distribution
function was corrected for the fictitious confining potential of the circular
geometry (see SI). In the dilute interaction limit, pairwise interactions
dominate, and the negative natural logarithm of the radial distribution
function is the interaction potential (potential of mean force) plus a
constant, as shown in Figure 4b.

We thank Patricia Bassereau, Ben Freund, Kerwyn Huang, Greg Huber, Sarah
Keller and Udo Seifert for stimulating discussion and comments on the
manuscript, and Jenny Hsaio for help with experiments. TU and RP acknowledge
the support of the National Science Foundation award No. CMS-0301657, NSF
CIMMS award No. ACI-0204932, NIRT award No. CMS-0404031 and the National
Institutes of Health Director’s Pioneer Award. WK acknowledges support from
NSF CAREER Award CMMI-0748034.

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# An evolutionary approach to Function

Phillip Lord
School of Computing Science
Newcastle University
Newcastle-Upon-Tyne
United Kingdom
NE1 7RU
<EMAIL_ADDRESS>
###### Abstract

Background: Understanding the distinction between function and role is vexing
and difficult. While it appears to be useful, in practice this distinction is
hard to apply, particularly within biology.

Results: I take an evolutionary approach, considering a series of examples, to
develop and generate definitions for these concepts. I test them in practice
against the Ontology for Biomedical Investigations (OBI). Finally, I give an
axiomatisation and discuss methods for applying these definitions in practice.

Conclusions: The definitions in this paper are applicable, formalizing current
practice. As such, they make a significant contribution to the use of these
concepts within biomedical ontologies.

##  1 Background

Large parts of modern biology are aimed at answering questions about function.
For example, much of the Gene Ontology deals with molecular function(Ashburner
et al., 2000). In dealing with the social aspects of science, roles are
similarly important. It is clear, therefore, that function and role are
important concepts in biomedical ontologies and are prime candidates for
inclusion in an upper ontology. A coherent, consistent and shared definition
for function and role is likely to decrease the effort required to integrate
independently-developed ontologies.

One upper ontology, the Basic Formal Ontology (BFO)(bfo, ), currently in use
by a number of groups, includes a definition of function. However, this
definition is not naturally applicable to biology; it is not clear, for
instance, that Gene Ontology molecular functions are also BFO functions, as
discussed later (Section 2.1). An alternative is available within the General
Formal Ontology (GFO), which provides an ontology of function(Burek et al.,
2006). This ontology provides a more extensive framework for describing
functions but, in itself, does not define biological function. Despite this,
there is a reasonable degree of informal agreement among biologists as to the
meaning of the word. Conversely, while formal and informal definitions for
role seems clear, many people have difficulties in applying it in practice.

In this paper, I address two key issues relating to the modeling of function
and role coherently for biomedical ontologies: firstly, how I unify the
definitions of function as they apply to artifacts and to life; and, secondly,
how do I differentiate between roles and functions? I consider illustrative
examples where the answers are reasonably clear and evolve a definition from
them. I then consider examples of the application of current definitions from
BFO in a practical biomedical use. Finally, I offer an axiomatisation in OWL
stemming from my definitions and consider how they could be applied in
practice.

##  2 Results

###  2.1 Biological Function

First, consider the current definition of function provided by BFO (see Def:
1).

###### Definition 1

Function is a realizable entity the manifestation of which is an essentially
end-directed activity of a continuant entity being a specific kind of entity
in the kind or kinds of context that it is made for

As a simple example, a hammer (_the continuant entity_) was made to hammer
nails (_the function_) in a hammering process (_the end-directed activity_).
This definition is problematic for biological systems. The problem here is
simple: most biological systems were not made or designed for any purpose.

Although, it has not been incorporated into BFO yet, there is a potential
definition for “biological function” which would become a new child of
function (Def: 2)(Arp and Smith, 2008).

###### Definition 2

A biological function is a function which inheres in an independent continuant
that is i) part of an organism and ii) exists and has the physical structure
as a result of the coordinated expression of that organism’s structural genes.

As an example, a foot (_the independent continuant_) is part of an organism,
exists, and develops in a controlled way as a result of gene expression. There
are some difficulties with this definition. Consider the following examples: a
differentiated tumour has its structure through coordinated expression of its
genes, it exists and it engages in end-directed activity; a male ant has its
structure as a result of gene expression, and engages in end-directed
activity, however, it is not part of an organism. From this, I conclude that a
differentiated tumour does, indeed, have a function (growing). A male ant,
however does not have a function. Also consider molecular function: the
physical structure of a protein is independent of the expression of an
organisms structural genes – only its presence depends on this. A protein,
therefore, does not have a function, by this definition. So, there are two key
problems: the definition does not work for entities above or below a certain
size; and most biological entities have their structure as a result of
coordinated expression. I offer the following alternate definition (Def: 3).

###### Definition 3

A biological function is a realizable entity that inheres in a continuant
which is realized in an activity, and where the homologous structure(s) of
individuals of closely related and the same species bear this same biological
function.

The definition given uses the notion of homology; evolution is key to our
understanding of biology and it is appropriate that it should be used to
define biological function. If a biological structure has a function, then
this function will have evolved along with the structure; so, other structures
with a common evolutionary descent will display the same behaviour. This
definition also mirrors closely normal biological practice; the most common
way to determine the function of an unknown structure is to look for function
of a homologous structure.

It should be noted that this definition of biological function is _not_
circular, although it has itself as part of its definition; rather it is
recursive; a chimp hand and a human hand can have the same function because of
each other. It does require that a structure must have a homolog for it to
have a function. It does not require that these homologous structures be
extant.

Applying this to our examples: the tumour now has no function because it has
no homologs (different tumours arise as independent events and share no common
ancestor). Likewise, the activity of the male ant now clearly is a function,
as many different, related organisms behave in a similar way. Finally, a
protein may have a function depending on the activity of its homologs. In
short, this definition results in the same conclusions as our biological
understanding.

###  2.2 Relating the Functions

As well as biological function, another subclass, artifactual function has
been suggested(Arp and Smith, 2008). Next, I consider the relationship between
function and these biological and artifactual subclasses. Taking an
illustrative example, consider the sole of my foot and the sole of my shoe.
They appear to operate to: provide a frictional surface to enable motion;
provide padding to reduce shock to everything above; be tough enough to resist
abrasion. They would appear to have the same function; indeed, like many
artifacts, we would guess that the shoe owes much of its design to mimicry of
biology. It would seem, therefore, sensible that instances of Shock Resistance
could be either a biological or artifactual function. The alternative would be
to duplicate many functions under both subclasses (“biological shock
resistance”, and “artifactual shock resistance”) with very similar
definitions. Some functions such as reproduction have to be a biological
function, while others liquifying iron can never be. In short, whether an
instance of function is biological or artifactual should be determined from
the nature of the entity in which it inheres, rather than the process by which
it is realized. I therefore offer a simple definition of function which
reflects this (Def: 4).

###### Definition 4

A function is a realizable entity which is a biological function or an
artifactual function.

For the purposes of this paper, I note that the definition given for function
earlier (Def: 1), can serve as a reasonable definition for artifactual
function.

It is also interesting that this definition covers some unusual but
nonpathological examples. Take a bacterium whose colonies change colour
depending on the presence of a toxin and which was produced using synthetic
biology techniques. The components have all evolved, but the organisation has
not. Is the detection of the toxin then a biological or artifactual function?
This is clearly a difficult, if uninspiring, question but given Def: 4, one
that can avoid by simply describing it as a function; perhaps more
intuitively, it can be described as both a biological and an artifactual
function, suggesting strongly that these two classes should not be disjoint.

###  2.3 Roles

Next, having considered the definition of function and its applicability to
biology, I consider the issue of roles. The current definition (Def: 5) is
complex; put more simply, it suggests that the entity having that role can be
involved in an activity but that it was not necessarily intended for, nor
necessarily has the structure for this.

###### Definition 5

A realizable entity the manifestation of which brings about some result or end
that is not essential to a continuant in virtue of the kind of thing that it
is but that can be served or participated in by that kind of continuant in
some kinds of natural, social or institutional contexts.

Consider the relationship between role and function. Again, I shall use a
simple biological example, in this case of a man walking on his hands. By our
earlier definition (Def: 3), “to walk on hands” is _not_ a function. While the
homologous structure is, indeed, used for walking on in all closely related
species, most humans do not walk on their hands. It would, therefore appear to
be a role. In this context the hand has a role of Shock Resistance. This
realizable entity also appears in the hands of many other primates; in this
case, however, it would appear to be a function of the primate hand, as it is
a function of their feet.

I am left with a similar conclusion as previously. Just as Shock Resistance
maybe either a biological or artifactual function, I must also conclude that
the individuals of the same class can be a role, depending on the nature of
the relationship between the independent continuant and realizable entity.

###  2.4 OBI as a case study

So far in this paper, I have considered a number of illustrative examples and
used these to draw conclusions about definitions for functions and roles. This
methodology is appropriate, but has the limitation that the choice of other
examples may have led us to different conclusions. In this section, therefore,
I will consider the use that OBI (Ontology for Biomedical Investigations)(The
OBI consortium, 2009) has made of function and role (Analysis was performed on
OBI rc-1, (release 2009-11-06)). I choose to use OBI as it was built after BFO
and with knowledge of it; many of the ontologies available from OBO were
started without its use or knowledge.

Function Bearer |  OBI Function
---|---
Human |  Perturb, Measure, Separation (3), Sort
Computer |  Information Processor (3), Consume Data
Highly Generic |  Freeze, Heat, Environment Control, Mechanical, Record, Contain, Transfer, Cool, Connection, Synthesizing, Excitation (2), Ionization, Energy Supply (1)
Distant Galaxy |  Magnify
Out of Scope |  Molecular Function (3)
Table 1: OBI Functions considered as Roles. I provide suggestions for entities
that could engage in the same end-directed activity, but without being
designed for the purpose. Function has been omitted from the OBI term names.
Numbers indicate child terms which have been omitted for brevity. I do not
consider Information Processor to be the function of a computer, as the
definition is more specific than the term suggests.

Considering first the functions of which OBI has 38. Can these functions be
fulfilled by an entity which was not designed for the purpose? As shown in
Table 1 most of them can, often by considering a highly generic device (like a
computer) or organism (like a human). Some are highly generic in themselves
and can be performed by many things (heat for example). I find no cases where
a function could not be fulfilled by an entity which was not designed for the
purpose. I consider “Molecular Function” to be out-of-scope for this section
as it is not clear whether it fulfils the BFO definition of function.

Next, I consider OBI roles. There are many more roles than functions – around
90 in fact. Due to the size, here I consider whole branches of the role
hierarchy.

There are a set of roles relating to reagents and their states. While label
role (defined as a reagent role realized in a detection of label assay) seems
sensible, it would appear that S35 CTP, used to label DNA, is manufactured
specifically for this purpose. It certainly does not occur in nature. It would
appear to fulfil the requirements for a function.

Next, let us consider reference substance roles (a role which can support the
observation of similarities, differences, relative magnitude or change). In
many cases, biological assays use a reference which is not manufactured.
However, consider $\lambda$-HindIII fragments or a calibration standard. These
would all appear to function as an reference and have been produced
specifically for this purpose.

Finally, there are a number of roles for molecules or organisms: antigen role,
pathogen role and primer role. In an age where we can engineer the production
of DNA, protein and organisms, it is not clear that these can only ever be
roles and not functions.

As a result of this analysis, I suggest a modification to the current
definition of role (Def: 5). Both roles and functions can become apparent
(_realized_) in natural, social or institutional contexts. That such a context
exists does not provide a clear differentation between role and function; the
critical distinction relates to whether the entity in question was designed to
be or has homologs that are engaged in a given process. I suggest, therefore,
this alternate and simpler definition for role (Def: 6).

###### Definition 6

A realizable entity the manifestation of which brings about some result or end
that is not essential to a continuant because of its kind.

In short, from this case study, I conclude that the role/function distinction
is not clear. While OBI has a specific intent in mind with its application of
the distinction (broadly and not exhaustively, social or experimental roles,
device or instrument functions), this distinction is not the distinction made
in the current definitions of role and function in BFO; further given that
most functions could also appear to be roles, and many roles appear also to be
functions, I suggest that the distinction made in the current definitions is
not useful in the context of OBI.

The earlier theoretical analysis seems to be confirmed in practice within OBI.
This suggests that the limitations in the definitions drawn from the earlier
illustrative examples are general and not simply as a result of the specific
examples chosen.

###  2.5 An axiomatisation for function and role

I have produced an axiomatisation in OWL of functions and roles as defined in
this paper, available as described in the abstract; due to space
considerations I report here key differences between this and the BFO
axiomatisation.

  * •

There is an explicit relationship between RealizableEntity and Process.
Subclasses use a more specific relationship. So, a ToAbsorbShock function may
only be realized by a AbsorbtionOfShock process, if it is realized at all.

  * •

Function and Role are defined classes. Stating that an instance of
ToAbsorbShock is_function_of instance of FootSole implies, therefore, that the
former is a function.

  * •

Most leaves of RealizableEntity are direct children of RealizableEntity, with
a few exceptions (ToReproduce is a child of BiologicalFunction).

The definitions could be extended further; for simplicity, I have not added
classes to differentiate between organisms and artifacts. These could be added
to automate the population of BiologicalFunction and ArtifactualFunction.

At the current time, it remains an open question whether Role, Function and
its children should be disjoint. The key example of the function of a
synthetic biological organism suggests that Biological and ArtifactualFunction
function should not be disjoint (as it appears to be both), but I have no
example which suggests whether Role and Function should be disjoint. In
axiomatisating the examples given, these disjoint statements make no practical
difference.

Many ontologies are built using the OBO format; while this has a slightly
weaker semantics than OWL it is possible to represent much of OWL in OBO
format(Golbreich et al., 2007). The axiomatisation presented here can be
represented using OBO union and intersection of to describe classes, which is
usually translated as a definition. The universal link between
RealizableEntity and Process has no natural equivalent. However, as this link
has its own specific relationship which is restricted to this use, problems
caused by the lack of an inexact semantic equivalent are likely to be
relatively minor.

The axiomatisation presented here is related to that produced by others;
Dumontier(Dumontier, 2008) focuses more on roles, while Burek et al.(Burek et
al., 2006) provides for a more complex representation, covering issues such as
preconditions.

###  2.6 Applying the Definitions in Practice

Finally, I consider whether these definitions are _applicable_ ; for a given
set of entities how do we decide whether we have a function (of either
subclass) or a role.

The definition of an artifactual function easily allows its application:
first, we determine whether the entity in question is an organism or part of
one (which it should not be); second, we could ask whoever produced the entity
what it was designed for. Of course, the second may not always be possible, in
which case, we can guess from its design what its purpose is. In most cases,
these questions will provide a clear answer.

For biological function, the situation is less clear. Whether an entity is an
organism or part of one is, in practice, likely to be straightforward for
extant entities; otherwise, we can apply palentological techniques. Likewise,
identification of closely related species and homologous structures is well
known as it forms the basis of taxonomy. While developing an exact definition
for “closely related” is outside the scope of this paper, it is possible.

The definition that I introduce for Function in this paper (Defn 4) is
conjunctive; it is either biological or artifactual. Here I have given little
evidence that these are the only kind of function. Fundamentally, these two
arise from very different mechanisms. There could be other appropriate
subclasses of function; the most obvious possibility would be Chemical
Function. However, artifacts are designed by humans who understand, mimic and
improve on biology by building tools. It is this mimicry that we wish to
reflect with a common definition joining biological and artifactual function;
this is not true for Chemical Function.

To determine whether something is a role, it is possible to make a
determination on the basis of whether the context is optional(Arp and Smith,
2008). However, this optionality is a difficult criterion; firstly, all
RealizableEntity’s are optional in the sense that they might never be realized
and, secondly, the optionality can depend on how specifically we define the
bearer. A hammer is not designed to hammer nails, as claimed earlier, it is
designed to hit things; a nail hammer is designed to hit nails, a toffee
hammer to hit toffee, a warhammer to hit anyone who irritates you. In
practice, a role can be considered to be a negative definition; if there is a
continuant and an end-directed activity that the continuant can be involved
in, and this involvement is known not to fulfil the definition of either
function, then we have a role.

In this paper, I have considered OBI and found that the distinction between
role and function is hard to apply; this is not true for all ontologies. For
example, consider the Gene Ontology. In many cases, the homology will be
considered as a standard part of the operating procedure(goe, ) in determining
the function of a gene product; regardless, the evidence codes would allow us
to make the distinction. We can conclude, therefore, that when the Gene
Ontology is used to annotate a protein, this describes a biological function
rather than a role.

###  2.7 Life is hard

So far, I have considered a set of examples and how the definitions might be
applied, including examples from OBI which have not been preselected. However,
categorising life is hard; here, I consider some examples which present
difficulties for the definitions I have given and the implications of these
examples.

In the first example, which I term a _drop out_ species, consider a human
walking on their hands. Earlier (Section 2.3), I have suggested that this
should be considered a role. Most of the primates do, however, walk on their
hands. However, given that the homologous structure of closely related species
use the structure for the same purpose, the definition of function (Def: 3)
would appear to apply. It is for this reason that the definition specifies
that (most) individuals of the same species must also demonstrate this
behaviour (This definition differs slightly from that given at Bio-Ontologies
2009). In short, in the absence of most individuals in a species using a
structure in a specific process, we should not use consider this structure to
have a function.

The second example, I term a _drop in_ species. Again, using a human example,
I use my larynx for vocalisation and talking. Most primates, likewise,
vocalise with their larynx; therefore, according to the given definition, this
is a function of the larynx. However, speech is considered unique to humans,
and therefore, their larynx; given that homologous structures in closely
related organisms do not bear this realizable entity, which is part of the
definition for biological function (Def: 3), I am forced to conclude that this
is a role of the larynx and not a function.

In short, while sharing a realizable entity within a species is NOT sufficient
to allow the conclusion that this entity IS a function, NOT sharing a
realizable entity within a species is sufficient to conclude that this entity
is NOT a function.

One solution to this difficulty is to state that where most individuals in a
single species use a structure within a given process, this alone is
sufficient to conclude that the structure has a function. Simply, most humans
talk with their larynx, therefore this would be a function. I counter this,
however, with the example that most humans use their fingers to operate their
mobile phones, so we would be forced to conclude that this would also be a
function. As this seems opposed to normal biological intuition and usage, I
conclude, the presence of most individuals in a species using a structure in a
specific process, is _not_ sufficient to conclude that this structure has a
function.

It is also possible that this difficulty could be resolved with greater
knowledge or changes in biology. Def: 3 does not require species be extant; if
a close, but extinct, relative of humans were shown to speak with their
larynx, or if humans speciated while maintaining their speech, again, I would
conclude that this represented a function.

While human speciation seems unlikely, it is much more relevant to other taxa.
Bacteria, in particular, evolve rapidly. There are many genes and proteins in
bacteria which are unique to a species, family or lineage(Siew et al., 2004).
In this case, the requirement for closely-related species seems to rule out
the presence of a function. Again, this seems opposed to normal biological
intuition and usage. I would counter this with two arguments. First, unlike
primates, our knowledge of the extant bacterial species is very limited. The
lack of knowledge of another speaking primate species is good evidence that no
such species exists; the lack of knowledge of a close relative for a given
bacterial species is not. Second, any definition which relies on a notion of a
species is only as good as the definition of species; for bacteria, there is
considerable debate about the utility of a species classification(Rosselló-
Mora, 2003); my definition of function will need to evolve along with our
understanding of bacterial ecology and gene flow; it may be necessary, as has
been suggested with definitions for bacterial species(Rosselló-Mora, 2003), to
have different definitions of BiologicalFunction tailored to different parts
of the taxonomy.

The evolution of a definition of role and function for proteins is difficult.
At the level of the protein, I side with Dumontier(Dumontier, 2008), who
suggests that the role/function distinction may be redundant; broadly,
proteins can do anything their structure allows, and only do things their
structure allows. The definitions given in this paper have a consistent
interpretation at the level of the protein; this avoids the necessity of
deciding at which level of granularity to stop making the role/function
distinction. We can make the distinction at all levels if we choose, but we
are not forced to do so, at those levels of granularity where it is not
useful.

It is important to note that arguing against a role/function distinction for
proteins is not to dismiss the experience of biologists in the analysis of
function assignment for genes. In this sense, the word “function” is being
used to describe an association between a protein and a process that a protein
molecule may be involved in; in short, the word “function”, in this case, can
be considered to be a synonym for “realizable entity”.

##  3 Conclusions

Here, I have taken an evolutionary approach to function and role by
considering examples and using this to derive definitions which are as
consistent as possible with current use within biomedical sciences. These
definitions have been encoded in an axiomatisation which should enable the use
of these definitions in a machine-interpretable way.

The applicability of these definitions is a key advantage; the current
distinction being made between function and role is a hard one to understand
and apply. My definition distinguishes between the two based on the nature of
the relationship to the independent continuant in which they inhere. I suggest
that it is very hard to make the distinction at the class level; my study of
OBI shows that very few of the functions and roles clearly fall into one
category or another. For an individual continuant bearing a realizable entity,
this distinction appears to be much more straightforward.

I also provide a definition of biological function, something that is
currently lacking in BFO. I have paid close attention to current biological
usage; the definition is close to the process used to determine function.
Moreover, it is highly applicable; all parts of the definition are measurable.

The desire for an applicable and measurable definition is also the reason that
I have avoided a definition based on the outcome of selective pressure; this
is hard to test in most circumstances, requiring expensive evolutionary
studies, and impossible for extinct species. Selective pressure can also be
transient. Consider industrial melanism(Majerus, 1998); should melanic
coloring be considered to gain its function during periods of pollution and
lose it in post-industrial periods? By way of analogy, should a spanner
measured in inches be considered to lose its function following metrication?
Serendipitously, it also avoids difficult questions about artificial
selection; we can state clearly that cows do not have a function of producing
beef, though this is the outcome of selection.

Importantly, my definition of biological function works across multiple levels
of granularity: from organisms and organism parts through to genes and
molecules; this is not true of previous definitions(Arp and Smith, 2008),
which cover only anatomy. It is, however, not clear how useful the
role/function distinction is for proteins and genes, as discussed earlier
(Section 2.7); It is for this reason that I have used homology rather than
orthology as the basis for the definition, as the latter is limited to the
genetic scale, where the distinction is least useful.

Finally, my definitions also do not allow distinctions that may often be made
between different types of function. For example, most biologists would
consider motion the most important function of muscle, while heat production a
byproduct; or, for a more pathological example after Hoehndorf et al. (2009),
most biologists would consider blood circulation to be a function of the
heart, but “making loud thumping noises” not to be. This is a concern which
could be best addressed by incorporating a degree of social ascription into
the categorisation of realizable entities within biology; although it is
outside the scope of this paper, this would provide a valuable and useful
addition to the current ontological practice.

In summary, I believe that the definitions and axiomatisation given in this
paper make a significant contribution to the use of role and function in
biomedical ontologies. They should enable a consistent use of these classes,
because they consider current usage of the terms and the applicability of
these definitions. I seek not to change current use but to formalize it.

##  4 Competing Interests

The author has no competing interests.

##  5 Acknowledgements

Thanks to Frank Gibson, Allyson Lister, James Malone, Helen Parkinson, Matt
Pocock and Robert Stevens for many useful discussions on the contents of this
paper. Thanks to the audience at Bio-Ontologies 2009 for the example of
rapidly-evolving bacteria.

To the memory of Mike Majerus who showed me how fundamentally strange
evolution can be.

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