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\begin{center}
{\Large CONDENSED MATTER PHYSICS OF BIOMOLECULE SYSTEMS \\ IN A DIFFERENTIAL GEOMETRIC FRAMEWORK} \\
\vspace{1cm}
\vspace{0.5cm}
Henrik Bohr\\
Department of Physics, Quantum Protein Centre \\
The Technical University of Denmark, Building 309 \\
DK-2800 Lyngby, Denmark \\
\vspace{0.5cm}
John H. Ipsen \\
Memphys, Department of physics \\
South Danish University, SDU \\
Campusvej 55, DK-5230, Odense M, Denmark \\
\vspace{0.5cm}
Steen Markvorsen \\
Department of Mathematics \\
The Technical University of Denmark, Building 303S \\
DK-2800 Lyngby, Denmark
\end{center}
\vspace{1cm}
\noindent
\section*{Abstract}
In this contribution biomolecular systems are analyzed in a framework of differential geometry
in order to derive important condensed matter physics information.
In the first section lipid bi-layer membranes are examined with respect to statistical properties and topology,
e.g. a relation between vesicle formation and the proliferation of genus number.
In the second section differential geometric methods are used for analyzing the surface structure
of proteins and thereby understanding catalytic properties of larger proteins.
%\vspace{1cm}
\section*{SECTION I: Thermodynamics and topology of closed membranes}
%\vspace{1cm}
%\noindent
%{\large Abstract} \\
In this section we shall describe the lipid
bilayer membrane system in differential
geometrical terms by a curvature elasticity
Hamiltonian (Canham-Helfrich) and analyse it
especially concerning it's topological features.
Phase transitions can be studied when varying the
coefficients ${\kappa}, {\bar{\kappa}}$
representing respectively the bending rigidity
and the Gaussian curvature. It is found that the
Hamiltonian has no lower bounds for
${\bar{\kappa}}$ positive, indicating that
handles and tunnels will be formed in large
numbers spontaneously in the phase region of
(${\kappa}, {\bar{\kappa}} \geq 0$). For
${\bar{\kappa}} \leq 0$ apparently there are two
phases where either small vesicles are formed in
large numbers or large, single vesicles appear,
with a second order phase transition in between.\\
%\vspace{2cm}
%\noindent
\section{Introduction}
Aggregates formed by small amphiphillic molecules in water display a
remarkable structural richness, e.g. micellar, hexagonal and bilayer
structures [~\ref{ref:Luz}]. Furthermore monolayer structures can be formed
in water-air or water-hydrocarbon interfaces. Amphiphiles constitute
a very extensive class of compounds, which count common substances
like soaps, alcohols and lipids.
The lipid bilayer structures are of particular interest because they play an
essential role in the organization of biological
cells. Hydrated lipid bilayer systems display a
wealth of polymorphic transitions, however their
possible significance in biological membranes are
still unrevealed. Although the discussion in this
section is restricted to lipid bilayer structures
it may be applicable to a range of other
amphiphillic surfactant systems.
Theory has been of limited help in the
characterization at the large-scale structural
transition properties of lipid bi-layers in
excess water. The stabilisation of the lipids in
a bilayer structure is understood in the
framework of a thermodynamics theory describing
the interplay between molecular interaction free
energy, molecular geometry and entropy
[~\ref{ref:Israel}]. This analysis has been
supplemented by thermodynamics considerations
based on electrolyte double-layer theory which
can give a description of the stability of simple
bilayer shapes like cylinders and spheres.
A popular phenomenological theory for the
description of shapes of individual lipid
bi-layers is the Canham-Helfrich model
[~\ref{ref:Canham}]. This model has even proven
to be successfull in the description of shape
transformations of biological membranes like
erythocytes [~\ref{ref:Deurling}]. Furthermore this
model has served as the basis for recent studies on
the effects of thermal undulations
\\undulations?\\
on the forces between bi-layers
[~\ref{ref:Helfrisch2}] and the possibility of
order-disorder transitions in macroscopic
conformations of membranes
[~\ref{ref:Peliti},~\ref{ref:Kantor},~\ref{ref:Helfrisch3}]. However, a
full statistical mechanical treatment of this
model is not possible before the partition
function over all shapes and topologies can be
handled analytically or numerically. This is the
issue the present paper is concerned with.
In this contribution we demonstrate that it is possible, within a mean-field approach,
to obtain important information about the phase behaviour of the theory when
only closed membranes are considered. The information is concerned with the
membrane stability against changes in surface connectivity.
%\vspace{2cm}
%\noindent
\subsection{Topology in membrane phenomenology}
A range of experimental techniques have been
applied in the characterization of the phase
behaviour and morphologies of lipid-water
systems. The existence of the lamellar bilayer phases
and a number of bilayer phases with bicontinous
structures exhibiting cubic symmetries have
been complimented and confirmed with
NMR [~\ref{ref:Lindblom}] and freeze-fracture
electron microscopy [~\ref{ref:Gulik}]. In the
more diluted regimes of the phase diagram the
characterization of the phases is complicated due to the
absence of long range order in the phases and
coexistence of a large number of bilayer
structures. Direct visualization by microscope
tecniques probably gives the best insight in the
nature of the phases in this regime
[~\ref{ref:Miller},~\ref{ref:Servus}].
A considerable effort has been directed toward
characterization of the phases in terms of
surface geometries [~\ref{ref:Hyde}]. The
multi-lamellar and cubic phases have been
characterized in terms of infinite periodic
minimal surfaces (IPMS), e.g. the surfaces having
zero mean curvatures and separating the ambient space into periodic
subspaces. An IPMS has thus an associated point group symmetry and
characteristic dimensions of its unit cell. The topology of an IPMS
can be very complicated, e.g. represented by the number of genus per
unit cell. Properties of minimal surfaces can be derived from complex
analysis through their representation by Weierstrass-polynomials. However,
IPMS is still not a fully determined group of surfaces, and the
non-periodic minimal surfaces with non-trivial topology has only recently
been explored. It is thus evident that surfaces are difficult to treat
in a statistical mechanical frame if they are
assumed to be minimal. Another difficulty in dealing with minimal surfaces in
membrane physics is that a physical principle, which dictates the
crystalline properties of the IPMS, is not known. No packing condition
or internal symmetri property of the constituents can guide us, as
in the case of molecular crystals. With these difficulties we find that
minimal surfaces at present do not provide a good starting point for the
description of membranes undergoing phase transitions involving topology.
We shall in this section restrict ourselves to closed membranes,
which actually never can be described as minimal surfaces in $ R^3 $. \\
Our original motivation for studying the differential geometrical
model of membranes has been to extend the previously found solutions
to simple non-trivial topological aspects, but it turned out that our
solutions would predict much more complex
topological structures than first anticipated. One might therefore
ask if such topological complexity, i.e. a large conglomerate of
handles and tubuli, has ever been observed in biological membranes.
In figure 1 examples from intra-cellular membrane structures are shown, e.g.
the Golgi apparatus, the Endoplasmic Reticulum, etc., where complicated
topological structures of tunnels and pipes are interwoven in each
other. Such structures are difficult to observe experimentally but can
within a certain accuracy be derived from e.g. x-ray diffraction data.
\begin{figure}
\centering
\includegraphics[keepaspectratio=true,scale=0.75]{golgi1a.jpg}
\caption{Schematic picture of a typical part of the endoplasmatic reticulum
in animal cells[~\ref{ref:Texas}].}%\label{fig1:...}
\end{figure}
%\vspace{2cm}
%\noindent
\subsection{Model}
In this subsection the Canham-Helfrich model of
membrane elasticity will be briefly described.
This model considers only fluid membranes which
at length-scales much larger than the molecular
distances and the bilayer thickness can be
modeled as a mathematical surface without any
internal structure. The lipid bi-layers exhibit a
number of low-temperature solid-like phases with
in-plane order of the lipid molecules, but they
do no demonstrate the deluge of large-scale
structural transitions displayed by the fluid
membranes. The model is based on the following
Hamiltonian that takes the form
\begin{equation}
{\cal H}= \mu \int dA + \frac{\kappa}{2} \int dA (\frac{1}{r_1} + \frac{1}{r_2}
-\frac{2}{r_0 })^2
+ \bar{\kappa} \int dA \frac{1}{r_1
r_2} \quad ,
\end{equation}
where the integrations are performed over the
surface area, $dA=dx_1dx_2$, $x_1,x_2$ being the
embedded coordinates of the membrane surface.
Here $r_1$ and $r_2$ are the local principal
curvatures of the surface and $r_0$ is the
spontaneous curvature, which can arise in
bi-layers with an intrinsic asymmetry between the
mono-layers of the bilayer. In this work we
consider full symmetry between the two bilayer
halves, which is the case when the bilayer is
composed of a single molecular constituent. The
notion of spontaneous curvature will thus be
omitted in the following. The mean curvature
$\frac{1}{r_1} + \frac{1}{r_2}$ and the gaussian
curvature $\frac{1}{r_1 r_2 }$ are surface
invariants, i.e. independent of the chosen
parametrisation of the surface. The surface
tension $\mu $, which couples to the surface
area, must be considered as a chemical potential
for the lipids in the membrane due to the fixed
cross-sectional areas of the lipids. In most
thermodynamic problems involving interfaces the
chemical potential controls the interface.
When all the lipids in the lipid-water mixture
form bilayer structures with a fixed
cross-sectional area per lipid molecule, $\mu $
is negligible since the free lipid molecules are
insoluble in water. $\mu $ may be non-negligible
in realistic cases when a lipid bilayer phase is
coexisting with a non-bilayer phase, when
different bilayer phases are coexisting or in the
case of black films. Other terms may be included
in Eq. (1) . If the membrane has boundaries a
line tension term
\begin{equation}
\mu_L \int_{\rm boundary} d \ell
\end{equation}
must be added. However the line tension $\mu_L $ is so large that
even the presence of small boundaries are suppressed for free membranes.
Boundaries can occur if the membrane can be attached to hydrophobic or hydrophillic
elements of the experimental setup. Here, we do not consider these cases and just
assume that the membranes are without boundaries. Furthermore anharmonic terms
are neglegted in Eq.(1).
The model parameters $\kappa $ and $\bar{\kappa} $ are difficult to obtain experimentally.
However, some concensus have been reached regarding the value of $\kappa $ for artificial
membranes.
%\vspace{2cm}
\subsection{The Willmore functional}
In this subsection we discuss some results from the
mathematical literature concerning the properties
of the functional which appears as the second term
in Eq.(1), the Willmore functional. The Willmore
functional is written as
\begin{equation}
W(\Sigma ) = \frac{1}{2} \int_{\Sigma } H^2 dA
\end{equation}
where $H = \frac{1}{r_1} + \frac{1}{r_2} $ is the
mean curvature and $dA$ is the area element of a
surface $\Sigma $. Here $\Sigma $ is {\em any}
compact surface in ${\rm\bf R}^3$ and we assume
that it has no boundaries and no
self-intersections. The functional $W$ is
invariant under conformal mappings of the ambient
3-space. Thus, if $\tilde{\Sigma }$ is the image
of $\Sigma $ under a M\"{o}bius transformation
(an isometry, a scaling or an inversion in a
sphere with center not in $\Sigma $), then
$W(\tilde{\Sigma })=W(\Sigma )$
[~\ref{ref:Weiner}]. Recently, considerable effort
have been directed toward a solution of the
Willmore problem for surfaces of any genus
(concerning the infimum of $W $ and the related
variational problem). A few results relevant for
our purpose will be given.
Following L. Simon [~\ref{ref:Simon}] we write
$\beta_g = \inf W(\Sigma )$, where the infimum is
taken over compact genus g surfaces without
self-intersections. The following inequality is
fundamental:
\begin{equation}
8 \pi \leq \beta_g < 16 \pi
\end{equation}
Equality holds on the left if and only if $g = 0$ and $\Sigma $ is a round
sphere [~\ref{ref:Weiner}]. The right hand side inequality was observed independently
by U. Pinkall and R. Kusner, see [~\ref{ref:Kusner}].
Simon then showed [~\ref{ref:Simon}], that
if we consider $e_g = \beta_g - 8 \pi $, then
\begin{equation}
e_g \leq \sum_{j=1}^q e_{\ell_j}
\end{equation}
for any integers $q \geq 2$ and
$\ell_1,...,\ell_q $ with $\sum_{j=1}^q \ell_j =
g $. Further he proved the {\em existence} of
$W$-minimizers in the following sense: For any
genus $g$ there exists a genus $g$ surface
$\Sigma$ with $W(\Sigma ) = \beta_g $, unless
equality holds in Eq. (5) in which case there
exists a sequence $\Sigma_k $ of genus g surfaces
and a genus $g_{\circ}$ surface $\Sigma_{\circ}$
(with $g_{\circ} \leq g$) such that $W(\Sigma_k )
\rightarrow W(\Sigma_{\circ}) =
\beta_{g_{\circ}}$ for $k \rightarrow \infty $.
Thus for a given surface the minimization of W
may cause a drop in genus number. However, it is
not known whether
equality actually can occur in Eq. (5). \\
%\vspace{2cm}
%\noindent
\subsection{Thermodynamics of surface topology}
Going back to the original Hamiltonian Eq.(1) we
have in the last chapter given bounds on the
second term, the Willmore functional. The third
term is easily evaluated by using the
Gauss-Bonnet theorem:
\begin{equation}
\int_{\Sigma} dA \frac{1}{r_1 r_2} = 2 \pi \chi
\end{equation}
When the surface is without boundaries the Euler
characteristic $\chi $ is simply related to the
genus number by $\chi = 2 \pi (2-2 g) $.
From the previous section it is clear that the
Willmore functional restricted to compact
embedded surfaces $\Sigma$ without boundaries is
related to ${\cal H}$: $W(\Sigma ) =
\frac{1}{\kappa}({\cal H}-\bar{\kappa}{\chi
}(\Sigma ))$. In particular $\inf_g {\cal
H}(\Sigma_g) = {\kappa} \inf_g W(\Sigma_g) - 4
\pi \bar{\kappa}(1-g)$, where $\inf_g W$
represents the infimum of $W(\Sigma_g)$ for all
boundary-less compact, embedded surfaces
$\Sigma_g$ with genus $g$.
The next step is to sum up the contribution of the Willmore functional from each topological genus
class. We shall be taking the extremum of the Willmore functional from each genus class. It's measure contributes to a number summed up to either almost 0 or to a finite number that could become infinite
dependent on the degeneracy of the extremum. The sum would be correspondingly 0 or infinite.
Hence, in order to obtain a meaningful result of the genus sum we shall assume each Willmore
contribution in each class to be finite or that there is a finite degeneracy of the functional.
We have reasons to believe that the Willmore functional is defined up to homotopy in finite exemplar.
The assumption of finiteness is also rendered probable by computer simulation where surfaces can
be triangularized and thus make a computer estimate of the Willmore functional possible.
We are now in the position to set up the partition function for a
single closed membrane made out of A lipids. Note, by introduction of a fixed membrane area we
break the conformal invariance explictly.
\begin{eqnarray}
Z(A) &=& \sum_{g=0}^{G_A} {\rm Tr}_g(\exp(-\beta {\cal H})) \nonumber \\
&=& \sum_{g=0}^{G_A} \exp(-4 \pi \bar{\kappa} (1-g)) {\rm Tr}_g (\exp(-\beta \frac{\kappa}{2} W))
\end{eqnarray}
The trace ${\rm Tr}_g$ represents the summation
of all physical distinct surfaces $\Sigma_g$ with
fixed area corresponding to $A$. Surfaces which
are identical apart from a reparametrisation of
the surface are not physically distinct. $G_A$
represents a cut-off in the number of genus for a
membrane of size $A$. $G_A$ exists if the
diameter and the sectional curvatures have upper
bounds. This is indeed the
case for a closed membrane due limited number of
lipids $A$ involved and material parameters
determined by size and physical properties of the
molecular constituents. A first approximation to
$Z(A)$ can be obtained by restricting ${\rm
Tr}_g$ to surfaces which realize the minimum of
$W$. This is - according the previous
considerations - the case for one surface for
each $g$. The structure of this surface is not
known for general $g$ at present, which makes it
impossible to go beyond this simple thermodymanical
level of description. We will further reduce the
number of degrees of freedom and assume that the
membrane is motionally hindered, so g is the
only available degree of freedom and the minima
of $W(\Sigma )$ are non-degenerate up to homotopy.
{\em case 1.}:
\begin{equation}
Z(A) \approx Z^{\rm SP} (A) =
\sum_{g=0}^{G_A} \exp(-4 \pi \beta \bar{\kappa} (1-g)) \exp(-\beta \frac{\kappa}{2} \inf_g (W))
\end{equation}
By use of Eq.(4) and Eq.(7)
\begin{equation}
Z_1(A) \leq Z^{\rm SP}(A) \leq Z_2(A) \quad ,
\end{equation}
where
\begin{equation}
Z_c (A)= \exp(-4 \pi \beta (2 \kappa +
\bar{\kappa})) +
\exp(-8 \pi \beta c \kappa ) \frac{1-\exp(4 \pi \bar{\kappa} \beta G_A )}{1-\exp(4 \pi \beta \bar{\kappa })}
\ \ \ \ c=1,2
\end{equation}
For $G_A \rightarrow \infty$, $Z_c$ and $Z^{SP}$ are analytic for $\bar{\kappa} < 0$, and
$Z_c $ is solely controlled by the second term in Eq.(9) for $\bar{\kappa} \rightarrow 0^-$ .
From Eq.(8) it is evident that $Z^{SP} $ is also governed by a singularity of this nature as
$\bar{\kappa} \rightarrow 0^- $. $Z_c $ can thus be considered as a good approximation for $ Z^{SP} $
under these conditions. Note that $Z_1 $ in this limit is essentially the partition function for a
quantum harmonic oscillator.
However, $1 \ll G_A < \infty $ is the regime of interest in the description of the
physical system. Here $Z_c $ and $Z^{SP}$ are analytic for $\bar{\kappa} \neq 0$. The inequality
in Eq.(8) holds term by term in an expansion of $Z_c$ and $Z^{SP}$ in $\exp(4 \pi \bar {\kappa} (1-g)) $
like Eq.(7). It is then trivial to show that the inequality holds term by term in an expansion
in $\bar{\kappa }$. The expansion coefficients are thermal expectation values which will
be considered in the following.
The free energy derived from $Z_c$, $F_c =
-\beta^{-1} \ln(Z_c (A))$, and its thermal
behaviour may be analyzed. The obvious
orderparameter in this problem is the averaged
genus $$ or the average Euler-characteristics
$<\chi>$=$4 \pi (1-)$.
\begin{eqnarray}
<\chi > & = & -\frac{\partial F_c }{\partial (\beta \bar{\kappa })} \nonumber \\
& = & \beta^{-1} \frac{1}{Z_c (A)} \frac{\partial Z_c (A)}{\partial (\beta \bar{\kappa })} \\
& = & 4 \pi \beta^{-1} \frac{1}{Z(A)} (-\frac{a}{x} + b \frac{-G_A x^{G_A}
+ (G_A -1) x^{G_A +1} + x}{(1-x)^2})
\end{eqnarray}
We have here used the notation $x=\exp(4 \pi \beta \bar{\kappa})$, $a = \exp(-8 \pi \kappa \beta)$
and $b = \exp(-8 \pi \kappa \beta c)$.
The fluctuations in $< \chi >$ can be expressed as
\begin{eqnarray}
\sigma( \chi ) & = &< \chi^2 > - < \chi >^2 \nonumber \\
& = & -\frac{\partial^2 F_c }{\partial (\beta \bar{\kappa })^2} \nonumber \\
& = & \beta^{-2}( \frac{1}{Z_c (A)^2} \frac{\partial^2 Z_c (A)}{\partial^2 (\beta \bar{\kappa })}
- (\frac{1}{Z_c (A)} \frac{\partial Z_c (A)}{\partial (\beta \bar{\kappa })})^2)
\end{eqnarray}
$ <\chi >$ display a sudden, but continous change at $\bar{\kappa} = 0$ from
$<\chi > \approx -4 \pi$ corresponding to a sphere for $\bar{\kappa} < 0$ to $<\chi > \approx -4 \pi G_A$
for $\bar{\kappa} > 0$, see fig. 2. This transition is governed by strong fluctuations in
$g$, which is manifested by a peak in $\sigma (\chi ) (\approx (2 \pi G_A)^2)$ at $\bar{\kappa}=0$.
{\em Case 2.}
The simplest extension of {\em case 1} is to consider a collection of immobile closed
membranes which only interact through exchange of lipids giving rise to a vesicle size
distribution. The partition function thus takes the form:
\begin{equation}
Z = \sum_{N_A} \frac{e^{-\nu \sum_A N_A A}}{\prod_A (N_A A)!} \prod_A Z(A)^{N_A}
\end{equation}
where the sum runs over all possible vesicle size
distributions $\{ N_A \} $. The total amount of
lipid in the system is controlled by the chemical
potential $\nu $. A has a lower cut-off which is
determined by the molecular details
[\ref{ref:Israel}].
{\em Case 3.}
This case represents a collection of closed
membranes, which can exchange lipids and are free
to move in space. Only the translational degrees
of freedom will be taken into account, because
handling of the rotational degrees of freedom
require detailed information about the structures
of the Willmore surfaces for all $g$, which is
not available.
\begin{figure}[h]
\centering
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_C.pdf}
\caption{The figure shows 3 phase diagrams in the parametres $\beta \kappa$ and
$\beta \bar{\kappa}$ of the partition function for the lipid system. Fig. 1a, 1b and 1c
correspond respectively to case 1, case 2 and case 3.}%\label{fig2:...}
\end{figure}
In figure 2 we summarise these thermodynamical results by showing
the various phases of case 1, 2 and 3 in phase
diagrams with the parameters $\kappa$ and
$\bar{\kappa}$, the coefficients of the bending
rigidity and the Gaussian curvature respectively.
A phase transition line along $\bar{\kappa} = 0$
is separating a topological phase, in the upper
region of the diagram, with infinitely many
handles (high genus number $g$, case 1) and a
phase ($g = 0$) with many vesicles, case 2 and 3,
in the lower region of the phase diagram. This
region is furthermore divided by a phase
transition line, approximately along $\kappa =
-2\bar {\kappa}$, into a phase containing many
smaller vesicles and a phase with one single
large vesicle. In figure 3 we have focused on the case 2,3 and shown
the average order parameter $<-1/2\chi>$ and the fluctuation of $\chi$
as a function of $\bar{\kappa}$ and furthermore the size distribution of
vesicles at different $\bar{\kappa}$ and fixed ${\kappa}$. These distributions
are exactly what we expect of the theory in case 3.
%\vspace{2cm}
%\noindent
\subsection{Discussion and Conclusion}
In the previous section the biophysical problem on the thermodynamic
properties of
closed membranes is related to a variational problem on Willmore surfaces.
This mathematical problem is still unresolved but the results obtained so far
gives sufficient information to provide valuable results on the
phase behaviour of membranes. Three simple cases was considered:
{\em case} 1 an ensemble of independent and
motionally restricted membranes. In this case the system display
an abrupt, continous change in $$ at $\bar{\kappa} = 0$. Although
it is accompanied by strong fluctuations in $$, the transition
is neither 1. order or 2. order, but rather $\infty$ order in the sense
that $\frac{\partial^n F}{\partial (\beta \bar{\kappa} )} \propto (G_A)^n
\rightarrow \infty$ for $n \rightarrow \infty$ at $\bar{\kappa} = 0$. In
an ensemle of weakly interacting membranes this may be changed to
a 1. order or a 2. order transition.
The principles behind the interplay between three
dimensional structure and the structural
transitions of proteins and their biological
function are to a large extent understood. A
similar relationship for the biological membranes
is still considered at a hypothetical level
[~\ref{ref:Cullis}]. Whether the extended lipid
polymorphism has any significance in biological
system is still unclear.
Of great interest in this contribution are the
topological properties of membranes. It is
observed [~\ref{ref:goldstein}] that large-scale structures of
lipid membranes exhibit great topological
complexity which is realized in biological
membranes, for example in the Golgi-apparatus,
see figure 1, inside the cell. Here are lots of
handles and tubuli connecting different
compartments between layers. The purpose of this
complex topological structure is a need for
filtration of proteins in the cellular liquid. An
interesting question is then how one could
explain dynamically the formation of handles or
tunnel structure and division into small
vesicles. Such an explanation requires a theory
of topology dynamics.
\begin{figure}[h]
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_A.pdf}
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_B.pdf}
\caption[...]{A) Figure 3 shows profile of genus g, or actually $-1/2 \chi$,
along line $\kappa = 3.0$ which has discontinuity around $g = 0$ Profile is for
both $c = 1$ and $c = 2$ Left-upper figure 2a shows average order parameter $<-1/2\chi>$
as a function of $\bar{\kappa}$ while Left-lower figure 2b shows the fluctuations of $\chi$ as
function of $\bar{\kappa}$. B) Right figure (3) shows several size distributions of vesicles with
different values of $\bar{\kappa}$ and fixed $\mu \kappa$ }\label{fig3a:...}
\end{figure}
%\begin{figure}
%\caption[...]{...}\label{fig3b:...}
%\end{figure}
The topological complexity appearing in
biological membrane structures, e.g. the
intra-cellular Golgi apparatus, can largely be
explaned by the phase diagram of figures 2 and 3, where
an "explosion" of handles and tubuli occur and
only limited by material constraints such as a
finite lipid size. The phase transition around
$\kappa = -2\bar{\kappa}$, between single
vesicle structure and many small vesicles can
also be observed in experiments [~\ref{ref:Litman}]. It seems
crucial for cell growth, in an early state of
life, that the membrane structure is balancing
between the two situations as if life tends to
stay in non-equilibrium.
\section{SECTION II: The topology of 2-dimensional protein surfaces}
Previously we considered smooth, biomolecular systems at the typical size
of hundreds to thousands of nanometers. We shall now zoom down to
particular biomolecules, proteins typically of the size of five to ten nanometers,
that also have a well-defined differential geometric structure that explains aspects
of their biofunctionality and material science features.
\subsection{Construction of surfaces for proteins from their coordinates}
It is not a trivial task to construct a smooth surface from a
3-dimensional protein structure, especially if the surface
has to contain all the protein backbone. In order to see that,
one should visualize the protein as a cluster of atoms connected
to each other by an unbranched chain that winds in and out of
the center of the cluster.\\
In X-ray crystallography as well as in NMR the problem is to
construct the 3-dimensional protein structure from spectroscopic
data that provide distances between some of the atoms in the
protein. Such construction is an expansion of 2-dimensional
information in the form of distance maps to 3-dimensional
structural information which requires large processing
capacity and some extra chiral information or chain topology.
Nevertheless it has been one of the most important processing
tasks of this century since it gave science all the known
3-dimensional structures of bio-molecules, as well as other
molecules, gathered together in huge databases. There has
now been devised various short-cuts to the problem of
2d to 3d expansion [~\ref{ref:harvel},~\ref{ref:jbohr}].\\
Before discussing how to construct surfaces one should decide what
kind of surface one wants. The most obvious one is the "outer"
surface or also called the {\it solvent accessible surface} for a
protein.\\
This solvent accessible surface is most commonly
obtained by rolling a ball on the protein
structure given by coordinates thereby obtaining
a subset of the protein coordinate set that could
be in contact to a solvent with the size of the
rolling ball. This is of course all done in a
simulation by a computer. The ball size
can then be varied in order to get into grooves
and cavities or, oppositely, in order to avoid
them by making it smaller, respectively, larger.
There are many standard programs for this
procedure. In this review we have produced some
pictures with the help of the Connolly packet
[\ref{ref:connolly}]. Once the outer solvent
accessible surface has been made one can then
construct (a triangulation of) the {\it
electrostatic potential surface} or the {\it Van
der
Vaal contact surface} as shown in Fig. 4, and hence find the curvature tensors.\\
%\vspace{8.5cm}
\begin{figure}
\centering
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_G.pdf}
\caption[...]{A construction of the solvent accessible surface
for the protein, lysozyme. The darker color represents the outer
surface while the lighter color represents the inner surface. A
ligand is embedded in the surface.}%\label{fig.4}
\end{figure}
%\vspace{1cm}
\subsection{Differential geometry of the protein surface}
We start again by considering the constructed
2-dimensional protein surface embedded in a
3-dimensional Euclidean space. We shall only be
considering embedded surfaces with no
intersections. The surface $\{\bar{x}(\sigma)\}$,
which can be described by a parametrization
$\sigma(\sigma_1,\sigma_2)$, is completely
determined, up to Euclidean transformations, by
the two tensors, the metrical tensor $g_{ij}$ and
the extrinsic curvature tensor (the second
fundamental form) $K_{ij}$. The {\it induced}
metric tensor $g_{ij}$ is of course:
\begin{equation}
g_{ij} = \partial_i \bar{x} \partial_j \bar{x}
\end{equation}
\noindent where $\partial_i =
\frac{\partial}{\partial\sigma_i}$. The extrinsic
curvature tensor, $K_{ij}$ is:
\begin{equation}
K_{ij} = \bar{n} \partial_i \partial_j \bar{x}
\end{equation}
\noindent where $\bar{n}$ is the unit normal
perpendicular to the tangential surface, $\bar{n}
\cdot \partial_i \bar{x} = 0 $ for $i = 1, 2$.
$\bar{x}, g_{ij}$ and $K_{ij}$ are related
through the Gauss-Weingarten equations:
\begin{equation}
\partial_i \partial_j \bar{x} = \Gamma^{\alpha}_{ij}
\partial_{\alpha} \bar{x} + K^{\alpha}_{ij} \bar{n} \quad
\mbox{where} \quad \Gamma^{\alpha}_{ij}
g_{\alpha \kappa} = \partial_i \partial_j \bar{x}
\partial_{\kappa} \bar{x} \quad .
\end{equation}
If we want to construct a surface model with an energy function
it must be invariant with respect to changes of the coordinate
system. Therefore the terms of an eventual energy function must
contain surface invariants and they can only be constructed from the
the curvature, metrical and torsional tensors up to third order
in the derivatives. The first invariants to be included in an
expansion of the free energy of the surface are:
\begin{equation}
1, \ \ g^{ij}K_{ij} \quad \mbox{and} \quad
K^i_jK^i_j \quad ,
\end{equation}
\noindent where $K^i_i$ is the mean (exterior)
curvature and $G=K^i_iK^j_j - K^j_iK^i_j$ is the
Gaussian curvature. These quantities can also be
expressed in term of the local principal radii of
curvature $r_1$ and $r_2$:
\begin{equation}
K^i_i = (\frac{1}{r_1} + \frac{1}{r_2}), \ \ G =
\frac{1}{r_1r_2} \quad .
\end{equation}
The next invariants will contain 2-forms of higher order
derivatives.
One can now write the Hamiltonian function as in section 1 but this time for the protein
surface that contains these surface invariants:
\begin{eqnarray}
H &=& \mu \int d^{\,2}\sigma \sqrt{det(g)} + k_1
\int d^{\,2}\sigma
\sqrt{det(g)}(K^i_i)^2 + k_2 \int d^{\,2}\sigma \sqrt{det(g)}G \\
\nonumber &+& ... \mbox{higher order terms}
\quad .
\end{eqnarray}
\subsection{Minimal surfaces of proteins}
We shall in this chapter discuss the importance of minimal
surfaces for protein functionality and especially enzymatic
reaction kinetics. In the last chapter we saw how surfaces
for proteins could be constructed from the set of atomic
coordinates by triangularization. As it turns out several
important proteins have a solvent accessible surface that
resembles that of minimal surfaces and their enzymatic
functioning is very dependent on that.\\
It is not surprising that protein surfaces often appear as minimal
surfaces since, as e.g. in the case of beta-sheets, a network of
hydrogen bonds is forcing the surface to maintain constant zero mean
curvature such that a bend in one direction will cause an opposite
bend in another direction. This can be understood by applying the Poisson
equation that seek to maintain constant average density on
the protein surface.\\
%\vspace{5cm}
\begin{figure}
\centering
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_D.pdf}
\caption{A picture of the two most well-known
minimal surfaces (shown cut-out with boundary), (a) the
catenoid and (b) the helicoid. From ref. [\ref{ref:hyde}].}%\label{fig.5}
\end{figure}
%\vspace{1cm}
First, some words about minimal surfaces arising
from infinite systems. Minimal surfaces have
their mean curvature overall equal to zero:
$K_{i}^{i}=0$. In general the minimal surfaces
are infinite but they are often considered and
drawn with a boundary, such as for the catenoid
or helicoid, see figure 5. One of the most famous
minimal surfaces is the helicoid, which can be
generated as a ruled surface. The simplest
minimal surface is the plane. A minimal surface
generally has a Weierstrass representation where
the coordinates of a point in 3D Euclidean space
on the surface is given by the function ${\cal
F}$ of complex variables $\omega = \sigma +i\tau$
in the plane produced from the minimal surface by
a conformal parametrization:
\begin{eqnarray}
x &=& {\rm{Re}} \int (1 - \omega^2)
{\cal{F}}(\omega)\, d\omega ,\\ \nonumber y &=&
{\rm{Im}}
\int (1 + \omega^2) {\cal{F}}(\omega)\, d\omega ,\\
\nonumber
z &=& {\rm{Re}} \int 2\omega {\cal{F}}(\omega)\, d\omega .\\
\end{eqnarray}
\section{The surface geometry of protein reaction kinetics}
When looking at protein functionality in general, and especially for
enzymatic reactions, it is not surprising that the geometry of
the surface of the protein plays an important role. By the surface we
of course again mean the solvent accessible surface, SAS. This dependence
on the surface geometry is especially seen when the functionality
is about binding to a substrate, antibody-antigen interaction,
enzymatic hydrolysis of peptide bonds, charge transfer etc.
We shall in the following consider the particular process of
substrate (ligand) binding or ligand transferring and relate that
to the minimal surface property.\\
\subsection{Minimal surfaces and protein-ligand interaction}
Coming back to protein surfaces there are obvious
cases where the SAS of a
protein seems to resemble a minimal surface, at
least partly, and which seems to have profound
reasons for the functioning of that particular
protein. Such cases are often involving
particular topological patterns of secondary
structures. The word {\it topology} is here used
in the meaning of a graph theoretical sense where
the various secondary structure elements are
connected to each other in a particular way.
Typically these patterns consist of an
alternating, symmetric occurrence of alpha
helices and beta strands, $\alpha
/ \beta $. This is
seen in the $\alpha / \beta $ barrel proteins,
such as the triose phosphate
isomerase where the beta strands, see figure 6,
intertwine the alpha helices and form a
particular core structure with a barrel structure
and which can be described by the center portion
of the catenoid minimal surface. The open end of
the catenoid is then supposed to contain the
active site while the tunnel of the catenoid is
formed by the hydrophobic residues of the beta
strands which are tilted relative to the axis of
the tunnel. The beta sheets are straight meaning
that the normal curvature is zero and their
relative twist is equal to the geodesic torsion.
The value of the twist gives the magnitude of the
Gaussian curvature, G. The less symmetrical $\alpha
/ \beta $ protein, the redox favodoxin, has
similarly a beta sheet core that is
well described by the helicoid minimal surface. Thus, we base our analysis
of enzymatic processes on the curvature tensors K and G calculated from
triangularization of the SAS. \\
%\vspace{4cm}
\begin{figure}
\centering
\includegraphics[keepaspectratio=true,scale=0.75]{Henrik_E.pdf}
\caption{A constructed picture of the protein,
Triose Phosphate Isomerase, schematically
embedded into a catenoid surface
[\ref{ref:ander}]}.
\end{figure}
\noindent
We shall now, following the argument in the book
of Hyde et al. [\ref{ref:hyde}], see how these
minimal surfaces can be important for the
enzymatic processes the two mentioned proteins
are involved with. Most commonly the enzymatic
processes of a protein involve hydrolysis of some
sort or binding to a substrate, e.g. a particular
ligand. In the latter case it is important for
the understanding of the process and its
effectiveness that one can explain how the
diffusion of the ligand is steered through the
protein rather than being a matter of "random
encounter chemistry" which never would be able to
explain the extremely high yield of protein
catalysed reaction kinetics. The driving
mechanism in the so-called "forced diffusion" of
the catenoid modelled beta-barrel protein is the
guiding gradient of the Gaussian curvature which
attain its most negative value in the waist of
the catenoid. Thus a ligand approaching the
protein from infinity will be smoothly guided
through the catenoid tunnel of the protein
towards the waist and the active site. The
minimal surface will guarantee smooth and swift
diffusion of the ligand towards and through the
protein. Indeed, the negative Gaussian curvature
of any minimal surface locally speeds up the
Brownian diffusion process in comparison with the
corresponding diffusion on a flat surface, see
[\ref{ref:mm}]. However, a spherically cut off
part of any minimal surface also has larger area
than a similarly cut off part (disk) in the
plane. It turns out, as a consequence of
minimality, that the larger area balances
precisely the faster Brownian motion, so that the
Brownian motion mean exit times are exactly
identical on the two domains: the sperically cut
off part of the minimal surface and the disk,
respectively. The specificity of the reaction is
given by the particular combination of the
geometry and side-chain composition of the ligand and protein.
Concerning construction of minimal surfaces we can refer to [\ref{ref:mm}]
[\ref{ref:cm}].\\
%\vspace{1cm}
\subsection{Conclusion}
%\vspace{1cm}
%\subsection{Summary}
In section 2 we have analyzed constructed protein surfaces that can be
triangularized in order to provide differential geometric information.
Usually the protein folding problem is seen as a matter of going from 1 dimensional
representations to 2 dimensional (secondary) structures and then going to the full
3-dimensional (tertiary) structures. However, when going to 2-dimensional surface
structures one can by means of minimal surfaces understand enzymatic activities.\\
{\large\bf Acknowledgement}\\
We wish to thank the employees at department of
physics at DTU and the group at Memphys of SDU
for support. Especially Dr Gerhard Besold is
acknowledged.\\
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\end{document}