# Proteins: the physics of amorphous evolving matter

Jean-Pierre Eckmann 1,2, Jacques Rougemont1, Tsvi Tlusty 3,4 1 Département de
Physique Théorique,Université de Genève, CH-1211, Geneva 4, Switzerland  2
Section de Mathématiques, Université de Genève, CH-1211, Geneva 4, Switzerland
3 Center for Soft and Living Matter, Institute for Basic Science (IBS), Ulsan
44919, Korea  4 Department of Physics, Ulsan National Institute of Science and
Technology (UNIST), Ulsan 44919, Korea

###### Abstract

Proteins are a matter of dual nature. As a physical object, a protein molecule
is a folded chain of amino acids with multifarious biochemistry. But it is
also an instantiation along an evolutionary trajectory determined by the
function performed by the protein within a hierarchy of interwoven interaction
networks of the cell, the organism and the population. A physical theory of
proteins therefore needs to unify both aspects, the biophysical and the
evolutionary. Specifically, it should provide a model of how the DNA gene is
mapped into the functional phenotype of the protein.

We review several physical approaches to the protein problem, focusing on a
mechanical framework which treats proteins as evolvable condensed matter:
Mutations introduce localized perturbations in the gene, which are translated
to localized perturbations in the protein matter. A natural tool to examine
how mutations shape the phenotype are Green’s functions. They map the
evolutionary linkage among mutations in the gene (termed epistasis) to
cooperative physical interactions among the amino acids in the protein. We
discuss how the mechanistic view can be applied to examine basic questions of
protein evolution and design.

Sections marked with ∗ contain more technical material and can be omitted at
first reading.

###### Contents

  1. I The protein problem: a theoretical physics perspective
  2. II Biology as a challenge to theorists
  3. III Proteins as information machines
    1. III.1 Handling reading errors
    2. III.2 Folding
  4. IV Mechanical views on protein evolution
  5. V Condensed-matter theory of proteins
    1. V.1 Lattice models
    2. V.2 The lattice Laplacian

##  I The protein problem: a theoretical physics perspective

The macromolecules that make living matter – lipids, hydrocarbons, nucleic
acids, and in particular proteins – are among the most studied objects of
Nature. Proteins comprise the central nano-machinery of the cell, whose
numerous functions include the formation of structural elements, catalyzing
metabolic reactions and conveying biochemical signals alberts1998cell;
Fersht1999; Howard2001; Goodsell2009; Whitford2013a. For their significance in
life, proteins and the genes that encode them have been extensively
investigated using various experimental methods, such as crystallography,
biochemical assays, mass spectrometry, fluorescence imaging, electron
microscopy, directed evolution and deep sequencing Rambo2013; Cohen2001;
Collins2011; Mandala2018; Barrera2011; Mehmood2015; Ha2012; Mardis2013;
Chapman2011; Fernandez-Leiro2016. In parallel, sophisticated computational
models, such as molecular dynamics, have been developed to predict the
structure, function and folding of proteins Karplus2002; Karplus2005;
Adcock2006; Dror2018; Scheraga2018; Isralewitz2001.

These experiments and simulations provide valuable data on protein structure,
dynamics and genetics. However, there remain two inherent challenges: (i)
Sparsity of data – the protein is the outcome of long evolutionary search in a
high-dimensional space of gene sequences, which is impossible to sample, even
by high-throughput experiments. (ii) Complexity of interactions – The function
of a protein arises from collective many-body interactions in the
heterogeneous amino acid matter, which are hard to probe and model.

In light of these challenges, we focus in this colloquium on a complementary
theoretical approach that links the protein problem to the realm of condensed
matter physics. Rather than using realistic simulations predicting the
dynamics and function of concrete proteins, we shall discuss minimal models
that allow, under several simplifying assumptions, to examine basic questions
of protein evolution, especially how the collective physical interactions
within the protein direct its evolution.111The main body of this colloquium is
based on, and expands, ideas from papers Tlusty2007; Eckmann2008; Tlusty2008a;
Tlusty2010; Tlusty2016; Tlusty2017; Dutta2018.

The structure of many proteins is known at a resolution of a few angstroms and
there are detailed computational models of the forces between the amino acids.
Here, however, the protein will be examined at a coarse grained level, in the
spirit of lattice Lau1989; Shakhnovich1991 and network Chennubhotla2005
models. The protein will be described as a connected network whose nodes
represent the amino acids. Furthermore, from this conceptual point of view, it
suffices to assume that there are only two types of amino acids instead of the
usual twenty.22221, when counting the rare pyrrolysine Hao2002;
Srinivasan2002. (for example the classical HP model Lau1989 used in Sec.
LABEL:sec:pnasmodel, in which amino acids can only be either hydrophobic (H)
or polar (P)). In a real protein, forces between the amino acids are a
complicated combination depending for example on their polarity,
hydrophobicity, charge and shape. At the coarse grained level, it again
suffices to consider instead just simple springs between pairs of neighboring
sites. This is akin to using harmonic approximations in mechanics, which
provide a generic understanding and a good physical insight.

With this kind of simplifications, one can translate certain questions of
biology to analogous questions in the physics of amorphous networks. Among the
rich set of methods in this classical subject of physics, some tools seem
particularly well adapted to the protein problem. The approach is based on the
dual nature of the protein; it is a physical object whose formation and
physical interactions are also represented in the ‘dual’ gene, a sequence of
symbols from a four-letter alphabet of the DNA bases, ‘A’, ‘C’, ‘G’, ‘T’.
Evolution progresses by introducing mutations, that is, permanent
modifications of this sequence. There are local mutations (nucleotide
substitutions, short insertions and deletions) besides larger scale
modifications (e.g., translocations, inversions, duplications). A natural
approach to study protein evolution is to model the effect of mutations on the
physical properties of the amino acids network.

Local mutations amount to short jumps between neighboring sequences in the
genotype space, differing by one letter only, while large-scale mutations are
equivalent to longer jumps. Both classes of mutations can be described in
terms of alterations of the mechanical properties of the amino acid network.
However, we shall focus on the class of local mutations. Practically, local
mutations are easy to treat with classical techniques of condensed matter, for
instance via Green’s functions, since they induce localized perturbations in
the spring network. More importantly, it is possible to statistically sample
the genotype space with continuous trajectories progressing by consecutive
local mutations. This will be the main axis of this colloquium. Along the
evolutionary trajectory, mutations come in three flavors: The ones leading to
some sort of functional catastrophe or significant disadvantage, and therefore
get eliminated by selection; others which improve the properties of a protein
and finally, the large ‘neutral’ majority which do not induce any significant
change in the function of the protein Kimura1983; Neher2011. In this manner,
the ‘learning’ evolutionary process reduces the problem of improving a protein
from an exhaustive combinatorial search approach into a biased random walk.
This drastically reduces the dimension of the space which one needs to
explore.

The condensed-matter approach to the protein problem may be viewed as an
example of a potentially general framework that may be used to examine other
strongly-coupled biological systems. For example, one may analyze metabolic
and genetic networks in terms of localized perturbations and Green’s
functions. Such analysis may suggest common underlying principles. It might as
well turn out that biology is more contingent and depends on the history of
the evolutionary process, but at least the few examples we describe give us
hope that a rational approach, based on the laws of physics, may be useful in
some cases.

Biological molecules are far from being the spring networks we use as a model.
Still, similar abstractions proved successful in many areas of physics. For
example, the dynamical systems of the so-called Axiom A class Eckmann1985 are
systems with a very special, yet simple, structure. And although most systems
do not belong to the Axiom A class, it proved very useful to consider that
they behave ‘as if’.

There is a long history of studies in similar spirit of abstraction and
simplification, starting with the conformal maps of D’Arcy Thompson
Thompson1942, through the morphogenetic studies based on the theory of
catastrophes by René Thom Thom2018. In the 21st century researchers have much
more data available on biological systems. This allows to test hypotheses
against measurements, infer from the data other questions to investigate, and
suggest possible experiments to confirm or refute the theory. We close the
introduction by two citations which reflect the general outlook of this
colloquium:

Misha Gromov, in [Gromov2013, Abstract]

> When you read a textbook on molecular/cellular biology you are enchanted by
> the logical beauty of biological structures. You want to share your
> excitement with your colleagues, but…you find out you are unable to do it:
> there is no language in the 21st century mathematics that can express this
> beauty. You feel there must be a new world of mathematical structures
> shadowing what we see in Life, a new language we do not know yet, something
> in the spirit the ‘language’ of calculus we use when describing physical
> systems.

Giovanni Jona-Lasinio, in Jonalasinio2012:

> Theoretical physics was recognized as an independent field of research only
> at the end of the 19th century, shortly before the great conceptual
> revolutions of relativity and quantum mechanics. Today theoretical physics
> has multiple facets. I think that the time has come for a more precise
> characterization of the research field of theoretical biology, and for an
> assessment of its scope. [Translated from Italian]

We are convinced that such outlooks are important and our work should be
viewed as an attempt in this general direction, in the hope that readers will
be encouraged to proceed along this path.

##  II Biology as a challenge to theorists

Biological research has been extremely active in the past decades and
experimental results have flourished to vastly improve our understanding of
living matter. The challenge for theorists is to find subtopics which are at a
stage where theoretical abstraction can be fruitful.

Here we focus on the relation between genes and the functions of proteins:
genes (in DNA) code for amino acid chains that fold into the three-dimensional
configurations of functional proteins. This sequence-to-function map is hard
to decrypt since it links the collective physical interactions inside the
protein to the corresponding evolutionary forces acting on the genome
Koonin2002; Xia2004; Dill2012; Zeldovich2008; Liberles2012. Furthermore,
evolution selects the tiny fraction of functional sequences in an enormous,
high-dimensional space Povolotskaya2010; Keefe2001; Koehl2002, which implies
that proteins form non-generic, information-rich matter, outside the scope of
standard statistical methods. Therefore, although the structure and physical
forces within a protein have been extensively studied, the fundamental
question of how a functional protein originates from a linear DNA sequence
still provides research challenges, in particular how functionality constrains
the accessible DNA sequences.

To examine the geometry of the sequence-to-function map, we devise below a
mechanical model of proteins as amorphous evolving matter.333In his book “What
is Life?” schrodinger1944, Schrödinger uses the term ‘aperiodic crystal’ to
describe material which contains genetic information. This is of course a very
interesting forethought, but since the advent of quasiperiodic crystals, the
term ‘amorphous’ leads to a more precise classification. Rather than
simulating concrete proteins, we construct models which describe the hallmarks
of the genotype-to-phenotype map (the translation of the gene to the protein).
These models are sufficiently simple so that large-scale simulations can be
performed, which allow to average over stochastic noise inherent to
evolutionary dynamics. Furthermore, we restrict our approach to models in
which the function of a protein arises from large-scale conformational
changes, where big chunks of the protein move with respect to each other.
These motions are central to certain functions Koshland1958; Henzler-
Wildman2007; Savir2007; Schmeing2009; Savir2010a; Huse2002; Savir2013, For
example, allosteric proteins are a type of ‘mechanical transducers’ that
transmit regulatory signals between distant sites Perutz1970; Goodey2008;
Lockless1999; Ferreon2013.

We end this section by mentioning a few papers which have dealt with similar
issues, and which highlight the increasing interest in connecting biological
questions with methods from solid state physics.

Common to these studies is a mechanical perspective on protein function. The
motivation originates from many observations of proteins whose functions
involve collective patterns of forces and coordinated displacements of their
amino acids Daniel2003; Bustamante2004; Hammes-Schiffer2006; Boehr2006;
Karplus2002; Henzler-Wildman2007; Huse2002; Eisenmesser2005; Goodey2008;
Savir2010a. In particular, the mechanisms of allostery Monod1965; Perutz1970;
Cui2008; Daily2008; Motlagh2014; Thirumalai2018; Koshland1966, induced fit
Koshland1958, and conformational selection Grant2010 often involve global
conformational changes by hinge-like rotations, twists or shear-like sliding
of protein subdomains Gerstein1994; Mitchell2016; Mitchell2017.

A now-standard approach to examine the link between function and motion is to
model proteins as elastic networks of amino acids connected by spring-like
bonds. Early studies that apply this class of models are from the 1980s and
90s Levitt1985; Tirion1996, and in the last two decades the methods have been
further developed and applied to many proteins Chennubhotla2005; Bahar2010;
Lopez-Blanco2016. Decomposing the dynamics of the network into normal modes
revealed that low-frequency ‘soft’ modes capture functionally relevant large-
scale motion Tama2001; Bahar2010a; Haliloglu2015, especially in allosteric
proteins Ming2005; Zheng2006; Hawkins2006; Arora2007; Tehver2009; Wrabl2011;
Greener2015.

Recent work associates the soft modes of protein conformations with the
emergence of weakly connected regions as described above, but also ‘cracks’
Miyashita2003, ‘shear bands’ or ‘channels’ Mitchell2016; Mitchell2017;
Tlusty2016; Tlusty2017; Dutta2018; Rocks2019 that enable low-energy
viscoelastic motion Qu2013; Joseph2014. Such contiguous domains evolve in
models of allosteric proteins Hemery2015; Flechsig2017; Tlusty2017.

A source of inspiration for linking proteins to the physics of amorphous
matter are the papers by the late Shlomo Alexander, especially Alexander1998;
Alexander1982. In these works, Alexander highlighted the essential role of
‘floppy modes’ in the mechanical spectrum of amorphous solids. Also relevant
are studies by Thorpe and Phillips on constraint theory and rigidity
percolation in glasses, such as Thorpe1985; Phillips1985. Those works
highlighted the ability to control the rigidity and accessible zero-energy
modes of mechanical networks by balancing the number of degrees of freedom and
the number of constraints, as was noted by Maxwell in 1864 Maxwell1864.

The link between the dynamical spectra of proteins and amorphous matter has
been further explored in a recent series of works on mechanical metamaterials.
The emergence of long-range allosteric response was used in Rocks2017 as a
design principle for ‘programmable’ metamaterial made of amorphous spring
networks Rocks2018. A similar random network approach was applied in Yan2017
to design elastic materials with tailored mechanical response. These works
suggest that tunable amorphous materials have the flexibility required to
produce elaborate designs, as recently demonstrated by mimicking the cyclical
conformational motion of protein motors Flechsig2018. These promising
approaches to metamaterial design are discussed elsewhere, for example in
Ronellenfitsch2018; Kim2018; Rocklin2017; Baardink2018.

The present Colloquium focuses on a different aspect: understanding
fundamental properties of the protein evolution – in particular the genotype-
to-phenotype map – within the framework of condensed matter theory.

##  III Proteins as information machines

The building plan of a protein is determined by its corresponding gene, via
the genetic code. The gene is a 1-dimensional string in an alphabet of 4
letters: the nucleotides ‘A’ (adenine), ‘C’ (cytosine), ‘G’ (guanine), and ‘T’
(thymine) (see Alberts2017, Ch. 6). The protein is a (folded) chain of amino
acids (AA) which is translated from the gene according to the genetic code:
each three successive letters (each non-overlapping triplet, called a codon)
maps to a single AA. In principle, this would allow for $4^{3}$ possibilities,
but in general there are only 20 different AA’s, making the code redundant, as
we shall discuss in Sec. III.1.

We view the gene, i.e., the 1-dimensional string of letters as the tape of a
Turing machine Turing1936; Herken1992; Condon2018. Since any alphabet can be
recoded in binary (for example, each of the 4 nucleotides can be recoded as a
2-bits number), one can always think of it as a string of ‘0’s and ‘1’s. The
proteins (and the transcription-translation machinery, which is itself made of
proteins) would be the computer, which is able to read and interpret the
string.

This particular machine is an example of a self-reproducing Turing machine
Neumann1966, since the replication of the genome can be achieved by genome-
encoded proteins. In addition, these machines are evolving when the genes are
mutated. In other words, the machine can modify its own tape (see also
Tlusty2016). A further study in this direction is Dyson1970, but there are
many more, see e.g., Freitag2004.

###  III.1 Handling reading errors

Translation of the gene into its corresponding string of amino acids requires
a specialized machinery, which includes the ribosome Alberts2017.444In
addition to the ribosome, the machinery includes two sets of molecules, tRNAs,
which carry the amino acids, and aminoacyl-tRNA synthetases, which charge the
tRNAs with amino acids. The translation is preceded by a transcription step in
which the DNA gene (a segment of the genome) is copied into a mRNA (a single
molecule). The translation machinery ‘reads’ the code through chemical
affinity, and might therefore mis-read the tape. Most amino acids are encoded
by more than one codon, and this hard-coded redundancy of the genetic code
helps to reduce the impact of such misreadings (see Tlusty2007; Tlusty2008;
Tlusty2008a; Tlusty2008b; Tlusty2010 for a theoretical study and Eckmann2008
for an illustration).

As noted above this system allows for $64=4^{3}$ different codons (number of
triplets from an alphabet of 4 nucleotides), but they generate only 21
different symbols.555Some terminology: The individual symbols (A, C, G, T)
refer to nucleotides. The triplets of 3 nucleotides form the 64 codons. The 64
codons code for 20 AA and the stop symbol (which does not generate an AA). One
of the AA is Methionine (codon ATG) which marks the start of a protein. The
geometric aspects of this arrangement of 21 among 64 possibilities can be
understood in graph-theoretical terms: One presents the 64 codons as the nodes
of a codon-graph, and two nodes are connected by a link if the corresponding
codons differ in only one symbol. Note that swapping ‘C’ and ‘T’ in the
codon’s third position always results in the same AA (Fig. 1) and we can
therefore reduce the graph to $48=4^{2}\cdot 3$ nodes.666This graph is
difficult to draw, as each node has $8=3+3+2$ neighbors which differ in
exactly one position. So a representation would have to be in 8 dimensions.
Recall that in a cube in 3-dimensions, every corner has 3 neighbors. In the
codon-graph, each amino acid is coded as a simply connected region, as shown
in Fig. 1, with the exception of Serine (ser) (Arginin (arg) is disconnected
in the 2D table, but not in the graph). Such an arrangement minimizes the
ratio of surface by area for each region. This reduces the probability of
coding the wrong AA, under the assumption that most reading errors involve
only one-letter differences.

Figure 1: A representation of the genetic code, as function of the measured
polarity of each codon (values from Haig1991; Woese1965). The smoothness of
the landscape shows that moving from one AA to its neighbor does not change
the polarity too abruptly.

Additionally, amino acids with similar chemical properties (for example
polarity) tend to be neighbors in this graph. This can be visualized by
plotting the measured polarity as a function of the codon, which produces a
relatively smooth landscape. The smoothness manifests the chemical similarity
between neighboring amino acids, and implies that most misreadings change the
polarity of AA only moderately. We note that, unlike the 2D landscape of Fig.
1, an ideal representation should wrap the surface so that each AA would have
8 neighbors (and can therefore be embedded only in high dimension).

For the connection between the numbers 21 and 48, an inequality can be given
in terms of the genus of the codon-graph Tlusty2007; Tlusty2007a; Tlusty2010
(this uses results from Colin1993; Banchoff1965). Without going into further
detail we conclude: _The optimal code must balance contradicting needs for
tolerance to errors (with the smoothness of the mapping between codons and
chemical space) and chemical diversity, which is essential for the versatility
of protein function_.

###  III.2 Folding

Having translated the gene into a linear chain of amino acids (the backbone,
see Fig. 2) via the genetic code (and modulo translation errors), this chain
will spontaneously fold into a 3-dimensional shape which gives rise to its
function. How this folding proceeds is an important and difficult question,
which we shall not address here. Instead we will assume that a certain folding
pattern is preserved (see Petsko2004 for a discussion of these issues). This
assumption is practical, as we shall be mostly interested in how the function
of the protein changes under point mutations of the gene, i.e., bit flips of
the code in the tape. Such mutations often do not seriously affect the overall
shape of the protein (see also Bussemaker1997).

Figure 2: Schematic illustration of protein folding: Proteins are
polypeptides, linear heteropolymers of AAs (colored spheres), linked by
covalent peptide bonds (red sticks), which form the protein backbone. These
peptide bonds are much stronger than the non-covalent interactions among the
AAs (side chains) and do not change when the protein mutates.

We can next model the function of this folded amino acid chain and we will
show that there is yet another level of redundancy besides the redundancy of
the genetic code and the robustness of the folding. we shall see that there
are many mutations which have no effect on performance. _Namely, there is high
redundancy in the AA sequences that are mapped to the same or similar enough
protein function_. we shall quantify this property in terms of dimension
Grassberger1983; Eckmann1992.

##  IV Mechanical views on protein evolution

Consider a protein interacting with a small molecule. Presence of the latter
often induces a conformational change at some distance from the interaction
site. One important example is the class of allosteric proteins for which an
active site is regulated by binding at another site, resulting in a
reconfiguration of the active site. More specifically, we shall examine the
role of large-scale, functionally-relevant dynamical modes, and their link to
long-range genetic correlations.

Before reviewing the literature on this issue, we illustrate such a mechanical
effect on a particular example: human glucokinase (which is involved in sugar
metabolism), see Fig. 3. The data were obtained from crystallographic
structure of two conformations of that protein: the first (PDB777PDB = protein
data bank, https://www.rcsb.org. accession 1v4s) corresponds to the binding of
glucose to its active site and is compared to the conformation in the absence
of glucose (PDB 1v4t) [Kamata2004].

The backbone, see Sec. LABEL:sec:backbone, is shown as a light blue curled
tube, and the arrows indicate the displacement from one shape to the other
(any Galilean motion between the two is eliminated). The color of the arrows
indicates up/down motion relative to a horizontal plane. The red coloring in
the twisted tube shows the high shear region separating two low-shear domains
that move as rigid bodies (shear calculated by the method of Mitchell2016;
Rougemont2099).

On a conceptual level, one can simplify the figure as shown in Fig. 4. The
protein seems to have a central shear band and two external flaps which
perform a rotating motion when a ligand attaches to the protein. This kind of
mechanical phenomenology is accessible to the language of physics.

Large-scale motions take part in several basic biological functions and
mechanisms. For example, in the induced fit Koshland1958 and conformational
selection Bahar2007; Grant2010 mechanisms, the presence of a substrate induces
reshaping of the enzyme to properly align the catalytic groups in the active
site. Such reshaping is a dynamic mechanism of specific recognition that
allows the selection of a target ligand among similar competing molecules
Savir2007; Savir2013. In allostery, reconfiguration of the active site is
regulated by binding at a secondary, allosteric site, often via long-range
mechanical interactions Motlagh2014; Thirumalai2018. In this Colloquium, we
describe simple physical models for the emergence of these mechanisms via
evolutionary tuning of the protein’s mechanical response.

Figure 3: The motion and deformation between two states of glucokinase in
state 1v4s and 1v4t. The arrows are scaled up for better visibility. They are
colored green, resp. red depending on whether they move down or up relative to
a plane passing horizontally through the center of the protein. Galilean
motions have been eliminated. The red coloring of the tube corresponds to
concentration of shear and is the same as in the leftmost panel of Fig. 5. See
Sec. LABEL:sec:shear. Figure 4: A schematic interpretation of Fig. 3. Emphasis
is given to the two moving pieces, with a hinge between them. This kind of
hinge will be called the ‘fluid channel’ or ‘shear band’.

Like their dynamic phenotypes, proteins’ genotypes (their gene sequences), as
explained in Sec. III, are remarkably collective. The history of protein
evolution can be traced by gathering evolutionary related proteins in
different species (homologous proteins) and aligning their sequences. Genes of
these proteins sometimes display long-range correlations Goebel1994;
Marks2011; Jones2012; Lockless1999; Suel2003; Hopf2017; Poelwijk2017;
Halabi2009; Tesileanu2015; Juan2013. The correlations indicate epistasis, the
compensatory mutations that take place among residues linked by physical
forces or common function. As an example Rougemont2099, consider again
glucokinase. We aligned about 120 variants of this molecule and asked where
along the gene have mutations preferentially occurred (Fig. 5).

Figure 5: Evolutionary and mechanical properties of glucokinase. Left: Shear
(darker is more shear), discussed in Sec. LABEL:sec:shear. Center:
Conservation (darker = fewer mutations), discussed in Sec.
LABEL:sec:conservation. Right: Correlation (dark red = mutations correlate),
discussed in Sec. LABEL:sec:epistasis.

Still, the relationship between sequence correlation, epistasis and selection
pressure are not fully understood. As discussed in Sec. I, the two main
challenges are the intricacy of the physical forces among the amino acids, and
the high dimensionality of the of the genotype-to-phenotype map Koonin2002;
Povolotskaya2010; Liberles2012. These inherent difficulties motivated the
development of complementary approaches which utilized simplified coarse-
grained models, such as lattice proteins Lau1989; Shakhnovich1991 or elastic
networks Chennubhotla2005. Network and lattice models have been recently used
to study the evolution of allostery in proteins and in biologically-inspired
allosteric matter Tlusty2016; Tlusty2017; Hemery2015; Flechsig2017; Rocks2017;
Yan2017. Our aim here is different: to construct a simplified condensed-matter
model in terms of how the mechanical interactions within the protein shape its
evolution.

##  V Condensed-matter theory of proteins

This section will review a theory of proteins in terms of evolvable condensed
matter. we shall discuss the conceptual roots of this approach in the physics
of amorphous matter (mainly glasses) and spectral theory. We will introduce
the basic setting of modeling proteins as evolving amino acid networks. The
emergence of function is associated with the evolution of a weakly connected
region, which enables a low-energy ‘floppy’ mode to appear. This minimal
network approach allows one to examine basic questions of protein evolution.

we shall discuss two different models in this review. One will be called the
‘cylinder-model’ and the other the ‘HP-model’. The first model is simpler, but
the second comes somewhat closer the biological reality. Before distinguishing
the two models, we describe their common features.

###  V.1 Lattice models

Our protein is modeled by a finite (regular) lattice in 2 (or 3) dimensions.
We assume that the lattice forms a cylinder (periodic boundary conditions) or
an open rectangle (open boundary conditions) of width $w$ and height $h$ (see
the examples in Sec. LABEL:sec:cylinder–LABEL:sec:pnasmodel).

It is important to note that $w$ and $h$ are _finite_ while otherwise quite
arbitrary. This is so because the protein should not be viewed as a problem of
thermodynamic limits, but rather in the context of small amorphous objects.
This being said, other aspects of the geometry seem less important. The points
may also be chosen as lying on small perturbations from the regular lattice to
avoid effects of lattice symmetry on the spectrum. The number of AAs should
typically be in the range 200–2000, corresponding to the typical size of the
protein.

Amino acids interact via electrostatic forces, van der Waals forces, hydrogen
bonds, disulfide bonds and hydrophobicity Petsko2004; Fersht1999. All these
are short range interactions, which amount to _local_ coupling between lattice
points. we shall therefore assume that each AA interacts with its nearest and
next nearest neighbors. For example on a hexagonal lattice, with nearest and
next-nearest neighbors linked, the number of connections (the node’s degree)
is at most 12; all nodes in the protein interior have 12 links (i.e., bonds)
while those at the boundary have fewer (but at least 3), see Fig. LABEL:fig:1.

Finally, the coupling itself is modeled by harmonic springs carried by each
graph link Born1954; Alexander1998; Chennubhotla2005; Tirion1996. Its strength
is determined only by the types of AAs at each end of the link.

###  V.2 The lattice Laplacian

The lattice and its links may be viewed as an abstract graph. This means that
one can define gradients and Laplacians Biggs1993; Chung1997.888The first book
is more combinatorial, and the second introduces more spectral concepts. In
the graph, there are $n_{a}=w\times h$ amino acid nodes, indexed by Roman
letters, and $n_{b}$ bonds, indexed by Greek letters.999If a bond $\alpha$
connects nodes $i$ and $j$ ($i\neq j)$, we also write $\alpha=(i,j)$.

First, one endows every bond in the graph with an arbitrary but fixed
orientation, and then the incidence matrix of a graph is the $n_{b}\times
n_{a}$ matrix defined by

$\nabla_{\alpha i}=\left\\{\begin{array}[]{@{}rl@{}}-1~{},&\text{if $i$ is the
initial vertex of edge $\alpha$}~{},\\\ 1~{},&\text{if $i$ is the final vertex
of edge $\alpha$}~{},\\\ 0~{},&\text{if $i$ is not in $\alpha$}~{}.\\\
\end{array}\right.$

Remark that for any function $f$ on the vertices, the map $f\mapsto\nabla f$
is the co-boundary mapping of the graph, namely

$\nabla f(\alpha)=f(j)-f(i)~{},$

where $\alpha$ is the link connecting $i$ to $j$.

As in the continuum case, the Laplace operator $\Delta$ is the product
$\Delta=\nabla^{\textrm{T}}\nabla$, where ${}^{\textrm{T}}$ denotes the
adjoint. The non-diagonal elements $\Delta_{ij}$ are $-1$ if $i$ and $j$ are
connected and $0$ otherwise. The diagonal part of $\Delta$ is the degree
$\Delta_{ii}=z_{i}$, i.e., the number of nodes connected to $i$. Note that
this is a discrete graph Laplacian, and no coordinates are involved so far.

We next embed the graph in a Euclidean space $\mathbb{R}^{d}$ ($d=2$), by
assigning positions $r_{i}\in\mathbb{R}^{d}$ to each AA, i.e., to each lattice
point $i=1,\dots,n_{a}$. This is coded as a $n_{a}\times d$ real matrix
${\mathbf{r}}$. Finally, to each bond $\alpha$ we assign a spring with
constant $k_{\alpha}$ which we view as the diagonal elements of an
$n_{b}{\times}n_{b}$ matrix ${\mathbf{K}}$:

${\mathbf{K}}_{\alpha\beta}\,=\,k_{\alpha}\delta_{\alpha\beta}~{}.$ (1)

This defines a deformable spring network which has an internal energy, an
equilibrium configuration.

To account for the energy cost of deformations in the lattice protein, one
considers the elastic tensor $\mathbf{H}$ (or Hamiltonian) which we now
describe in detail [Chung1992, pp. 618–619]. The quantity $\mathbf{H}$ is a
tensor because the deformations are not scalars, but vectors in
$\mathbb{R}^{d}$. We first denote by ${\mathbf{n}}_{\alpha}$ the (normalized)
direction vectors for each bond $\alpha=(i,j)$:
${\mathbf{n}}_{\alpha}=\left({\mathbf{r}}_{i}-{\mathbf{r}}_{j}\right)/\left|{\mathbf{r}}_{i}-{\mathbf{r}}_{j}\right|$.
Then, we define the ‘embedded’ gradient tensor ${\mathbf{D}}$ (of size
$n_{b}\times n_{a}\times d$) which is obtained by multiplying each element of
the graph gradient $\nabla$ by the corresponding vector ${\mathbf{n}}$:

${\mathbf{D}}_{\alpha i}\,=\,{\mathbf{n}}_{\alpha}^{\textrm{T}}\nabla_{\alpha
i}~{},{\immediate\hbox{{(}2)}}q:mba{=}e1{2}$