# Thermodynamics of biological processes.

**ABSTRACT** There is a long and rich tradition of using ideas from both equilibrium thermodynamics and its microscopic partner theory of equilibrium statistical mechanics. In this chapter, we provide some background on the origins of the seemingly unreasonable effectiveness of ideas from both thermodynamics and statistical mechanics in biology. After making a description of these foundational issues, we turn to a series of case studies primarily focused on binding that are intended to illustrate the broad biological reach of equilibrium thinking in biology. These case studies include ligand-gated ion channels, thermodynamic models of transcription, and recent applications to the problem of bacterial chemotaxis. As part of the description of these case studies, we explore a number of different uses of the famed Monod-Wyman-Changeux (MWC) model as a generic tool for providing a mathematical characterization of two-state systems. These case studies should provide a template for tailoring equilibrium ideas to other problems of biological interest.

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**ABSTRACT:**The antibodies produced initially in response to most antigens are high molecular weight (MW) immunoglobulins (IgM) with low affinity for the antigen, while the antibodies produced later are lower MW classes (e.g., IgG and IgA) with, on average, orders of magnitude higher affinity for that antigen. These changes, often termed affinity maturation, take place largely in small B-cell clusters (germinal center; GC) in lymphoid tissues in which proliferating antigen-stimulated B cells express the highly mutagenic cytidine deaminase that mediates immunoglobulin class-switching and sequence diversification of the immunoglobulin variable domains of antigen-binding receptors on B cells (BCR). Of the large library of BCR-mutated B cells thus rapidly generated, a small minority with affinity-enhancing mutations are selected to survive and differentiate into long-lived antibody-secreting plasma cells and memory B cells. BCRs are also endocytic receptors; they internalize and cleave BCR-bound antigen, yielding peptide-MHC complexes that are recognized by follicular helper T cells. Imperfect correlation between BCR affinity for antigen and cognate T-cell engagement may account for the increasing affinity heterogeneity that accompanies the increasing average affinity of antibodies. Conservation of mechanisms underlying mutation and selection of high-affinity antibodies over the ≈200 million years of evolution separating bird and mammal lineages points to the crucial role of antibody affinity enhancement in adaptive immunity. Cancer Immunol Res; 2(5); 381-92. ©2014 AACR.Cancer immunology research. 05/2014; 2(5):381-92. -
##### Article: Foundations for modeling the dynamics of gene regulatory networks: a multilevel-perspective review.

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**ABSTRACT:**A promising alternative for unraveling the principles under which the dynamic interactions among genes lead to cellular phenotypes relies on mathematical and computational models at different levels of abstraction, from the molecular level of protein-DNA interactions to the system level of functional relationships among genes. This review article presents, under a bottom-up perspective, a hierarchy of approaches to modeling gene regulatory network dynamics, from microscopic descriptions at the single-molecule level in the spatial context of an individual cell to macroscopic models providing phenomenological descriptions at the population-average level. The reviewed modeling approaches include Molecular Dynamics, Particle-Based Brownian Dynamics, the Master Equation approach, Ordinary Differential Equations, and the Boolean logic abstraction. Each of these frameworks is motivated by a particular biological context and the nature of the insight being pursued. The setting of gene network dynamic models from such frameworks involves assumptions and mathematical artifacts often ignored by the non-specialist. This article aims at providing an entry point for biologists new to the field and computer scientists not acquainted with some recent biophysically-inspired models of gene regulation. The connections promoting intuition between different abstraction levels and the role that approximations play in the modeling process are highlighted throughout the paper.Journal of Bioinformatics and Computational Biology 02/2014; 12(1):1330003. · 0.93 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**It is becoming increasingly clear that the functionalities of an organism are mostly derived from regulation of its gene repertoire. Specialized cell types are created from pluripotent stem cells by regulating expression of genes. In eukaryotes, genes are primarily regulated by gene regulatory networks consisting of highly sequence-specific transcription factors and epigenetic modifications. The former mode of regulation is more readily reversible and non-heritable across cell generations, whereas the latter mode is less reversible and heritable. In this article, we explore the relationship between cell differentiation and the two modes of regulation of gene expression, focusing primarily on pluripotent and multipotent stem cells. Recent studies suggest that stem cells execute different gene expression programs, probably driven by one or more gene regulatory network(s). It is now also evident that as stem cells differentiate to more specialized progeny cells, rewriting of epigenetic marks occurs in parallel with the change in the pattern of gene expression. A conceptual framework is put forward in which it is proposed that the cell fate determining gene regulatory network in a pluripotent or multipotent cell has the capability to exist in multiple stationary states with each stationary state dictating a particular pattern of gene expression. We also propose that the broad pattern of gene expression in each stationary state, termed the lineage biased state or LIBS, resembles that of a more differentiated progeny cell. The differentiation process leading to a particular progeny cell involves rewriting of epigenetic marks that result in upregulation of genes in a LIBS and silencing of genes involved in alternative LIBS; thus selecting a particular pattern of gene expression and making a lineage commitment. © 2014 IUBMB Life, 2014.International Union of Biochemistry and Molecular Biology Life 02/2014; 66(2). · 2.79 Impact Factor

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Thermodynamics of Biological Processes

Hernan G. Garcia★, Jane Kondev†, Nigel Orme‡, Julie A. Theriot§, and Rob Phillips¶

★Department of Physics, California Institute of Technology, Pasadena, California, USA

†Department of Physics, Brandeis University Waltham, Massachusetts, USA

‡Garland Science Publishing, New York, USA

§Department of Biochemistry, Stanford University School of Medicine, Stanford, California, USA

¶Department of Applied Physics, California Institute of Technology, Pasadena, California, USA

Abstract

There is a long and rich tradition of using ideas from both equilibrium thermodynamics and its

microscopic partner theory of equilibrium statistical mechanics. In this chapter, we provide some

background on the origins of the seemingly unreasonable effectiveness of ideas from both

thermodynamics and statistical mechanics in biology. After making a description of these

foundational issues, we turn to a series of case studies primarily focused on binding that are

intended to illustrate the broad biological reach of equilibrium thinking in biology. These case

studies include ligand-gated ion channels, thermodynamic models of transcription, and recent

applications to the problem of bacterial chemotaxis. As part of the description of these case

studies, we explore a number of different uses of the famed Monod–Wyman–Changeux (MWC)

model as a generic tool for providing a mathematical characterization of two-state systems. These

case studies should provide a template for tailoring equilibrium ideas to other problems of

biological interest.

1. Introduction: Thermodynamics is Not Just for Dead Stuff

Thermodynamics has long been a key theory in biology, used in problems ranging from the

interpretation of binding both in vitro and in vivo to the study of the conformations of DNA

whether under the action of optical traps in well-characterized solutions or in the highly

compacted state of the cellular interior. Despite this long tradition, there is often the

sneaking suspicion that because thermodynamics (perhaps more properly referred to as

thermostatics) is a theory of equilibrium that tells us how to reckon the “terminal privileged

states” of systems (Callen, 1985), it is somehow irrelevant for thinking about the behavior of

living cells which are demonstrably not in equilibrium. While the terminal state of a living

system is death, there are many problems for which an equilibrium treatment is not only a

good starting point, but may be the most appropriate tool for the problem of interest.

In a now classic article, Eugene Wigner spoke of the “unreasonable effectiveness of

mathematics in the natural sciences,” (Wigner, 1960), expressing surprise at the truth of

Galileo’s earlier assertion that “Mathematics is the language with which God has written the

universe.” In the time since Wigner’s article, many others have taken liberties with his

theme by noting the seemingly unreasonable effectiveness of other specific ideas in a much

more general context than they were originally intended, and now it is our turn to add our

names to the list. Indeed, the unreasonable effectiveness of equilibrium ideas for inherently

out-of-equilibrium problems has already been developed by Astumian for specific cases

such as a colloidal particle falling through water and a single molecule being stretched by an

atomic force microscope (Astumian, 2007). This chapter complements that of Astumian by

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exploring the perhaps surprising effectiveness of equilibrium thermodynamics in thinking

about a wide range of biological problems.

Our chapter has several goals. First, we describe the key theoretical foundations required for

the application of equilibrium statistical mechanics models to problems spanning from

ligand-gated ion channels to the action of enhancers in transcriptional regulation. In

addition, we address conceptual issues related to the applicability of equilibrium concepts by

using arguments about separation of time scales to determine when equilibrium ideas can be

appropriately used in a living biological context, even though the cell as a whole is not in

equilibrium. With these theoretical preliminaries in hand, we carry out a series of illustrative

case studies from the last decade or so that show the broad reach of equilibrium ideas to a

number of topics that are both timely and exciting. One of our main goals is to argue that

equilibrium ideas are a good jumping-off point for thinking quantitatively about a range of

problems in cell biology. In particular, they often lead to mathematical formulae that can be

explicitly tested in biological experiments to arrive at a deeper understanding of a proposed

mechanism. These ideas will be made explicit in the examples to follow.

2. States and Weights from the Boltzmann Rule

For all of the biological examples we wish to examine, the problem formulation plays out

the same way. Our starting point is the notion of a “microstate,” one of the many distinct

ways that the microscopic objects making up our macroscopic system can be arranged. For

example, if we are interested in the disposition of a fluorescently labeled DNA molecule on

a surface, there are many different ways in which the molecule can lie down on the surface,

as shown in Fig. 2.1. Each one of these conformations is a distinct microstate but they all

share the common feature that the molecule is adsorbed on the surface (Maier and Radler,

1999). Similarly, if we have a collection of ligands in solution, both the positions and the

momenta of the different ligands can be shuffled around without changing the overall

concentration and temperature, for example. Again, each such arrangement corresponds to a

different microstate. The job of statistical mechanics is to compute the relative probabilities

of all the microstates consistent with the constraints imposed on the system. The constraints

are defined by macroscopic variables like temperature, mean distance between the ends of

the DNA, or the concentration of ligands in solution. For problems of biological interest, the

challenge is to determine what set of microstates are biologically equivalent, and then to

enumerate these micro-states and calculate their probabilities. For example, a receptor in the

presence of many molecules of ligand in solution may be considered “activated” if any one

of the individual ligand molecules is bound, although these would all be considered distinct

microstates. In practice, it would be tedious or impossible to actually enumerate the

microstates for any real system, but the toolkit of statistical mechanics provides elegant

methods to accurately estimate their numbers and probabilities, even for complex living

systems.

For thinking about processes in the living world, one relevant constraint is the assumption of

fixed temperature, which is equivalent to imposing the constraint of constant mean energy.

For some biological systems, such as endothermic animals, this approximation is almost

true, and in nearly all biological systems, the temperature changes very slowly compared to

the rapid molecular transformations that we consider here. This is one example of the

importance of the separation of time scales in the application of thermodynamics concepts to

biological systems; as long as we can treat temperature as being nearly constant, we can

vastly simplify the task of determining the probabilities of the microstates in the system. In

this case, statistical mechanics provides us with an elegant and compact formula for the

probabilities of all the microstates in the form of the celebrated Boltzmann formula, namely,

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(2.1)

where Ei is the energy of the microstate i, β = 1/kBT, kB is the Boltzmann constant, and T is

the temperature. The denominator in this expression is obtained by summing over the

Boltzmann factors (exp(−βEi)) for each of the distinct microstates and is known as the

partition function. The key intuition provided by this formula is that the probability of every

microstate of the system is solely determined by its energy. For many biological

experiments, it is easier to determine the probability of a state (e.g., the concentration of

ligand-bound receptors) than to directly measure its energy. Within this framework, the two

properties can be conveniently interconverted.

Perhaps the simplest problem of biological interest to which these ideas can be applied is

that of a “two-state” ion channel like that shown in Fig. 2.2. In such models, it is assumed

that the channel has only two states, closed and open, and the probabilities of these two

states can be read off from the fraction of the time spent in each state, as is shown in Fig.

2.2A. There are several underlying assumptions explicit in this treatment, including the idea

that the channel has no “memory” of how long it has been open or closed, and the idea that

all the channels in a population are functionally equivalent. In other words, the system is

assumed to be ergodic, such that the average open probability for a single channel examined

over time should be the same as the average fraction of channels in a population that

happens to be open at any given instant. For cases where these assumptions are reasonable

(or nearly reasonable), statistical mechanics tells us how to compute the probabilities of each

of the states from their energies, or equivalently to compute their energies from their

probabilities. In this chapter, we will repeatedly resort to the same cartoon depiction of the

Boltzmann rule by showing a cartoon of the states and their corresponding Boltzmann

weights which are obtained by exponentiating the energy of the relevant state, as shown in

Eq. (2.1), and multiplying the Boltzmann factor by its associated multiplicity. For a channel

like the one being considered here, the corresponding states and weights are shown in Fig.

2.2B. Using these ideas, we see that the probability of the open state is obtained as the ratio

(2.2)

where εopen and εclosed are the energies of the open and closed states, respectively. This

expression can be rewritten in the alternative fashion

(2.3)

where Δε = εclosed − εopen. The functional form introduced above is used widely in the

fitting of opening-probability curves (Keller et al., 1986; Perozo et al., 2002; Zhong et al.,

1998) and is shown in Fig. 2.2C. Our interest here was simply to note the way in which this

functional form arises completely naturally from the ideas of statistical mechanics.

Of course this is a deliberate oversimplification, as an ion channel that is opening and

closing must go through a continuum of multiple structural states in between. However,

inspection of the time trace in Fig. 2.2A reveals that the amount of time spent during these

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transitions is relatively brief compared to the time that the channel typically dwells in either

the open or closed states, so for purposes of estimating probabilities, we may make the

useful simplification that the system exists primarily in just these two states. Furthermore,

we acknowledge that any one state, for example, “open,” may in reality represent several or

many structurally distinct substates that are equivalent as far as their biological function is

concerned, that is, the amount of current that passes through them. One of the most useful

properties of the thermodynamic framework for the analysis of biological systems is its

flexibility with respect to the precision with which the states are defined; depending on the

exact question being asked, the investigator can choose how finely to delineate the various

states of the system. Overall, we argue that extremely simple models such as the two-state

ion channel seem to fit experimental data unreasonably well, and furthermore provide

extremely useful intuition as a starting point for thinking about highly complex systems.

There are many different kinds of ion channels, characterized not only by their selectivities

for different ions, but also by the classes of driving forces that gate them (Hille, 2001).

Regardless, from the two-state statistical mechanics perspective adopted here, the difference

in gating mechanisms from one channel to the next is embodied in the dependence of Δε on

the driving force, whether it is the voltage applied across the membrane, the concentration of

some ligands, or the tension in the membrane. This is where the power and utility of the

statistical mechanics approach becomes clear. All of the different environmental influences

that may affect the opening and closing of the ion channel may be characterized with respect

to their effect on the energy (or equivalently, on the probability) of the closed versus open

state. So, the expectations for the behavior of a channel with multiple different ligands, or a

channel affected by both ligand binding and voltage, can be described quantitatively within

this framework, using energy as a universal currency. Formally, it is straightforward to

predict quantitatively how a channel with multiple environmental influences is expected to

respond when the several different factors operate independently of one another. If the

factors such as ligand binding and voltage are not in fact independent, that will be revealed

by the failure of the data (measurement of open probability as a function of these two

variables) to fit the simple model, and the actual energy of the coupling between the factors

can then be calculated.

3. Binding Reactions and Biological Thermodynamics

3.1. Thermodynamic models of binding

One of the poster children for the usefulness of equilibrium thermodynamics and its

statistical mechanics partner ideas in biology is the study of binding reactions (Dill and

Bromberg, 2003; Hill, 1985; Klotz, 1997). To illustrate our points, we focus on several key

case studies. First, we use simple ideas about binding reactions to highlight a few key points

about transcriptional regulation. With these ideas in hand, we turn to a class of models that

have served as a centerpiece in the analysis of biological cooperativity, namely, the Monod–

Wyman–Changeux models (MWC; Monod et al., 1965) which, we will argue, serve in the

same capacity in biology that the Ising model introduced to describe the magnetic properties

of materials does in physics (Brush, 1967; Plischke and Bergersen, 2006). Both the MWC

model and the Ising model make the extremely useful simplifications that, first, the

individual elements within a complex system can exist only in a countable number of

discrete states (rather than in a continuum), and that an individual element can sometimes

change its state. For the simplest cases, such as the spins making up a magnet or ion channel

opening, the number of discrete states is just two, but as we will see below, this same

framework can be readily expanded to include more than two states.

As a biological case with very broad applicability, we start by considering binding problems

in which several different molecular species can exist either separately or in complexes. As

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shown in Fig. 2.3A, the simplest receptor–ligand binding system can exist in one of two

classes of states, or one of two macrostates. Either the receptor is unoccupied or occupied by

a ligand molecule. However, for each of these macrostates, there are many different

microscopic realizations of the system since the ligands can be distributed in many different

ways throughout the solution. For simplicity, we introduce a model of the solution known as

a “lattice model” in which the solution is divided into a huge number Ω of boxes and the

configurations of the solute molecules are captured by their placement on these lattice sites.

This idea is captured in the “multiplicity” column in Fig. 2.3A which tells us the number of

distinct ways of arranging our L ligands in the lattice model of the solution adopted here. In

reality, of course, the unbound ligands are not confined to boxes in the solution volume; they

may exist at any location. However, the lattice model provides an unreasonably effective

approximation to a continuous solution in the limit where the number of possible lattice

positions is taken to be very large, and it greatly simplifies the statistical mechanics task of

enumerating the microstates (Dill and Bromberg, 2003). To find the total statistical weight,

we simply multiply the multiplicity of the two macrostates times their associated Boltzmann

factors which depend upon their corresponding energies (i.e., upon the energy of binding

εbound and the energy of being in solution εsolution).

With the statistical weights in hand, we can now compute the probability of either of the two

macrostates as its statistical weight divided by the sum of the statistical weights of all of the

possible microstates. In particular, this leads to a formula for the probability of the receptor

to be occupied by a ligand of the form

(2.4)

where Δε = εbound − εsolution is the energy loss of the ligand upon binding to the receptor and

we have assumed that the number of ligands L is much less than the size of the solution

represented by the number of boxes in the lattice model, Ω. The factor L/Ω accounts for the

loss in translational entropy of the ligand upon binding. As written, this equation describes

the probability of receptor occupancy as a function of the number of ligands in our lattice

model of solution. This probability is plotted in Fig. 2.3B as a function of several choices of

Δε. However, to make contact with concentrations, it is convenient to rewrite this expression

by using the volume per elementary box in our lattice model (v) and occupied by ligand as a

function of ligand concentration [L]. In particular, we can write the number of ligands L as L

= [L]Ωv, in which case the equation takes on a familiar form

(2.5)

where

which the receptor has a probability of being occupied of 1/2.

is the equilibrium dissociation constant which provides the concentration at

In most interesting biological systems, the concentration of ligand will change over time

(e.g., because of changes in cellular signaling), so the system is not truly in equilibrium.

However, this is another instance where the separation of time scales is important. As long

as the rate at which the ligand concentration changes is relatively slow compared to the

individual rates of ligand binding and unbinding, the system can be considered to be nearly

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in equilibrium at each moment in time, with the probability of ligand binding simply

adjusting as its concentration changes slowly.

Often in binding problems that are biologically interesting, the simple functional form

defined above is not consistent with the data. This is usually the case when, for example,

more than one ligand may bind to the same receptor simultaneously, or when ligand binding

causes receptor dimerization. The general biochemical problems of understanding

cooperativity and allostery have historically received a great deal of attention (Cui and

Karplus, 2008). Below, we will argue that these more complex situations may also be

analyzed usefully within this same formal framework. Indeed, the classic MWC model for

allostery and cooperativity is a statistical mechanical model that considers molecules that

intrinsically exist in a distribution of possible conformational states and assigns these

different states different binding affinities (Cui and Karplus, 2008; Gunasekaran et al.,

2004). But first, with the basics of the statistical mechanics of single-ligand binding under

our belt, we are now equipped to attack a specific problem of biological interest, the

regulation of gene expression.

3.2. Thermodynamic models of transcription

Regulation is one of the great themes of biology. Few are left unimpressed after watching

the ordered cell divisions and differentiation that attend embryonic development, which

serves as a great reminder of what has been dubbed “the regulatory genome” (Davidson,

2006). The roots of regulatory biology are largely to be found in the study of prokaryotes

(Ackers et al., 1982; Jacob et al., 2005; Ptashne and Gann, 2002), and these simple single-

celled organisms continue to provide valuable insights into transcription and other processes

of the central dogma of molecular biology (Buchler et al., 2003; Michel, 2010; Wall et al.,

2004). One of our arguments is that the systems that were the early proving ground for our

understanding of regulation, namely, questions centering on bacterial metabolism and the

bacteriophage life cycle, can now be used as a test bed for a more stringent, systematic, and

quantitative attack on questions in regulation. One of the earliest systematic uses of

thermodynamic models for computing the properties of a regulatory network was carried out

by Ackers and Shea on the decision-making apparatus in bacteriophage lambda (Ackers et

al., 1982). More recently, those efforts were generalized to consider the question of how

various transcription factors by virtue of being present or absent from regulatory regions of

the DNA can conspire to yield combinatorial control of the expression of a particular gene

(Bintu et al., 2005a,b; Buchler et al., 2003). In the time since, these ideas have been used

even more aggressively for an ever-increasing set of regulatory architectures (Dodd et al.,

2005; Fakhouri et al., 2010; Giorgetti et al., 2010; Kuhlman et al., 2007).

To see the way in which these ideas play out most simply within the statistical mechanics

framework, consider the case of repression of transcription by a transcription factor

(repressor), as shown in Fig. 2.4. The idea is one of simple competition. The promoter can

either be unoccupied, occupied by RNA polymerase, or occupied by repressor, but not by

both simultaneously. The transcriptionally active state corresponds to that state in which

RNA polymerase is bound to the promoter. In the thermodynamic models, all attention is

focused on promoter occupancy, and it is assumed that the level of gene expression is

proportional to the probability of promoter occupancy by RNA polymerase (Straney and

Crothers, 1987). As with the examples worked out above for the two-state ion channel and

the simple binding problem, we can compute the probability of interest by resorting to the

states and weights diagram shown in Fig. 2.4 which tells us that the probability of promoter

occupancy is given by

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(2.6)

Here, the probability is expressed as a function of the number of poly-merases (P), the

number of repressors (R), the size of the genome NNS in base pairs, and the relevant energy

differences that characterize the binding of polymerase and repressor to promoter and

operator DNA, Δεpd and Δεrd, respectively. Details about how this formula is obtained in

analogy to the probability of the ligand binding to a receptor from Eq. (2.4) are shown in the

caption of Fig. 2.4.

From an experimental point of view, often the most convenient measurable quantity for

carrying out the kind of quantitative dissection that is possible using thermodynamic models

of gene expression is the fold-change, defined as the ratio of the level of expression in

strains that harbor the repressor molecule to the level of expression in strains that do not.

This definition can be generalized to an array of different regulatory architectures by always

computing the ratio of the level of expression in the regulated strain to that in an unregulated

strain. The prediction for the fold-change that follows from the thermodynamic model of

simple repression described above is fold-change = pbound(R)/pbound(R = 0). For repression,

the fold-change is always less than one, while for activation, the fold-change is greater than

one. As shown in Fig. 2.5, several different bacterial promoters have had their fold-change

systematically characterized, and we compare the measured value with the thermodynamic

models that are appropriate for the particular promoter. Such experiments lead to knowledge

of the parameters of the promoter architecture such as the relevant binding energies. Using

these parameters, falsifiable predictions about the gene regulatory input–output relations can

be generated (Bintu et al., 2005a).

The idea to use models based on equilibrium ideas to describe the transcriptional output of a

promoter might seem ill-conceived, given that transcription is an inherently out-of-

equilibrium process with key steps like the elongation stage of transcription leading to

mRNA production being essentially irreversible. Still, the key thing to keep in mind is what

makes equilibrium ideas useful in these settings is always the separation of time scales. For

example, even in the setting in which statistical mechanics and thermodynamics are

typically taught, that of an ideal gas, the gas is thought of as being held in a container that is

impermeable (i.e., molecules cannot escape). In reality, no such container exists! Still, if the

diffusion of the gas out of the container occurs on times scales that are much slower than the

rate at which the gas explores the volume of the container (i.e., the time for a molecule to

diffuse from one end of the container to the other), then we can consider the gas to be in

equilibrium. Similarly, if transcription factor and RNA polymerase binding and falling off

the DNA occur on time scales that are distinct from the time scales associated with initiation

of transcription, we can treat the different states representing combinations of transcription

factors bound to promoter DNA as being in equilibrium with each other. This is illustrated

in Fig. 2.6. A more intuitive way of restating this conclusion is that the rate of transcription

should depend on the concentration and activity of the transcription factors, a proposition

that is likely to be widely accepted. Here, we have simply developed the formal

underpinnings of this assertion.

3.3. The unreasonable effectiveness of MWC models

In the world of statistical mechanics, the Ising model has celebrity status and can be argued

to be one of the most useful conceptual frameworks in all of physics. One of the arguments

we want to make here is for a similar status for the MWC model in the context of biology

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(Monod et al., 1965). The biological essence of the MWC philosophy is that many of the

molecules of life, or complexes consisting of many molecules, can exist in several different

functional states (e.g., inactive and active), and their propensity to bind ligands is different

in those states. For a protein that is activated by ligand binding, the simplest picture is that

the free energy of the inactive state is intrinsically lower, making it more likely in the

absence of ligands. However, if the binding energy for ligands is greater when the molecule

is in the active state, then the presence of ligands can shift the equilibrium toward this state.

What this means in turn is that as ligands are titrated in, the active state will ultimately be

the thermodynamic winner. More generally, the same kind of enumeration of discrete states

can be applied to any other reversible biological transformation such as protein

phosphorylation and dephosphorylation, and transport into or out of a subcellular

compartment. There are many important and nuanced features of this idea, some of which

will be made mathematically explicit in the case studies to be given in the remainder of the

chapter.

3.3.1. MWC and hemoglobin: Where it all began—The MWC model in its various

forms has been applied in many different contexts. The most famous example and a story

told many times before concerns the application of these ideas to the binding of oxygen to

hemoglobin. Because hemoglobin can bind four separate oxygen molecules, there are at

least five distinct states of occupancy: empty, single-, double-, triple-, and quadruple

occupancy (we are glossing over the possible distinctions among the substates within these

states; e.g., a single hemoglobin tetramer with two bound oxygens may carry those oxygens

either on the two alpha chains, the two beta chains, or one of each). One of the most

important experimental findings about these binding probabilities is the existence of

cooperativity: one way of couching it is the idea that the Kd for adding the next ligand

depends upon how many ligands are already present. In this situation, the simple binding

curves such as those shown in Fig. 2.3B fit the experimental data very poorly. In this case,

people often resort to a richer binding curve known as a Hill function, which is a

generalization of the functional form shown in Eq. (2.5) to the case where the ratio [L]/Kd in

the numerator and the denominator is raised to the power n,

(2.7)

The parameter n is the so-called Hill coefficient and is usually associated with the degree of

cooperativity. For the hemoglobin case, the cooperativity concept was developed by Linus

Pauling in 1935 specifically as a way to explain the nontrivial shape of the observed binding

curve (Pauling, 1935). In this framework, the binding of one oxygen molecule to

hemoglobin alters its affinity for the subsequent binding of another oxygen molecule to

another site. While conceptually attractive and very useful for fitting experimental data, the

Pauling model for cooperativity and subsequent elaborations of it (Koshland et al., 1966)

require an explicit accounting for how each ligand affects the energetics of subsequent

binding events. This formulation becomes increasingly unwieldy if other kinds of

interactions are also considered. For example, the metabolic byproduct 2,3-

bisphosphoglycerate (2,3-BPG) is found at high concentrations in red blood cells and binds

to a site on the hemoglobin tetramer far from the heme groups, substantially decreasing the

affinity of hemoglobin for oxygen as part of the blood-based oxygen delivery system in

mammals (Benesch and Benesch, 1967). Incorporation of 2,3-BPG into a Pauling-style

model for hemoglobin (or, similarly, incorporation of the Bohr effect, etc.) requires a

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proliferation of coupling terms describing how the binding of each ligand affects the affinity

for every other possible ligand (Phillips et al., 2009a).

The MWC view of the cooperativity problem is fundamentally different. The original MWC

model took the approach of assuming that hemoglobin itself could exist in only two distinct

structural states: in one, the binding of oxygen to all the sites is weak, while in the other, it is

strong; there is also an energy penalty to be paid when switching from the state in which

oxygen is bound weakly to the one in which it is bound more strongly. The cooperativity in

this case arises from the fact that the penalty for binding one, two, three, or four oxygen

molecules tightly is the same regardless of the number of molecules. In other words, the

presence of one or more bound ligands simply alters the probability of the protein being in

each of the two structural states (or in the language of statistical mechanics, alters the energy

difference Δε between the two; Monod et al., 1965). Inclusion of 2,3-BPG in this framework

is straightforward; binding of 2,3-BPG also alters the population distribution between the

states, lowering the relative energy of the weak oxygen-binding state, and therefore driving

the population of hemoglobin molecules in that direction. For this first-order model, the

ligands can all be assumed to stabilize or destabilize each possible protein structural state

independently, and the effect of combining the various different ligands can be predicted by

calculating the linear combination of all of the binding energies with respect to the state

probabilities. Though the hemoglobin example was historically foundational, we believe that

the MWC framework for biological statistical mechanics can be even more usefully applied

to an unreasonably broad range of biological problems by virtue of its intrinsic ability to

describe systems that exist primarily in a countable number of discrete functional states.

3.3.2. MWC and ligand-gated ion channels: Cooperative gating—The general

applicability of the MWC philosophy is perhaps best illustrated with the example of ion

channels. This time our discussion is based on an ion channel that is gated by the binding of

ligands. Even though it is an oversimplification, we continue with the picture of ion

channels that have only two allowed conformational states, an open state which permits the

flow of ions and a closed stated which forbids any ionic current. Further, imagine an ion

channel like the nicotinic acetylcholine receptor that has two binding sites for ligands,

meaning that there are four possible states of occupancy when the channel is in a given state:

unoccupied by ligand, occupied by ligand on site 1, occupied by ligand on site 2, and

occupied by ligands on both sites 1 and 2. This is a reasonable first description of the

acetylcholine receptor involved in the neuromuscular junction, which is also one of the best-

studied ligand-gated channels, though detailed studies show that a faithful interpretation of

these channels requires more than this simplest of models provides (Colquhoun and Sivilotti,

2004). The interesting twist that results from exploiting the MWC framework is that the

binding energy for the ligands is different in the open and the closed state. All of these

eventualities are shown in Fig. 2.7A.

If we make the simplifying assumption that the binding energy for the two different sites is

identical, then the statistical weights of the different states can be written in the simple form

shown in Fig. 2.7A. The outcome of this model is that the open probability as a function of

ligand concentration has the simple but subtle form

(2.8)

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The parameters that come into play here include the energies of the open and closed states,

namely, εopen and εclosed, and the dissociation constants for the ligand when in the open and

closed states, namely, Kd(o) and Kd(c), while the concentration of the ligands themselves is

given by [L]. Note that this functional form bears some resemblance to that worked out

earlier for the simple two-state ion channel, but as a result of the fact that the concentration-

dependent terms come in a quadratic fashion, the dependence of the open probability on the

ligand concentration is sharper than revealed in our earlier model. This sharpness can be

explored by looking at the way that the open probability changes with concentration. Not

surprisingly and just as in the case of hemoglobin, more careful studies of the dynamics of

ligand-gated channels reveal behavior that is more nuanced than that captured in the

simplest MWC model (Colquhoun and Sivilotti, 2004). Nevertheless, the simple treatment

represents a very good first approximation to describing the system that can be used to build

intuition and refine the precision of the quantitative questions that can be brought to bear.

Within the same statistical mechanics framework, more sophisticated models can be

constructed by including more precisely defined structural states and including the

possibility for energetic coupling between the two ligand-binding sites (Colquhoun and

Sivilotti, 2004).

3.3.3. MWC and chemotaxis: Cooperativity in signal detection—One of the most

beloved microscopy videos in the history of modern biology was taken by David Rogers and

shows the purposeful motion of a neutrophil as it chases down a bacterium, Staphylococcus

aureus. This compelling directed motion, a few frames of which are shown in Fig. 2.8,

captures people’s imaginations because at first blush one cannot avoid a sense of amazement

that so many different processes can be so exquisitely synchronized on such short time

scales. Indeed, similar rich and complex behavior of the single-celled Paramecium led some

to wonder whether they were capable of some form of primitive thought (Greenspan, 2006).

One of the captivating features of the Rogers video is that the neutrophil “knows” which

way to go in order to track down its prey, revealing a specific example of the widespread

phenomenon of chemotaxis. Though eukaryotic chemotaxis is a field unto itself, the study of

chemotaxis in bacteria is, in many ways, the fundamental paradigm of signal transduction

and has also been fruitfully viewed through the prism of equilibrium statistical mechanics

(Berg, 2004).

The motion of a bacterium such as E. coli is characterized by “runs” and “tumbles” in which

the bacterium moves forward in a nearly straight path, reorients in the tumbling process, and

then heads off in a new direction (Berg, 2000). Bacterial chemotaxis refers to the way in

which bacteria will bias the frequency of their tumbles in the presence of a gradient of che-

moattractants (Cluzel et al., 2000). At the molecular level, this behavior is mediated by

surface-bound chemoreceptors and cytoplasmic response regulators that communicate with

the flagellar rotary apparatus (Falke et al., 1997). To illustrate how equilibrium statistical

mechanics has been used to study chemotaxis, we consider the simplified scenario shown in

Fig. 2.9. This watered-down version of the chemotaxis process centers on membrane-bound

receptors that can bind soluble chemoattractants in the surrounding medium. The receptor

communicates the presence of che-moattractants in the external milieu by modifying

response regulators within the cell through phosphorylation. More precisely, from the

standpoint of the statistical mechanics approach advocated here, the receptor can be either in

an inactive or an active state, with only the active state able to perform the posttranslational

modification of the response regulator. The balance of the active and inactive states of the

receptor is determined, in turn, by whether or not the receptor is occupied by a ligand. Just

as the balance between the open and closed states of the ligand-gated channel is altered by

the presence of a ligand, here, the kinase activity of the receptor is tuned by ligand binding.

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To compute the probability that a given receptor is activated and hence that the frequency of

tumbles is altered, we resort to precisely the same states and weights philosophy already

favored throughout the chapter. We begin with the simplest model of an isolated

chemoreceptor, as shown in Fig. 2.9. In this case, the states and weights are shown in the

figure and reflect the four eventualities that can be realized: the receptor is either inactive or

active and ligand-bound or not. When the ligand is bound, the entropy of the ligands in

solution is changed and there is an additional binding energy. This results in probability of

being active of the form

(2.9)

where we have introduced the energies εactive and εinactive to capture the energy of the

receptor in the active and inactive states, respectively, and Kd(active) and Kd(inactive) to

capture the equilibrium dissociation constant for the ligand to bind the receptor when in the

active and inactive states, respectively. The states and weights corresponding to this model

are shown in Fig. 2.10.

One of the most important outcomes of systematic quantitative experimentation on bacterial

chemotaxis is the recognition that the behavior is much more cooperative than indicated by

the simple formula derived above (Sourjik and Berg, 2002). The first level of sophistication

beyond the naïve model written above is to incorporate the idea that chemoreceptors exist in

clusters (Mello and Tu, 2005). In this case, as shown in Fig. 2.11A, the various weights

conspire to yield an expression for the probability of the active state as a function of the

concentration of chemoattractant, namely,

(2.10)

In this scenario, N individual receptor molecules within a cluster are envisioned as acting as

a unit, where the entire cluster can interconvert between the active and inactive states. This

translates into the sharpness of the transition from inactive to active shown in the plot in Fig.

2.11B. Conceptually, the cooperativity for ligand-based activation of the clusters of receptor

molecules can be treated in much the same way as the cooperativity for oxygen binding in

the MWC model for hemoglobin.

Structurally, the chemotaxis receptors can, in fact, be seen in trimeric clusters on the

bacterial surface (Briegel et al., 2009; Shimizu et al., 2000), in support of the validity of this

treatment. In fact, this picture is itself only the starting point of a much more sophisticated

set of models which acknowledge the collective action of many such receptors as the trimers

are arranged in structurally connected networks. Such models even account for the

possibility that different receptor types can interact, thus explaining the intriguing

experimental observation that the presence of a ligand for one type of chemotaxis receptor

can alter the apparent sensitivity of the bacteria to ligands for other receptor types. These

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models accomplish this without the need to postulate the existence of any unidentified

signaling pathways that would enable this kind of crosspathway communication (Keymer et

al., 2006; Mello and Tu, 2005). Yet, a further complication in the chemotaxis signaling

system is the fact that receptors can be reversibly methylated at several sites in response to

continuous stimulation, allowing adaptation over a wide range of ligand concentrations.

Within the MWC framework, these posttranslational modifications can also be incorporated

as effectively independent “ligands” that alter the probability that the receptors will be either

active or inactive, by altering the relative stability of the two states. A statistical mechanics

model based on these ideas for modifying the population distribution of simple two-state

receptors is unreasonably well-able to reproduce experimental data over a broad range of

conditions, including the prediction of system behavior for mutants where methylation is

either constitutively on or off at any of several of the possible modification sites (Keymer et

al., 2006).

3.3.4. MWC and eukaryotic transcriptional regulation: From nucleosomes to

enhancers—A less familiar example of the use of MWC-like models is to binding

problems involving DNA and its binding partners. In particular, in a recent set of papers, it

was suggested that by analogy to the inactive and active states of a protein, DNA could be

either inaccessible or accessible to binding by transcription factors (Mirny, 2010; Raveh-

Sadka et al., 2009). One concrete mechanism for how that idea might be realized in a

biological system is that the DNA could either be wrapped up in nucleosomes (inaccessible)

or open for interaction with other factors. In Fig. 2.12A, we show a schematic of the states

and weights for this case, with the “ligands” in this case now being DNA-binding proteins

such as transcription factors which bind to some enhancer. For the concrete case shown in

the figure, inspired by an enhancer in Drosophila, we consider an enhancer region

containing seven binding sites, all of which have the same affinity for the transcription

factor of interest (though this simplification is not at all crucial).

The idea embodied in the figure is once again that embodied in an MWC model. This means

that the system can exist in two overall states (accessible and inaccessible) and that the

affinity of the relevant ligands for their target sites depends upon which of the two overall

conformational states the system is in. An equivalent way of stating this is that the relative

population distribution and therefore relative stability for each of the two DNA

conformational states is influenced by the binding of the ligands. For the particular example

shown here, we were loosely inspired by the binding of the transcription factor Bicoid in its

role as an activator of a second gene known as Hunchback, two genes that play a specific

role in the much larger process of development in the Drosophila embryo (Gilbert, 2010).

For simplicity, we assume that each of the seven distinct bicoid target sites has the same

binding energy and that there is no cooperativity in the sense that the binding of one protein

does not alter the binding energy of a second molecule of the bicoid protein to one of the

other sites. As a result, the partition function can be evaluated simply in the closed form

shown here and results in the level of Hunchback activation given by

(2.11)

where [Bcd] and [Hb] are the Bicoid and Hunchback concentrations, respectively. The data

for the relationship between bicoid binding and hunchback expression has been explored in

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a recent paper (Gregor et al., 2007). Empirically, the authors of that study found that the

expression of Hunchback can be fit to a Hill function that depends upon the concentration of

Bicoid. An example of both the Hill function approach favored in that study and the MWC

functional form described here are shown in Fig. 2.12B. At this point, the quantitative

dissection of developmentally important enhancers in eukaryotes is still in the very early

stages, and there is a huge amount still to be done both in carrying out experiments that are

at once quantitative and revealing and in finding the right set of “knobs” that can be tuned in

both these experiments and the models that are developed in response. Our discussion is

meant simply to illustrate the types of questions that are currently being considered and the

way that simple thermodynamics are beginning to be used to answer those questions

(Fakhouri et al., 2010).

3.3.5. The biological reach of MWC models—Of the nearly 5700 citations at the time

of this writing of the original paper by Monod, Wyman, and Changeux (Monod et al., 1965),

many are concerned with the limits and validity of this class of models and how they can be

used to reflect on a broad class of biological problems with special interest in the fitting of

some class of data. Our intent here has mainly focused instead on what such models assume

about the molecules they describe and how to use simple ideas from equilibrium statistical

mechanics to compute the MWC expressions for binding probability.

It is important to realize that in all of the case studies set forth here, the key point is to

illustrate the style of analysis and not the claim that the particular models are the final word

on the subject in question. For example, our treatment of the ligand-gated ion channel, while

a useful starting point, has been found to miss certain detailed features of the gating

properties of these channels. Similarly, our introduction to the MWC approach for bacterial

chemotaxis has swept many of the key nuances for this problem under the rug. For example,

to really capture the detailed behavior of these systems requires positing a heterogeneous

clustering of the different types of chemoreceptors. As concerns transcriptional activation in

eukaryotic enhancers, the use of models like that presented here is in its infancy and may

end up not being the right picture at all. The key reason for promoting these models is that

they provide quantitative hypotheses about the processes of interest which can be used a

starting point for developing experiments that test them. As is often the case for the

application of simplified analytical models to biological systems, their most useful role can

be to help the investigator determine what information is missing. To a first approximation,

experimental data that are extremely well-fit by MWC models may be reasonably assumed

to operate more-or-less as discrete state systems, where the relevant separation of time

scales has rendered the equilibrium assumption of statistical mechanics to be close to

correct. In such cases, no further complexifications of the mechanism need be postulated to

explain the phenomenon at hand, at least within the limits of the available data which is well

fit by the simple model. In the more interesting and perhaps more common case where the

simplest statistical mechanics models reveal systematic differences from the data, new kinds

of experiments may suggest themselves that will account for the discrepancies and reveal

more insight into the workings of the system. Thus, a careful comparison of theory and

experiment can serve to uncover quantitative details of the mechanism, whether it be gene

regulation, ion-channel gating, or detection of chemoattractant.

4. The Unreasonable Effectiveness of Random-Walk Models

So far, our emphasis has been almost exclusively on binding problems. However, our

argument that equilibrium ideas have a broad reach in the biological setting transcends these

applications. To demonstrate that point, we close with a brief discussion of the power of

such thinking in the context of random-walk models in general and their uses for thinking

about polymer problems in particular. The random-walk model touches on topics ranging

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from evolution to economics, from materials science to biology (Rudnick and Gaspari,

2004). For our purposes, we reflect on the random-walk model in its capacity as the first

approach one is likely to try when thinking about the equilibrium disposition of polymers,

including those referred to by Crick as the two great polymer languages, namely, nucleic

acids and proteins. Though the particular case study we address here concerns proteins that

harbor tethered receptor–ligand pairs, the same underlying ideas can be applied just as well

to nucleic acids for thinking about the ubiquitous process of DNA looping in transcriptional

regulation, for example, (Garcia et al., 2007; Rippe, 2001).

There is a vast literature on the use of models from equilibrium statistical mechanics to

explore the properties of biological polymers (de Gennes, 1979; Grosberg and Khokhlov,

1997). As usual, the idea is to figure out what the collection of allowed microstates is for the

biological polymer of interest (an example for the conformations of DNA was given in Fig.

2.1). Perhaps the simplest example imagines the polymer of interest in much the same way

we would think of a chain of interlinked paper clips. In particular, we treat the polymer as a

chain of N segments, each of which has length a. We then posit that each and every

configuration has the same energy (and hence the same Boltzmann factor) and thus, the

problem of finding the probability of different configurations becomes one of counting their

degeneracies. For example, those macrostates, characterized by a particular end–end

distance which can be realized in the most different ways are the most likely. These ideas

and their generalizations have been used to consider many interesting problems (Phillips et

al., 2009b). One of the most celebrated examples that we will not elaborate on here concerns

the use of these ideas in the setting of single-molecule biophysics where it has now become

routine to manipulate individual proteins and nucleic acids. Indeed, the force-extension

properties of these biological polymers are so well described by ideas of equilibrium

polymer physics that stretching individual DNA molecules has become a way to calibrate

various single-molecule apparatus such as optical and magnetic traps.

To get a sense of how these ideas from polymer physics insinuate themselves into biological

binding problems hence building upon the earlier parts of the chapter, we consider the

simple competition between a tethered ligand–receptor pair and soluble competitor ligands.

This kind of motif exists in a number of signaling proteins and has also been the basis of

fascinating recent experiments in synthetic biology (Dueber et al., 2003). In particular, the

toy model introduced here mimics a synthetic receptor–ligand pair in which the actin

cytoskeletal regulatory protein, N-WASP, has been modified to include a single PDZ

domain, thus allowing N-WASP activity to be artificially brought under the influence of the

PDZ ligand. Furthermore, a copy of the ligand is also attached to the modified N-WASP,

with both ligand and receptor domains attached by flexible unstructured protein domains

that serve as tethers. As shown in Fig. 2.13A, there are three distinct classes of states

available to the system. In the first state, the tethered ligand and receptor are bound to each

other. In the second state, the receptor is unoccupied. In the third state, one of the soluble

ligands is bound to the receptor. The question we are interested in addressing is the relative

probability of the two different bound states and how they depend upon the concentration of

soluble ligands.

The intuitive argument is that the probability that the receptor will be occupied by a ligand is

a result of the competition between the tethered ligand and its soluble partners. As the

concentration of the soluble ligands is increased, it becomes increasingly likely that they

will form a partnership with the tethered receptor. To explore the nature of this competition,

we compute the ratio of the probabilities for the free and tethered ligands. For the purposes

of the model shown in Fig. 2.13A, we treat the tether using the simplest one-dimensional

model of a random walk since all we are trying to demonstrate is the concept, as opposed to

the quantitative details. What this means really is that we evaluate the entropic cost of loop

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formation using a one-dimensional model which makes it a simple counting exercise to

determine the fraction of conformations which close on themselves. Stated simply, if we

think of each monomer in the polymer as pointing left or right, then loop formation in this

context requires that the number of right and left-pointing monomers be the same. The key

point is that in the closed conformation, the two tethers have many fewer conformations

available to them in comparison with the case when they are no longer linked, and each side

is free to flop around on its own. The result of this competition as a function of the soluble

ligand concentration is shown in Fig. 2.13B, and is consistent with our intuition in the sense

that in the high concentration limit, the receptor is saturated by soluble ligands. The specific

concentration at which the tethered ligand and the soluble ligands have the same probability

of being bound to the receptor depends upon the looping probability. As the relative

flexibility and length of the tethers are varied experimentally, the quantitative predictions of

this simple model can be rigorously tested (Dueber et al., 2003).

5. Conclusions

Thermodynamics is unreasonably effective in the biological setting, but effective it is. As

noted by Einstein in his autobiography, “A theory is the more impressive the greater the

simplicity of its premises is, the more different kinds of things it relates, and the more

extended its area of applicability. Therefore the deep impression which classical

thermodynamics made upon me. It is the only physical theory of universal content

concerning which I am convinced that, within the framework of the applicability of its basic

concepts, it will never be overthrown.” (Schilpp, 1970).

Equilibrium thermodynamic ideas and their statistical mechanics partner concepts pervade

not only the in vitro domain of traditional biochemical binding reactions, but also permeate

our thinking for more biologically relevant in vivo examples ranging from gene regulation to

signaling networks to the physical limits on biological detection (Bialek and Setayeshgar,

2005, 2008). In this chapter, we have tried to articulate some of the fundamentals of

equilibrium models for a variety of different problems. Our analysis has focused more on the

conceptual underpinnings that on the specific and detailed ways that biological data is

greeted by these kinds of models. Here, we have described a few of our favorite examples of

the confrontation of the models and corresponding experiments, and many more can be

found elsewhere (Bintu et al., 2005a,b; Hill, 1985; Keymer et al., 2006; Klotz, 1997; Mello

and Tu, 2005; Phillips et al., 2009c). We have argued that conceptually part of the reason for

the effectiveness of equilibrium ideas in the biological setting is likely a matter of separation

of time scales, and the most unreasonably effective simplification underlying the MWC and

statistical mechanical treatment of these problems, that they exist primarily in a countable

number of interconvertible functional states rather than as a squishy continuum.

In his book “How the Mind Works,” Steven Pinker notes “The linguist Noam Chomsky

once suggested that our ignorance can be divided into problems and mysteries. When we

face a problem, we may not know its solution, but we have insight, increasing knowledge,

and an inkling of what we are looking for. When we face a mystery, however, we can only

stare in wonder and bewilderment, not knowing what an explanation would even look like. I

wrote this book because dozens of mysteries of the mind have recently been upgraded to

problems. Every idea in the book may turn out to be wrong, but that would be progress,

because our old ideas were too vapid to be wrong.” (Pinker, 2009). In our view, one of the

most important reasons for the potency of the quantitative slant which equilibrium models

are but one example of is that they are a tool for generating specific and detailed hypotheses

which are a step along the way to turning mysteries into problems and which give us an

opportunity to design experiments that can tell us whether we are wrong. The rigorous

framework of statistical mechanics provides no space for being vapid.

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