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Statistical Mechanics of Monod–Wyman–Changeux(MWC) Models
Sarah Marzen1, Hernan G. Garcia2 and Rob Phillips3,4,5
1  Department of Physics, University of California Berkeley, Berkeley, CA 947207300, USA2  Department of Physics, Princeton University, Princeton, NJ 08544, USA3  Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA4  Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA5  Laboratoire de PhysicoChimie Théorique CNRS/UMR 7083, ESPCI, 75231 Paris Cedex 05, France
Correspondence to Rob Phillips: Department of Applied Physics, California Institute of Technology, Pasadena, CA91125, USA. [email protected]://dx.doi.org/10.1016/j.jmb.2013.03.013Edited by C. Kalodimos
Abstract
The 50th anniversary of the classic Monod–Wyman–Changeux (MWC) model provides an opportunity tosurvey the broader conceptual and quantitative implications of this quintessential biophysical model. Withthe use of statistical mechanics, the mathematical implementation of the MWC concept links problems thatseem otherwise to have no ostensible biological connection including ligand–receptor binding, ligandgatedion channels, chemotaxis, chromatin structure and gene regulation. Hence, a thorough mathematicalanalysis of the MWC model can illuminate the performance limits of a number of unrelated biologicalsystems in one stroke. The goal of our review is twofold. First, we describe in detail the general physicalprinciples that are used to derive the activity of MWC molecules as a function of their regulatory ligands.Second, we illustrate the power of ideas from information theory and dynamical systems for quantifying howwell the output of MWC molecules tracks their sensory input, giving a sense of the “design” constraints facedby these receptors.
© 2013 Published by Elsevier Ltd.
Introduction
Modern biology has garnered deep insights from alarge collection of “model systems” ranging fromspecific molecules such as hemoglobin, Lac repressor and the nicotinic acetylcholine (nACh) receptorto organisms such as Escherichia coli and itsphages to the fruit fly Drosophila melanogaster andbeyond.1,2 Studies of “model” molecules such ashemoglobin have given rise, in turn, to very generalstatistical mechanical models that provide a simplelink between the structural conformation of thesemolecules and their regulation by external ligands.One such model, the Monod–Wyman–Changeux(MWC) model,3,4 is beginning to assume similarproportions in biology to those of the Ising model inphysics, which has been used to explain diversephenomena ranging from magnetism to the liquid–gas transition.5 As we describe in this article, the
00222836/$  see front matter © 2013 Published by Elsevier Ltd.
MWCmodel sheds light on a similarly broad swath ofbiological phenomena.A signature feature of any powerful model is its
ability to make convincing connections betweenapparently unrelated phenomena. A simple searchon the initials “MWC” on PubMed reveals the vastarray of different molecular situations in whichresearchers have appealed to the MWC model asan instructive conceptual framework. A similarsearch on the Web of Science in December of2012 reveals 1517 unique citations for the 1963paper of Monod, Changeux and Jacob and 6086unique citations for the 1965 paper of Monod,Wyman and Changeux.3,4 In speaking of the papersthat introduced these ideas, Monod noted “The firstpaper was, really, on the idea of indirect regulation.Which I think is the really important idea. The secondpaper is a physicalchemical interpretation of thisfact in terms of the geometry of the molecule”.6 In our
J. Mol. Biol. (2013) 425, 1433–1460
http://dx.doi.org/mailto:[email protected]://dx.doi.org/10.1016/j.jmb.2013.03.013

1434 Review: Statistical Mechanics of MWC Models
paper, we hope to explain why Monod christened theidea of indirect regulation embodied in the MWCmodel “the second secret of life”6 by mathematicallyexamining the general implications of the indirectregulation concept and its implementation in statistical mechanical language. It is interesting to notethat, of the two papers, the second paper4 is morecited than the first,3 despite Monod's claim of thegreater importance of the former.Indirect regulation, the subject of the first of this
important pair of papers and a key feature of theMWC model, arises when a macromolecule ofinterest has two classes of conformational states,which we will refer to generically as the inactive andactive states. The molecular decision of whether ornot the macromolecule is active is dictated by thebinding of some regulatory ligand or ligands that bindpreferentially to one state over the other, therebytilting the balance between the inactive and activestates.2,4,7,8 As we will see below, in the simplestvariant of the model, there are thus four distinctstates, active and inactive both with and withoutligand. This simplest picture can serve as the startingpoint for a whole suite of more complex modelsinvolving multiple binding sites and hence, cooperativity, different sets of intermediate states andmultiple ligands, for example.The organization of the paper is as follows. In
States and Weights in the MWC Setting, we showhow “statesandweights” diagrams illustrate all ofthe different microscopic states available to an MWCmolecule and how statistical mechanics can be usedto assign “statistical weights” to each such state.With these results in hand, we show how to computethe activity of an MWC molecule as a function of theconcentration of its governing ligand. In CaseStudies in MWC Thinking, we highlight a few of themany biological processes that can be described byanMWCmodel, including ligandgated ion channels,twocomponent signal transduction systems andgene regulation. Some of the unexpected predictions that arise from such models are highlighted inthese various examples. In An InformationTheoreticPerspective of the MWC Concept, we then considerhow information theory can be used to characterizethe ability of MWC molecules to “read” the state oftheir environment and to convert it into cellulardecisions. In Dynamical MWC, we introduce asimplified model of dynamics for MWC moleculesand analyze how well such molecules “read” anenvironment whose state is changing in time. InDiscussion, we close by reflecting on the MWCconcept as viewed through the prism of “toy models”in statistical mechanics and how it provides apowerful complement to more microscopically realistic perspectives that have emerged from structuralbiology. Some of the detailed steps of our derivationsare presented in several appendices that follow thereference list. We consider these appendices an
important part of the overall review since theyprovide details for results used in the literature thatare rarely presented pedagogically, if at all.The references cited throughout the paper are
intended to provide an entry into the vast literature onthe MWC concept with special emphasis on howphysical scientists have exploited the model. Specifically, we place less emphasis on fitting the datafor one particular molecule and more emphasis onthe general features of such molecules and thephysical constraints that they must face. Given themore than 7000 citations of the two papers thatlaunched the MWC world view, it is no surprise thatour list of references is representative rather thancomplete, and we apologize in advance to thosewhose important work has been neglected.
States and Weights in the MWC Setting
For the purposes of this article, we define an MWCmolecule as having two classes of structuralstates.3,4 In the case of hemoglobin, for example,these states correspond to the famed “T” (tense)and “R” (relaxed). For ligandgated channels suchas the nACh receptor or cyclic guanosine monophosphate (cGMP)gated ion channels such as thosefound in photoreceptors, these two states correspond to the closed and open states of thechannel.9 These same ideas can be used evenfarther afield such as to describe different structuralstates of chromatin in which the DNA molecule iseither “inaccessible” or “accessible”.10,11 In thissection, we will denote the two states of the receptorby “I” and “A”, referring to the inactive state and tothe active state, respectively. Our goals in thissection are twofold. First, we aim to discuss indirectregulation and cooperativity in the MWC model.Second, and perhaps more importantly, we aim toprovide a clear recipe for converting diagrammaticdescriptions of ligand–receptor binding (“states diagrams”) into testable equations. To achieve thelatter goal, we begin this section by describing thenoncooperative onesite MWC molecule. Then, wefocus our attention on the general nsite MWCmolecule from which we can understand how theindirect regulation inherent in the MWC model cangive rise to apparently cooperative interactions.An MWC molecule with a single binding site has
four possible states: the receptor can be inactive oractive, and the binding site can have or cannot havea bound ligand. Figure 1a provides a genericschematic of these four distinct molecular situationsand also lays the groundwork for a statisticalmechanical investigation of the relative probabilitiesof these different states. Each of the possiblemicroscopic states has probability proportional toits Gibbs factor, exp(−(Estate − nstate μ)/kBT), whereEstate is the energy of the microscopic state of

STATE ENERGYSTAT. MECH.
STATISTICAL WEIGHT
εI
εA
εI + εb – μ(I)
εA + εb – μ(A)
e–βεI
e–βεA
e–βεI eβμ0e–βεb(I) c
c0
THERMO.
e–βεI
e–βεA
(A)e–βεA c
Kd
10 3 10 2 10 1 100 101 10210 5
10 4
10 3
10 2
10 1
100
Pro
babi
lity
(0,0)(1,0)(0,1)(1,1)
10 3 10 2 10 1 100 101 1020
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
pac
tive
(Active,Bound)
e–βεb(I)c
c0
(a)
(b) (c)
p(0,0)p(1,0)
=eβ(εAεI)
p(0,1)p(1,1)
=eβ(εAεI)eβ(εb  εb )(A) (I)
e–βεI(I)c
Kd
eβμ0e–βεA e–βεb(A) c
c0
eβμ0
pac
tive
ec/c0–βεb
(I)
eβμ00 2 4 6 8
0
0.2
0.4
0.6
e–βεb(I)c
c0eβμ0
Fig. 1. Statesandweights diagram of the onesite MWC molecule. (a) Each of the four states has an associatedenergy, part of which is due to the conformational degrees of freedom of the molecule and part of which reflects the freeenergy of the binding process. (b) pactive as given in Eq. (2) as a function of concentration in units of the inactive state'sdissociation constant. The activity curve is shown on a log scale in the main plot and on a linear scale in the inset. (c) Thefour curves show the probabilities of each of the distinct states as a function of the ligand concentration. Each state islabeled by a pair of numbers. The first number of the pair is 1 if the receptor is active and 0 if the receptor is inactive; thesecond number of the pair is 1 if a ligand is bound and 0 if no ligand is bound. The parameter values used in the figure areΔε = εI − εA = −4 kBT and Δεb = εb(A) − εb(I) = −5 kBT.
1435Review: Statistical Mechanics of MWC Models
interest, nstate is the number of ligands bound to thereceptor in that state and μ is the chemical potentialthat here encapsulates the freeenergy cost ofmoving a ligand molecule from the solution to thereceptor.2,5 For the remainder of this article, wedefine the variable β ¼ 1
kBTas is typically done in the
statistical mechanics literature.We can decompose the energy of a state Estate
into two different contributions: the conformationalenergy of the receptor and the energy of binding aligand. We label the conformational energy of theactive receptor as εA, the conformation energy of theinactive receptor as εI, the energy of binding a ligandto the active receptor as εb
(A) and the energy ofbinding a ligand to the inactive receptor as εb
(I).Calculating the conformational energy of the receptor or the energy of binding a ligand from first
principles could be incredibly complicated, as itdepends on the details of the bonding interactionswithin the receptor, between the receptor and thesurrounding solution and between the receptor andligands. The freeenergy cost of removing a ligandfrom a dilute solution is encapsulated in the chemicalpotential μ,
μ ¼ μ0 þ kBT ln cc0 ð1Þ
where μ0 and c0 are an unspecified referencechemical potential and its corresponding (unspecified) reference ligand concentration. Using theprescription that each state has a probabilityproportional to exp(−β(Estate − nstate μ)), we find“weights” shown in Fig. 1a for each state. Fromthese weights, we can derive the probability of the

pactive
pactivemin
pactivemax
c*
G0(c)
Quantity
Activity curve and key conceptual parameters
Description Formula (Thermo.)
probability of the“active” state
average number ofbound ligands
minimum probability ofbeing in the active state
(assuming Kd < Kd )
maximum probability ofbeing in the active state
ligand concentration at thetransition point
(pactive(c*)=(pactive+pactive)/2)min max
min
∂log(
∂log c
)pactive–pactive
minmaxpactive–pactive
c=c*
static gain ∂pactive
∂c
( )( )G0(c)=
(A) (I)
(assuming Kd < Kd )(A) (I)
Formula (Stat. Mech.)
(a) (b)
(c)
pactivemax
pactivemin
10 100 10210 2
10 1
100
pac
tive
One binding siteTwo binding sites
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
pac
tive
One binding siteTwo binding sites
c*
Fig. 2 (legend on next page)
1436Review
:StatisticalM
echanicsof
MWC
Models

1437Review: Statistical Mechanics of MWC Models
onesite receptor being active, which can becalculated as the sum of the weights of the activereceptor normalized by the total sum of all of theweights, namely,
pactive
¼e−βεA 1þ c
c0e−β ε
Að Þb −μ0ð Þ
� �
e−βεA 1þ cc0e−β ε
Að Þb −μ0ð Þ
� �þ e−βεI 1þ c
c0e−β ε
Ið Þb −μ0ð Þ
� �
ð2ÞFigure 1b shows that the probability of the active
state increases as ligand concentration increases;this curve is often called the “activity curve”.We can also derive the “binding curve” as the
average number of bound ligands, which for the onesite receptor is the sum of the weights of thereceptors with one ligand normalized by the totalsum of all of the weights,
Nboundh i
¼e−βεA c
c0e−β ε
Að Þb −μ0ð Þ
� �þ e−βεI c
c0e−β ε
Ið Þb −μ0ð Þ
� �
e−βεA 1þ cc0e−β ε
Að Þb −μ0ð Þ
� �þe−βεI 1þ c
c0e−β ε
Ið Þb −μ0ð Þ
� �
ð3ÞWhen the active state has a larger affinity for the
ligand than the inactive state, εb(A) b εb
(I), both pactiveand 〈Nbound〉 increase with the ligand concentrationc. These are consequences of the fact that anincrease in ligand concentration increases thestatistical weight of the bound, active state relativeto the weight of the unbound, inactive state.Figure 1c plots the probabilities of each of thesestates as a function of concentration. As expected,the inactive, unbound state dominates at low ligandconcentrations and the active, bound state dominates at high concentrations.Some readers may find Eqs. (2) and (3) unfamiliar
since activity curves and binding curves are oftenwritten in terms of a different set of parameters.Thermodynamic language is often used insteadof statistical mechanical language, employingdissociation constants†‡ K Að Þd ¼ c0eβ ε
Að Þb −μ0ð Þ and
K Ið Þd ¼ c0eβ εIð Þb −μ0ð Þ and the conformational equilibrium
Fig. 2. Table of key quantities that can be computed within tfor two MWC molecules: a onesite receptor with Δε = εI −difference in binding energy of ε Að Þb −ε
Ið Þb ¼ logK
Að Þd
K Ið Þd¼ −4kBT ; an
1 μM, and Kd(I) = 12.2 μM, giving a binding energy difference o
with concentrations on a log scale. The transition point concentshown with vertical and horizontal lines, respectively, for the onkey parameters of interest in both statistical mechanics andbinding sites on the receptor, L = e− βΔε is the conformationalconformational energy between the inactive and active state,Kfor ligand binding, K Að Þd ¼ c0eβ ε
Að Þb −μ0ð Þ is the active state's dis
concentration.
constant L ¼ e−β εA−εIð Þ.2 Note that, as commentedon previously, only energy differences are meaningful. With this parameterization, the activity curvefor the onesite MWC molecule has the form
pactive ¼1þ c
K Að Þd
1þ cK Að Þd
� �þ L 1þ c
K Ið Þd
� � ð4Þ
and the binding curve has the form
Nboundh i ¼c
K Að Þdþ L c
K Ið Þd
1þ cK Að Þd
� �þ L 1þ c
K Ið Þd
� � ð5Þ
The thermodynamic formulation is directly relatedto the original MWC parameters§. The activity curveand binding curve given in Eqs. (4) and (5) can inturn be fit to the MWC equation for pactive to find themicroscopic parameters of the MWC model, forexample, Kd
(A), Kd(I), L. For the rest of this paper, we
will use a combination of statistical mechanics andthermodynamic notation: dissociation constants Kd
(A)
and Kd(I) will be used in preference of c0e
−β ε Að Þb −μ0ð Þand c0e
−β ε Ið Þb −μ0ð Þ, respectively, and conformationalenergies εA and εI will be used in preference of theconformational energy equilibrium constant L. Ofcourse, this choice is a matter of personal taste; wefind that the notation favored here combines thebrevity of the original MWC notation4 with the clarityof biophysical understanding provided by statisticalmechanics notation.The concept of indirect regulation is already
present in a onesite MWC molecule. Typically, theinactive receptor is more energetically favorablethan the active receptor, εA N εI or L N 1. Thepresence of ligand shifts the balance toward theactive receptor because the binding of ligand to theactive receptor is more favorable than the binding ofligand to the inactive receptor, that is, εb
(A) b εb(I),
Kd(A) b Kd
(I). However, studying the onesite receptorcannot elucidate the phenomenon of cooperativity,
he MWC framework. (a) The activity curve on a linear scaleεA = −4 kBT, Kd(A) = 1 μM and Kd(I) = 148 μM, giving ad a twosite receptor with Δε = εI − εA = −4 kBT, Kd(A) =f ε Að Þb −ε
Ið Þb ¼ logK
Að Þd
K Ið Þd¼ −2kBT . (b) The activity curves from (a)
ration c* = 40.6 μM and effective Hill coefficient heff = 1 areesite receptor. (c) This table gives formulas for some of thethermodynamic language. Here, n is the total number ofequilibrium constant where Δε = εI − εA is the difference inIð Þ
d ¼ c0eβ εIð Þb −μ0ð Þ is the inactive state's dissociation constant
sociation constant for ligand binding and c is the ligand

1438 Review: Statistical Mechanics of MWC Models
in which the binding of one ligand appears toencourage or discourage the binding of the next.Thus, we turn our attention to a more general nsiteMWC molecule. Some details of cooperativitycalculations are confined to Supplemental Information, Appendix 1.As for the onesite MWCmolecule, the nsite MWC
molecule can be in either an active state or aninactive state, and in each state, each of the n sitescan either be empty or have a bound ligand. A fullstatesandweights diagram would therefore have2 × 2n states.The weights assigned to each state follow the
general pattern set forth in Fig. 1a, and deriving theformulas for key quantities such as pactive followssimilar logic to that of the n = 1 case. For the reader'sconvenience, a number of these key formulas arelisted for the general MWC molecule with n bindingsites in Fig. 2, including formulas for the activitycurve pactive and the binding curve 〈Nbound〉. Note thatwhen faced with some new problem, we find that it isoften much simpler to write down the various statesand their associated statistical weights in statisticalmechanical language and then to convert to a morefamiliar Kd language at the end. It is for this reasonthat we illustrate the results in both languages.As n increases, the activity curves become flatter
for high and low concentrations and steeper near the“transition point”, the halfway point between aminimally and maximally active receptor denotedby c* in Fig. 2. This steepness is a signature ofcooperativity, which means that the binding of oneligand seems to encourage or discourage thenext.8,12,13 In the MWC model, this phenomenon isthe result of indirect regulation rather than directregulation: the presence of ligand increases theprobability of the receptor existing in the state withhigher ligand affinity, thereby increasing the probability of the next ligand binding. There are certainlyother models that can explain cooperativity, often bypostulating direct energetic interactions betweenbound ligands.2 One popular and simple model ofcooperativity postulates that activity curves followthe Hill function2,12,13 given by
p Hillð Þactive ¼c=KAð Þh
1þ c=KAð Þhð6Þ
If the Hill function were derived from a statesandweights diagram similar to that in Fig. 1a, then therewould only be two states for this receptor withidentical binding sites of dissociation constant KA: areceptor with no ligands bound and a receptor with hligands bound. This is often not a realistic mechanistic explanation for cooperativity despite the Hillfunction's ubiquitous presence in the biologicalliterature. However, h is a useful proxy for thedegree of cooperativity in the system since itquantifies exactly the steepness of the activity
curve at the transition point. Similarly, one candefine an effective Hill coefficient, heff, for any activitycurve as twice the slope of the activity curve on alog–log scale at its transition point.12 A formula forheff in the case of the MWC model is given in Fig. 2.The effective Hill coefficient is a useful heuristic forcooperativity. If heff N 1, then the presence of onebound ligand increases the likelihood of the nextligand binding, a signature of positive cooperativity; ifheff b 1, then the presence of one bound liganddecreases the likelihood of the next ligand binding;and if heff ≃ 1, then the binding of one ligandindifferent to the binding of the next, a signature of nocooperativity. As shown in Supplemental Information, Appendix 1, the effective Hill coefficient for theMWCmolecule can be summarized as follows.Whenthe active state and the inactive state of an MWCmolecule with multiple binding sites have differentligand affinities, heff N 1, and when there is only onebinding site or when the two states have equal affinityfor the ligand, then there is no signature ofcooperativity, heff = 1. To model an MWC moleculewith negative cooperativity, one needs to introducerepulsive interaction energies between ligands, aswas done by Narula and Igoshin, for example.11
With the statistical mechanical preliminaries nowin place, the remainder of the paper is devoted tospecific case studies. Each such study is introducedeither to highlight some specific twist on thestatistical mechanical analysis (such as the presence of more than one binding site) or to examinemodern applications of the MWC concept to problems of current interest.
Case Studies in MWC Thinking
The MWC concept presented above has beenapplied to a very wide array of different molecularscenarios, as evidenced by the massive citation list.In the 1960s, the MWC concept and related
models were used to great effect as the basisfor thinking about several important “model”molecules,4,14,15 most famously, hemoglobin. For abeautiful review on the evolution of thinking onhemoglobin, see the work of Eaton et al.16 Recallthat the hemoglobin molecule binds oxygen molecules, which are then delivered to tissues throughoutthe body. Hemoglobin can bind up to four oxygenmolecules and therefore has a more complicatedstatesandweights diagram and activity curve thanthat of the simpler onesite receptor in Fig. 1. One ofthe signature features of the binding curve of oxygenon hemoglobin is its characteristic sigmoidal shapethat indicates the existence of cooperativity, described in detail in the previous section. Traditionally,one of the most well studied ways of characterizingcooperativity is through the Hill function as alreadydiscussed in Eq. (6). As we already noted in States

1439Review: Statistical Mechanics of MWC Models
and Weights in the MWC Setting, Hill cooperativity isvery strict in the sense that, from a statisticalmechanical perspective, it banishes the states ofpartial occupancy that are present in the MWCframework. One of the key insights of the MWCperspective is that it too can give rise to sigmoidalbinding curves, but on the basis of a differentunderlying picture of the allowed molecular states.Models of ligand–receptor binding that includeintermediate states can also give rise to sigmoidalbinding curves, but unlike the MWC model, thesemodels often posit direct energetic interactionsbetween bound ligands. In the remainder of thissection, rather than focusing on MWC classics suchas hemoglobin, we highlight the spectrum of morerecent examples of the MWC concept that havebeen applied to topics of great current researchinterest ranging from bacterial chemotaxis to theaccessibility of chromatin to DNA binding proteins.
MWC ligandgated channels
The cell membrane is richly decorated with a hostof different molecular species, many of which detectand respond to molecules present in the externalmilieu. Ion channels are one of the most importantexamples of such membranebound proteins thatrespond to external cues resulting in changes of thecellular state such as a change in the membranepotential. One mechanism by which ion channelscan detect environmental signals is through anMWClike mechanism: when a molecule binds tothe channel, it shifts its equilibrium such that theopen state is more likely than the closed one. Thereare a number of important ligandgated ion channels,but we will primarily focus on two examples: (i) thenACh receptors at neuromuscular junctions17 and(ii) cGMPgated ion channels that enable photoreceptors to amplify their response to light.9,18
STATE
closed
WEIGHT STATE WE
e–β(εb
e
e–β(εb
e–β(2εb
e–βε x e
e–βε x e
e–βε x e–
Wclosed = (1 + e –β(εb
Wopen = e –βε
1
open
closed –μ)
closed –μ)
closed –2μ)
closed –μ))2
(a)
Fig. 3. Toy model of a ligandgated ion channel. (a) The moand can exist in four distinct states of occupancy (i.e., empty, 2and open states. (b) Open probability for a cGMPgated chanMWC model with four binding sites.19
Figure 3a shows the states and weights for a toyMWC model of an ion channel in which we imaginethat there are two distinct binding sites for therelevant ligand. The four states on the left of Fig. 3acorrespond to all the different variants on the closedstate and the four states on the right correspond tothe different variants of the open state. For convenience, we have taken the conformational energy ofthe closed state to be zero while the conformationalenergy of the open state is given by ε. We can add upthe statistical weights for all of the open statespermitting us to compute the open probability as
popen ¼1þ c
K oð Þd
� �2
1þ cK oð Þd
� �2þ e−βε 1þ c
K cð Þd
� �2 ð7Þ
Much effort on the use of MWC models has beenaimed at the rigorous attempt to figure out theanswers to precise questions such as how manybinding sites are present in the MWC molecule ofinterest, whether or not those sites are heterogeneous and what are the precise values of themolecular parameters associated with the variousstates. Of course, these questions are all of greatinterest. We show an example of the kind of data thatengenders these discussions in Fig. 3b, whichshows the open probability of cGMPgated ionchannels as a function of the concentration ofcGMP. These channels are a key part of the signaltransduction pathway in the retina, undergoing agating transition from open to closed when photoreceptors are exposed to light.2,18,20 The key point forour discussion here is to note that the MWC conceptgives us a framework for thinking about how channelgating depends upon key parameters such as theligand concentration, the number of binding sites (asrevealed in the effective Hill coefficient) and a variety
IGHT
–βε
–β(εb
–β(εb
β(2εb
(1 + e –β(εb
open –μ)
open –μ)
open –2μ)
open –μ))2
100 101 102 103 104 10510−3
10−2
10−1
100
Ope
n pr
obab
ility
cGMP (mM)
(b)
del ion channel has two binding sites for the control ligand× single occupancy, double occupancy) for both the closednel as a function of the cGMP concentration and fit to an

1440 Review: Statistical Mechanics of MWC Models
of other quantities of interest, including those shownin Fig. 2.2,19,21
MWC and bacterial chemotaxis
A second recent application of the MWC modelthat illustrates its adaptability to new experimentalsituations is that of bacterial chemotaxis, the processwhereby bacteria are observed to move up gradientsof chemoattractant.22,23 In the time since thedevelopment of the MWC concept, one of the beststudied (at least in quantitative detail) examples ofsignal transduction in living organisms is provided bythis fascinating directed motion. Specifically, bacterial motility in these situations is characterized by
tumble
run
run
tumble
attractant
increasing concentration
NO ATTRACTANTOR REPELLENT
POSITIVECHEMOTAXIS
0.25
0.5
0.75
1
010 4 10 3 10 2 10 1
Added MeAsp (mM)
Nor
mal
ized
rece
ptor
act
ivity
(a)
(c)
Fig. 4. Bacterial chemotaxis. (a) A schematic showing the mtumbles. (b) A chemoreceptor and the bacterial flagellar motorbacteria, they are often on opposite poles. In the presence of lnot induced to alter its rotation direction. (c) Activity of the chconcentration with MWC parameters taken from Ref. 24.
“runs” during which the bacterium uses its flagella toswim in a roughly straight path, punctuated by“tumbles” during which the bacterium reorients andthen swims off in a new direction (see Fig. 4a).22,25
The circuit that mediates this bacterial decisionmaking has been subjected to detailed experimentalscrutiny, and recent fluorescence resonance energytransfer (FRET) experiments26,27 provide precisequantitative data on the signal transduction pathways involved in bacterial chemotaxis. For ourpurposes here, the key point is that the bacterialsurface is decorated with chemoreceptors that servethe role of detecting chemoattractants in the surrounding medium and then changing the state ofphosphorylation of its diffusible response regulators
(b)
clockwiserotation(tumble)
flagellarmotor
ligand
counterclockwiserotation (run)
inactivereceptor
activereceptor
no ligand
CheY
receptor
P
P
otion of a bacterium that consists of a series of runs andare shown in the same membrane region, although in realigand, CheY is not phosphorylated and hence the motor isemoreceptor in the limits of low and high chemoattractant

1441Review: Statistical Mechanics of MWC Models
(CheY). Once phosphorylated, CheYP then inducesthe bacterial flagellar motor to undergo a change ofrotational direction that leads to a tumble. Thisprocess is shown schematically in Fig. 4b. In thiscase, the chemoreceptor is actually inactive inthe presence of ligands. In the presence of chemoattractants, the bacterium is what Howard Berghas dubbed an “optimist” and would like to simplykeep going in the same direction (i.e., to not undergoa tumble).The MWC concept has been applied in both
clever and subtle ways to describe the response ofbacteria to chemoattractants through sets of differ
STATE
INACTIVESTATE
n class 1receptors
m class 2receptors
e
e–βεoff
e
e–βεo
Fig. 5. Statesandweights diagram for chemotaxis clusterdifferent states of occupancy of the chemotactic receptors wdiagrams (not shown) for the active state. The statistical weigbeen drawn out of solution to bind the chemoreceptors. The dare of realizing a given state of binding. For example, in thebound on one of the n class 1 receptors. The class 1 receptorsthe inactive state and ε(on) in the active state. The n class 1state and Kd
(on) in the active state; the m class 2 receptors havin the active state.
ent chemoreceptors as indicated schematically inFig. 5.28–31 The simplest MWC description of achemotactic receptor is identical with the scenarioshown in Fig. 1 except that, in this case, binding tothe inactive state has the lower Kd (i.e., Kd
(off) b Kd(on)
or εb(off) b εb
(on)), which means that, in the absence ofligands, the receptor is active and that binding ofligands renders the receptor inactive. This is shownin Fig. 4c. The simplest model of activity as afunction of chemoattractant concentration is givenby Eq. (2), but respecting the condition Kd
(off) b Kd(on)
or εb(off) b εb
(on) described above. This is consistentwith Berg's optimism principle in the sense that,
WEIGHT DEGENERACY
1e–βεoff
ne–βεoffc
Kd(off)
n(n1)
2–βεoff
c
Kd(off)( (
2
c
Kd(off)( (
2 c
κd(off)( (
2
m–βεoffc
κd(off)( (
n x mffc
Kd(off)( (
c
κd(off)( (
x mn(n1)
2
SUM ALL OF THESE UP
s. The various states shown in the figure correspond tohile in the inactive state. There is a corresponding set ofhts of the different states reflect how many ligands haveegeneracies correspond to how many different ways theresecond state shown in the figure, there is only one ligandand class 2 receptors have conformational energy ε(off) inreceptors have dissociation constant Kd
(off) in the inactivee dissociation constants κd
(off) in the inactive state and κd(on)

1442 Review: Statistical Mechanics of MWC Models
with chemoattractant present, the bacteria tumbleless often.Figure 5 shows how the simplest MWC concept
can be extended to account for several particularlyinteresting features of chemoreceptors in bacteria.For instance, multiple receptors are clumped together into clusters, an aspect of these receptors that hasbeen long recognized and recently studied systematically across different species.32–35 For the case ofclusters of size n and all of the same species, theactivity is given by
pactive¼e−βε onð Þ 1þ c
K onð Þd
� �n
e−βε onð Þ 1þ cK onð Þd
� �nþe−βε offð Þ 1þ c
K offð Þd
� �n ð8Þ
a simple extension of the models introduced alreadyand which can be developed in direct analogy withthe way we worked out the ion channel openprobability in Fig. 3. The consequence of thisclustering is an effective increase in cooperativitythat sharpens the response of the chemotactic twocomponent signaling system to chemoattractant withrespect to the response of a single chemoreceptor.The FRET experiments of Sourjik and Berg
measured the fluorescence signal change when theresponse regulator CheYP interacts with downstream signaling partners, thereby effectively measuring activity curves for a number of differentmutantsof the receptors that mediate chemoreception. Theseexperiments provided stringent constraints on anytheoretical explanations set forth to explain chemotactic activity.26,27 Indeed, models following the MWCconcept found that the only way to explain the datawas to consider that the receptor clusters arechemically heterogeneous, which means that aspecific cluster of chemoreceptors will contain receptors of more than one type that have differentbinding affinities for the same chemoattractant.Specifically, as shown in Fig. 5, if there are n copiesof the first receptor type and m copies of the secondreceptor type, when constructing the statesandweights diagram, we must sum over all possiblestates of activity and ligand occupancy. If we ascribebinding constants κd
(on) and κd(off) to the second
receptor type, the activity of the receptor cluster as afunction of chemoattractant concentration is given by
pactive
¼e−βε onð Þ 1þ c
K onð Þd
� �n1þ c
κ onð Þd
� �m
e−βε onð Þ 1þ cK onð Þd
� �n1þ c
κ onð Þd
� �mþe−βε offð Þ 1þ c
K offð Þd
� �n1þ c
κ offð Þd
� �m
ð9Þa result used to consider the activity data coming fromFRET experiments for a number of different chemoreceptor mutants in quantitative detail.28–31
As shown in this section, the MWC model hasbeen used to great effect in a number of differentsituations, producing powerful predictions and insights into cellular signaling. We now describe acompletely different implementation of the MWCconcept in the context of the behavior of genomicDNA.
MWC and genomic accessibility
The MWC model has been recently andperhaps unexpectedly applied to transcriptionalregulation.10,11,36,37
Genomic DNA can exist in a compact state (i.e.,nucleosome bound or in some higherorder chromatin configuration) that is inaccessible to variousmolecules, for example, to transcription factors thatactivate some gene of interest. However, sufficientlyhigh concentrations of transcription factors (i.e., theligand) can increase the favorability of the chromatinopen state, even though the open conformation ofchromatin incurs a free energy cost. As will bediscussed below, there are many variants on thisbasic picture in which combinations of transcriptionfactors lead to different logic functions such as AND,OR and so on.11 Our aim here is to illustrate theoverarching conceptual picture through severalspecific examples.As a first foray into DNA accessibility problems
from the MWC perspective, we consider the accessibility of a DNA segment wrapped within a singlenucleosome. To get a first impression of the kinds ofmolecular states of interest and how they can bedescribed using statistical mechanics, Fig. 6 showsa hypothetical eukaryotic promoter bound with somedisposition relative to a nucleosome. We note fromthe outset that, because of the rules of nucleosomepositioning,38–40 the real situation is more subtlethan this and that this example is intended only toillustrate the “indirect regulation” that could beexercised by the presence of nucleosomeboundDNA. As seen in the figure, the DNA segment ofinterest harbors both a promoter and a binding sitefor a transcription factor. When the promoter iswrapped within the nucleosome, the gene of interestis inactive. The four states of this promoter in thissimple model then correspond to inactive and activeconfigurations of the promoter and the transcriptionfactor binding site either unoccupied or occupied,with the transcription factor serving as the ligand inmuch the same way as other ligands did in previousexamples. Computing the probability of the activestate follows the developments described above andfurther details can be found elsewhere (see chapter10 of Phillips et al.).2,41
One of the most compelling discoveries to emergefrom the study of eukaryotic gene regulation,especially in multicellular organisms, is the existenceof binding sites contained on the DNA known as

Proteinbinding site
Promoter
Activator
STATE WEIGHT
e–βεC
e–βεO
e–βεC
e–βεO cKd
(O)
cKd
(C)
Fig. 6. MWC model of nucleosome accessibility.Statesandweights diagram for a toy model of nucleosome accessibility that illustrates how transcription factorscould alter the equilibrium of nucleosomebound DNA. εcand εo refer to the conformational energies of the closedand open states, respectively, and Kd
(C) and Kd(O) are the
dissociation constants for transcription factor binding inthose two states.
1443Review: Statistical Mechanics of MWC Models
enhancers that result in regulatory “action at adistance”. Interestingly, the MWC concept is alsouseful for characterizing these ubiquitous eukaryoticregulatory architectures. The concept of suchenhancers is that there are binding sites that arenot in genomic proximity to the promoter theycontrol. Depending upon the binding of transcriptionfactors to these enhancers, the genes will beexpressed to differing extents. A particularly intriguing aspect of these enhancers from the point of viewof more traditional views of gene regulation is theirextreme flexibility—in some cases, there seems tobe a generic indifference to the number of bindingsites, their specific position and even their chemicalidentity.42
For example, the embryonic development of thefruit flyD. melanogaster's body plan is determined bythe expression levels of a hierarchy of genes withsinglecell resolution43,44 along the anterior–posterior axis of the embryo. One such gene is evenskipped, which is expressed in seven stripes alongthe anterior–posterior axis of the embryo. Each oneof these stripes is controlled by an individualenhancer located up to 8 kb upstream or downstream of the actual eve gene.42 The enhancer thatcontrols stripe 2, for example, is located 1.5 kb
upstream from the gene and in its minimal formspans 480 bp.45 It contains several binding sites fortwo activators and two repressors. Specifically, it hasthree binding sites for the activator Bicoid despite thefact that the deletion of one of these sites does notcause any qualitative changes to the outputpattern.45 Perhaps more revealing in terms of theflexibility of these regions is the fact that thisenhancer sequence has undergone significantchanges throughout evolution while retaining itsfunction. For example, in Drosophila pseudoobscura, the same enhancer has lost and gainedbinding sites while the remaining binding siteshave changed in their affinities with respect to theD. melanogaster enhancer. The spacing betweensome of these sites has also changed in some casesby up to 80 bp.46 Nevertheless, when the D.pseudoobscura enhancer is introduced into D.melanogaster, not only does it result in a very similarpattern of expression but also it can even rescuemutations in the eve gene.46–48
Recent quantitative models that have had somesuccess in explaining these observations are predicated on the idea that these enhancers affect geneexpression levels by controlling chromatin accessibility. This is in stark contrast to a picture in whichtranscriptional cooperativity is attributed to directinteractions between transcription factors and thebasal transcription apparatus. Figure 7a shows aschematic example of how the MWC concept can beapplied to model chromatin state. In the “closed” orinaccessible state, the DNA is wrapped up in sometight nucleosomal configuration, here indicated byone of many hypothetical higherorder chromatinstructures (i.e., the putative 30nm fiber).1 While inthis state, the promoter of interest is hypothesized tobe unavailable for transcription. The concept of themodel is that RNA polymerase and transcriptionfactors can bind more easily to DNA when it is in itsopen or accessible state, indicated schematically inFig. 7 by DNA that is freely available in the “openchromatin” configuration.Even within the relatively simple scenario depicted
in Fig. 7a, there is already a great deal of conceptualand quantitative flexibility to account for a host ofdifferent regulatory architectures. For example, onecan imagine situations such as shown in Fig. 7b inwhich the transcription factors are more favorablybound in either the closed or the open conformation,thus stabilizing one state or the other. Similarly, onecan imagine both positive and negative cooperativitybetween the transcription factors themselvesthrough direct physical contacts, permitting theconstruction of various logic functions such asAND and OR functions (and many others).11 Fromthe perspective of the MWC model itself, the keyparameters that come into play are the difference inenergy between the closed and open conformations,Δε = εc − εo, the binding energies (or Kd values) for

open chromatinclosed chromatin STATE WEIGHT STATE WEIGHT(b)(a)
e–βεC
e–βεC cAKA
(C)
e–βεC cBKB
(C)
e–βεC cAKA
(C)
cBKB
(C)
e–βεO
e–βεO cAKA
(O)
e–βεO cBKB
(O)
e–βεO cAKA
(O)
cBKB
(O)
A
A
B
B
B
A
AB
Fig. 7. Schematic description of MWC chromatin. (a) The genomic DNA exists in two classes of state, one of which is“off” and the other one of which is “on” and permits transcription. Transcription factor binding controls the relativeprobability of these different eventualities. (b) States and weights for the binding of two transcription factors, here denotedby A and B, which occupy the open and closed conformations with different affinity. The concentration of transcriptionfactors A and B is given by cA and cB, respectively. The conformational energies of the closed and open states are given byεc and εo. The dissociation constant for A is KA
(C) when chromatin is in the closed state and is KA(O) in the open state, and the
dissociation constant for B is KB(C) when chromatin is in the closed state and is KB
(O) when chromatin is in the open state.
1444 Review: Statistical Mechanics of MWC Models
the relevant transcription factors in each of the statesand the effective Hill coefficient that can be tuned bychanging the number of binding sites for the DNAbinding proteins in question. In Fig. 7c, we show anexample of how the probability of being in the activestate depends upon the concentrations of the twospecies of transcription factor.
The Bohr effect generalized
A beautiful example of the unifying power of MWCmodels is the suggestion of an analog of the Bohreffect in the context of chromatin. The reader isreminded that the Bohr effect refers to the oxygenbinding properties of hemoglobin and how the affinityfor oxygen is tuned by changes in the pH, forexample, as shown in Fig. 8a. Originally, the Bohreffect was an empirical observation. In the languageof MWC models, however, the Bohr effect can bethought of in terms of how the binding curves arealtered as the difference in energy between the twoconformational states is changed. Mirny recentlydescribed an analogous, Bohrlike effect in generegulation using the MWC model of chromatin statein which (for example) changes in the histone–DNAaffinity can affect changes in the occupancy curve asshown in Fig. 8b.10
Specifically, we consider the example given inFig. 7b for the case in which the two binding sites areused by the same transcription factor. In this case,we can use the states and weights highlighted inFig. 7b to compute the probability that the DNA willbe in the closed (inactive) state as
pinactive ¼e−βεc 1þ c
K cð Þd
� �2
e−βεc 1þ cK cð Þd
� �2þ e−βεo 1þ c
K oð Þd
� �2 ð10Þ
Note that, for the case considered here, the twoenhancers bind the same transcription factor with thesame affinities, although those affinities are differentin the open and closed chromatin conformations.From Fig. 2, the average number of bound transcriptions factors is
Nboundh i ¼2e−βεc c
K cð Þd1þ c
K cð Þd
� �þ 2e−βεo c
K oð Þd1þ c
K oð Þd
� �
e−βεc 1þ cK cð Þd
� �2þ e−βεo 1þ c
K oð Þd
� �2
ð11Þ

Frac
tiona
l occ
upan
cy
1.0
0.8
0.6
0.4
0.2
0
Oxygen partial pressure (mmHg)
Partial pressure of oxygenat muscle at lungs
a bc d e
0 0.5 1 1.5 2 2.5 3 3.5 4x 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
TF concentration (M)
Nor
mal
ized
occ
upan
cy
0 20 40 60 80 100 120 140
Open energy(kBT)
Occupancy
24
TFnucleosomal
(a) (b)
Fig. 8. The Bohr effect and MWC models. (a) The Bohr effect and oxygen binding to hemoglobin as a function of pH.The hemoglobin binding curves are shown for five values of the pH: (a) 7.5, (b) 7.4, (c) 7.2, (d) 7.0 and (e) 6.8. The verticallines indicate the partial pressures experienced in muscle and in the lungs. (b) The “Bohr effect” in the context of chromatinshowing how the occupancy of a transcription factor on nucleosomal DNA changes as the histone–DNA affinity (forexample) is changed, as described in Eq. (11). For the figure shown here, we have Kd
(o) = 10−9M and Kd(c) = 100 Kd
(o). Theclosed state energy has been chosen as the reference energy and is taken as zero.
1445Review: Statistical Mechanics of MWC Models
As seen in Fig. 8b, the modulation of this bindingcurve as a function of the energy difference betweenopen and closed chromatin conformations, εo − εc,reflects the chromatin Bohr effect.Interestingly, the mutants considered in the bac
terial chemotaxis setting correspond effectively todifferent states of methylation of the chemotaxisreceptors.26,27 Like in the case of chromatin, ourview is that the theoretical models using the MWCconcept in that context too are yet another exampleof the “Bohr effect”, but now in the context ofchemotaxis.28–31 This discussion provides a primeexample of the unexpected biological insights thatcome from classifying biological topics on the basisof their conceptual proximity based on the underlyingphysics or mathematics, rather than on the basis ofbiological concepts.
An InformationTheoretic Perspective ofthe MWC Concept
The MWC model provides a simple conceptualmechanism whereby ligands can regulate “at adistance”. For instance, as described in CaseStudies in MWC Thinking, enhancers can affectthe expression of a distant gene.49 So far, we havefocused on the generic features of MWCmodels andhow to calculate molecular activity from pictures ofstates. However, a powerful advantage of ananalytically tractable model such as the MWCmodel is that it can be used to calculate quantitiesthat are difficult to measure but that still have greatconceptual value. Calculating these quantities canshed light on how a regulatory system works andwhy it works in the way that it does. In this section,
we shift away from a discussion of the MWC modelitself and ask more general questions about thecapacities of MWC molecules as regulators.Specifically, we discuss the recent information
theoretic description of MWC molecules as sensorsof ligand concentration.8,50,51 To see what ideas arein play, consider the case of bacterial transcription. IfE. coli are grown in media rich in lactose instead ofglucose, they produce an enzyme to digest thelactose. This production is mediated by transcriptionfactors that allow information about the environmental conditions (lactose and glucose concentrations)to affect protein production (βgalactosidase enzyme) by influencing the likelihood of an RNApolymerase molecule to transcribe the relevantgene. Evolutionarily, it seems highly beneficial foran organism to excel at gathering information aboutenvironmental conditions and using that informationto regulate protein production. However, such descriptions are qualitative, whereas we desire a wayto quantify “how well” the output of a sensor (e.g., βgalactosidase production) tracks noisy sensory input(e.g., lactose concentration).One quantification strategy is to make an educated
guess as to how the molecule's behavior affects theorganism's fitness. This approach is fraught with risk,as what initially appears to be noise often turns out tobe signal in biological systems,52–54 and biologicalintuitions for fitness functions are often based onthese guesses as to what is a signal and what isnoise. For example, it is of course interesting that thesensory systems described by MWC models can betuned to mimic the Boolean logic gates that underlietoday's computers,11,55 but there is no guaranteethat MWC molecules have been selected to mimicBoolean logic gates. There is a more general

1446 Review: Statistical Mechanics of MWC Models
quantitative framework, information theory, that doesnot require knowing exactly what computation isbeing done by the cell but that still allows us toquantify how well the sensor output tracks input.56
Many have already written excellent reviews ofinformation theory with a biological bent57–61; thus,we will just introduce the definitions that we need forthis section. Given a black box system (e.g., anMWC molecule) that takes a noisy input X (ligandconcentration) and returns a noisy output Y (whetheror not the receptor is in the active state), thenmeasuring Y provides information about the state ofX. There is a unique function that will quantify theinformation content of a probability distributionsubject to certain plausible assumptions about theform of this function.56,62,63 From this function, wecan specify the amount of information about Xgained by measuring Y as the “mutual information”I(X;Y)56,62,63 (see Supplemental Material, Appendix2 for details). There are several ways to calculatethe mutual information, but the one that we will focuson here uses the conditional probability of y given x,p(yx), and can be written as
I X ;Yð Þ ¼ ∫∫pðy xj Þ p xð Þlog2pðy xj Þp yð Þ dx dy ð12Þ
If Y tracks X well, then I(X;Y) is large; on the otherhand, if Y and X are independent, then I(X;Y) =0.62,63 What is the maximal amount of informationone can expect between input and output? Ananswer to this question can be captured mathematically by computing the “channel capacity”, which isthe mutual information for an optimal p(x), explicitly
channel capacity ¼ Iopt X ;Yð Þ ¼ maxp xð Þ
I X ;Yð Þ ð13Þ
See Supplemental Material, Appendix 2 for details.The quantities described above provide principles
for quantifying what is possible in a molecularsignaling system, and interestingly, some biologicalsystems seem to be operating very close to channelcapacity. A spectacular example of this appears tobe the expression of the Hunchback protein in theearly Drosophila embryo, which is activated by theBicoid transcription factor. It has been argued thatthe probability distribution of the Bicoid transcriptionfactor is optimized so as to maximize the mutualinformation between this input and the outputHunchback expression level.50 Case studies suchas these have motivated other investigators to studythe channel capacity of MWC models moregenerally,8 a topic we take up below.We stress here that not all MWC molecules are
nearoptimal sensors, nor should we expect them tobe. For instance, hemoglobin picks up oxygen in thebloodstream and deposits that oxygen in distanttissues. This task does not necessarily requiremaximizing the mutual information between oxygen
concentration and the number of oxygen moleculesbound to hemoglobin. However, many of the MWCmolecules described in Case Studies in MWCThinking are likely to be highperformance sensorsof their environment. The nACh receptor at theneuromuscular junction must turn a chemical signalin the form of diffusing acetylcholine (ACh) molecules into a corresponding electrical signal that cancontract a muscle fiber. If nACh receptors misrepresent an incoming signal, the consequences couldrange from an inability to stimulate the motor systemto an inability to stop moving. Similarly, the cGMPreceptor must turn a chemical signal based on thepresence of light in the environment into an electricalsignal. If our cGMP receptors in our photoreceptorsdo not accurately represent the incoming light signal,then we will not be able to see. These and other suchreceptors could encode information about ligandconcentration in terms of the average number ofbound ligands or the probability that the receptor is inthe active state. Hence, it makes sense to study thesensing properties of MWC molecules, and ourmotivation for doing such an analysis is inspired by arecent general analysis of the sensing properties ofMWC molecules.8
In the remaining portion of this section, we use atoy model of N independent ligandgated ionchannels (inspired by the example of nACh receptors) to illustrate how to quantify the ideaspresented in the previous paragraph. Though wewill use the specific language of ion channels, theconcepts apply much more broadly. Indeed, generalreflections of this kind were analyzed comprehensively and in more generality elsewhere.8 Here, wepresent an abridged version of their analysisspecialized to a toy model of a ligandgated ionchannel such as the nACh receptor; many of thecalculational details have been relegated to Supplemental Material, Appendix 3.
Model system: nACh receptors
To illustrate the power of these informationtheoretic principles, we now investigate in detail anMWCmodel of a ligandgated ion channel. Again, thehope is that calculations on this toymodel will providesome qualitative insight into their functionality.To see why ligandgated ion channels could
plausibly be conceptualized as sensors, we nowdescribe the nACh receptors that lie at neuromuscular synaptic junctions, which are a key componentof the communication pathway between the nervoussystem and the motor system. When our braindecides that a particular muscle should contract, forexample, to avoid a hot stove, a motor neuronreleases vesicles of ACh molecules across asynaptic gap to a muscle fiber. On the other side ofthis synaptic gap are many thousands of nAChreceptors, with a surface density of roughly 105

1447Review: Statistical Mechanics of MWC Models
receptors per square micron. The diffusing AChmolecules bind to the nACh receptors, stochasticallyopening some number Nopen of the total number ofreceptors N. Each open channel allows for an influxof sodium ions and an outflux of potassium ions,which results in a net depolarization of the musclefiber. When membrane potential reaches threshold,the muscle fiber contracts.64,65
Extensive studies have revealed that nACh receptors behave as MWC molecules, though the twobinding sites in a nACh receptor are not necessarilyidentical, and there are likely more than two states ofthe nACh receptor.66,67 However, despite its shortcomings, the twostate MWC model of the nAChreceptor's response to ACh is an excellent examplewith which to illustrate the informationtheoreticunderpinnings of decision making based uponinput–output functions.It is clear that, in order to operate effectively, nACh
receptors should respond quickly to commands fromthe nervous system. As described above, theresponse of nACh receptors should reflect the sizeof the stimulus. Ligandgated ion channels moregenerally face a difficulty similar to that of nAChreceptors: they convey information about somestimulus from the outside to the inside of the cellbody, and they use energy by virtue of maintaining adifference between the ion concentrations inside andoutside of cells. Inspired by the example of the nAChreceptor, we investigate the ability of ligandgatedion channels to turn an input, ligand concentration c,into an output, the number of open ion channelsNopen. Conceptually, we are asking how well theNopen output tracks the c input. The calculationsshown here are merely illustrative of the kinds ofcalculations that could be done to elucidate thefunctioning of a particular sensory system. There areplenty of systems, including this ensemble of nAChreceptors, whose input might be best described by amagnitude other than c and whose output might bebest described by a magnitude other than Nopen.In general, this problem is challenging since c and
Nopen are both fluctuating quantities, typical of anysuch microscopic variable in biology. Even if eachreceptor experienced exactly the same ligandconcentration, the number of open ion channels isstill subject to fluctuations. In particular, for everyligand concentration, there is a correspondingprobability of being open, popen(c). Recall fromEq. (7) and Fig. 3 that this probability is given by
popen cð Þ ¼1þ c
K oð Þd
� �2
1þ cK oð Þd
� �2þ e−βε 1þ c
K cð Þd
� �2 ð14Þ
where Kd(o) is the dissociation constant of the open
ion channel, Kd(c) is the dissociation constant of the
closed ion channel and ε is the energy differencebetween the closed and open ion channels. Evenwhen ligand molecules are absent, there is anonzero probability of being in the active state.This limit defines the minimum, baseline probabilityof being open given by
pminopen ¼ popen c ¼ 0ð Þ ¼1
1þ e−βε ð15Þ
Likewise, as the ligand concentration tends toinfinity, there is a nonzero probability of the receptorbeing in the inactive state and the ion channel beingclosed. This limit as c → ∞ defines a maximum valueof the probability of being in the open state, namely,
pmaxopen ¼ limc→∞ popen cð Þ ¼1
1þ e−βε Koð Þ
d
K cð Þd
� �2 ð16Þ
For example, for the nACh receptor, we have popenmin ≃
8 × 10−4 and popenmax ≃ 1 using characteristic MWC
parameters for this channel.68 Thus, there is aprobability that all N nACh receptors will be closed,but this probability is very small evenwhen [ACh] = 0.WithN identical and independent ion channels, the
conditional probability that Nopen of the ion channelsare open given a ligand concentration c is thebinomial distribution
pðNopen cj Þ¼ NNopen� �
popen cð Þ� �Nopen 1−popen cð Þ� �N−Nopen
ð17ÞThis binomial distribution has mean N open cð Þ ¼Npopen cð Þ and variance σ2Nopen¼Npopen cð Þ 1−popen cð Þ
� �.
For each value of c, the distribution p(Nopenc) ishighly peaked about its mean, as can be seen inFig. 9, a fact that we will use later on to evaluate thechannel capacity using a “smallnoise approximation”. However, there are still fluctuations in Nopenthat prevent the output Nopen from determining theligand concentration c noiselessly.As ligand concentration varies, the most likely
value of Nopen varies from N minopen ¼ Npminopen toN maxopen ¼ Npmaxopen. The “dynamic range”8 capturesthe range of this likely output,
N maxopen−N minopen ¼ N p openmax −pminopen� �
¼ N 11þ e−βε Kd
Kd
� �2 − 11þ e−βε[email protected]
1CA ð18Þ
The dynamic range already provides a first glimpseinto how well the output Nopen follows the input c;the larger the dynamic range, the better Nopen willbe able to distinguish between different values of cdespite the intrinsic fluctuations in Nopen. This

InputCommunication
channelOutput
acetylcholine
states and weights
channel open probability
(a)
(b) (c) (d)
10 100
10210
0
101
102
Nop
en
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
p(Nopen
[AC
h])
Nopen
p*(N
open
)
0 20 40 60 80 1000
0.5
1
1.5
p*([
AC
h])
(μM
)1
[ACh] (μM) [ACh] (μM)0 1 2 3 4 5 6 7 8 9 10
00.05
0.10.15
0.20.25
0.30.35
0.40.45
0.5
Fig. 9. Information transmission through a twosite MWC molecule. The case of nACh receptors is illustrated forconcreteness. (a) ACh (input) binds to ligandgated ion channels (communication channel), thereby influencing thenumber of open ion channels (output). (b) The probability distribution of ACh concentration that maximizes the mutualinformation between input ([ACh]) and output (Nopen) from Eq. (21). (c) The MWC ligand–receptor binding probabilitiesdetermine the conditional distribution of the total number of nACh receptors open as a function of ACh concentration,p(Nopen[ACh]) from Eq. (17), shown here as a heat map. (d) The probability distribution of Nopen that maximizes the mutualinformation between input ([ACh]) and output (Nopen). (b–d) Plots assume a total of 100 nACh receptors on the synapticcleft for visualization purposes, although this underestimates the number of nACh receptors on a typical synaptic cleft andall plots use MWC parameters characteristic of nACh receptors.68
1448 Review: Statistical Mechanics of MWC Models
quantity is shown in Fig. 10a as a function of thedifference in energy between the inactive state andthe active state (−ε) and the difference in ligandbinding affinity between the active and inactivestates ( logK
oð Þd
K cð Þd) for a fixed N = 105. In this plot, we
also show the point corresponding to the experimentally available data for the nACh receptor, justfor comparison.68 Figure 10d shows the dynamicrange of a twosite MWC molecule as a function ofthe number of receptors N using characteristicMWC parameters for the nACh receptor.68
We are now ready to calculate the mutualinformation between the concentration of ligandand the number of open channels. If the jointprobability distribution of c and Nopen is p(c,Nopen),then following the procedure outlined in Eq. (12),mutual information is defined as
I c;Nopen� �¼∫ XN
Nopen¼0pðNopen cj Þ p cð Þlog2
pðNopen cj Þp Nopen� � dc
ð19Þ
As shown in Supplemental Material, Appendix 2, byinvoking key approximations such as the “smallnoise approximation”, this can be simplified as
I c;Nopen� �
≃−∫p N open� �
� log2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πeσNopen
2q
þlog2p N open� �� �
dN open
ð20ÞNotice that, in this equation, we have replaced theprobability distribution p(Nopen) with the value of theprobability around the mean of the distribution,p N open� �
. This latter distribution is directly determined by the probability distribution of the ligandconcentration p(c) sinceN open ¼ Npopen cð Þ and thusthe probability distributions of c andN open are relatedby p cð Þ ¼ p N open cð Þ
� �dN opendc
.Without measuring the probability distribution of
ligand concentration p(c), we are unable to calculatethe mutual information given in Eq. (20). Hence, weinstead calculate the channel capacity by consideringthe distribution p(c) or, equivalently, the distribution

0 1 2 3 4 5 6 7 8 9 10x 104
0
1
2
3
4
5
6
7
8
9
10x 10 4
N
Dyn
amic
ran
ge
0 1 2 3 4 5 6 7 8 9 10x 104
1
0
1
2
3
4
5
6
7
8
N
05
10 105
00
0.2
0.4
0.6
0.8
1
Dyn
amic
ran
ge
0 2 4 6 8 10 105
00
2
4
6
8
Cha
nnel
cap
acity
(bi
ts)
Cha
nnel
cap
acity
(bi
ts)
x 10 5(b) (c)
(d) (e)
(a)
0 2 4 6 8 10 105
01
1.2
1.4
1.6
1.8
2
Effe
ctiv
e H
ill c
oeffi
cien
t
βε
βε
βε
Kdlog(o)
Kd(c)
Kdlog(o)
Kd(c)
Kdlog(o)
Kd(c)
Fig. 10. Sensor properties of an ensemble of independent MWC molecules with two binding sites. In plots (a), (b) and(d), the MWC parameters are characterized by − βε, the conformational energy difference between the open and closedstates (in units of kBT), and log
K oð ÞdK cð Þd
¼ −β ε oð Þb −ε cð Þb� �
, the difference in ligand binding energies between the open and closedstates (in units of kBT). (a) The effective Hill coefficient of a twosite MWC molecule plotted as a function of MWCparameters. (b) The dynamic range of 105 independent twosite MWC molecules plotted as a function of MWCparameters. (c) The dynamic range of N twosite MWC molecule with MWC parameters characteristic of a nAChreceptor,68 plotted as a function of the total number of receptors N. (d) The channel capacity of 105 independent twositeMWC molecules plotted as a function of MWC parameters. (e) The channel capacity of N twosite MWC molecule withMWC parameters characteristic of a nACh receptor,68 plotted as a function of the total number of receptors N.
1449Review: Statistical Mechanics of MWC Models
p N open� �
that maximizes the mutual information.We hope that the channel capacity can still giveinsight into theworkings of the system, as it did for the
Hunchback/Bicoid gradient.50 We note that someof the systems described by the MWC model areunable to alter the probability distribution of ligand

1450 Review: Statistical Mechanics of MWC Models
concentrations and are therefore unlikely to operateat channel capacity. For example, bacteria cannotusually control the probability distribution of chemoattractant in the environment so that their bacterialchemotactic receptors operate constantly at channelcapacity. However, the body can alter the probabilitydistribution of ACh concentration at the neuromuscular junction by altering the size distribution andnumber distribution of synaptic vesicles. In short, acalculation of the channel capacity will not always bemeaningful, but we suspect that this calculation canbe made relevant for ligandgated ion channels andother MWC molecules. The optimal p N open
� �can be
found using variational calculus, a step that isdescribed in Supplemental Material, Appendix 3.The form of p N open
� �that maximizes the mutual
information is
p� N open� � ¼ 1
Z1
σNopen N open� � ð21Þ
where Z is a normalization constant,
Z ¼ ∫N
min
open
Nmax
open dN openσNopen N open
� � ð22ÞUsing Eqs. (21) and (22) to simplify Eq. (20) yields
Iopt c;Nopen� � ¼ log2 Zffiffiffiffiffiffiffiffiffi
2πep ð23Þ
indicating that the channel capacity increases as thenoise of the output decreases. In SupplementalMaterial, Appendix 3, we compute Z explicitly, thuspermitting us to write the channel capacity of the Nidentical uncoupled ligandgated ion channels as
Iopt c;Nopen� � ¼ log2 sin−1
ffiffiffiffiffiffiffiffiffiffipopenmax
q−sin−1
ffiffiffiffiffiffiffiffiffipopenmin
q� �
þlog2ffiffiffiffiffiffiffi2Nπe
rð24Þ
where popenmax and popen
min are given byEqs. (15) and (16).A more general formula for the channel capacity of Nreceptors when each receptor has n binding sites isgiven in Ref. 8, and a more detailed derivation of thechannel capacity is given elsewhere.51,58,69,70
Dynamic range and channel capacity are closelyrelated to the previously described concept ofcooperativity. A receptor with a high degree ofcooperativity will have a “steeper” activity curvepopen(c) near its transition point. Intuitively, increasing the cooperativity increases the ability of thesystem of N independent receptors to differentiatebetween different ligand concentrations c near thetransition point. For a given popen
min , increasingcooperativity will increase popen
max , thereby increasingdynamic range and channel capacity according toEqs. (18) and (24), respectively. The left column of
Fig. 10 below shows how dynamic range, channelcapacity and effective Hill coefficient vary as afunction of MWC parameters, the conformationalenergy difference (ε) and the difference in bindingenergy ( logK oKc ). Recall that the effective Hill coefficient, defined in States and Weights in the MWCSetting and Fig. 2, is a measure of the degree ofcooperativity. All three quantities are highly correlated and are largest when the closed ion channel is farmore energetically favorable than the open ionchannel and when the open ion channel has muchhigher affinity for the ligand than the closed ionchannel. The results of Ref. 8 also show that all threequantities increase when n increases, since increasing the number of sites increases the effectiveHill coefficient.The right column of Fig. 10 shows that the dynamic
range increases linearly with the total number of ionchannels N but that channel capacity increasesmore slowly as the logarithm of N. Increasing thetotal number of ion channels always increases thechannel capacity, but increasing the channel capacity by n bits requires increasing the total number ofion channels (and the average number of ionchannels) by a factor of 4n. However, increasingthe total number of ion channels requiresmanufacturing proteins, which requires materialsand energy.2,71 Hence, increasing the channelcapacity might be potentially energetically expensive. Again, we emphasize that these calculationsare merely illustrative, and we are not claiming thatnACh receptors or any other receptors have thistradeoff between energy and information.
Optimization principle: Maximization of mutualinformation
Operating at channel capacity requires that theprobability distribution of ligand concentration takeson a peculiar form shown in Fig. 9. If the measureddistribution of ligand concentration matched thispredicted probability distribution, then this matchwould provide additional support for the claim thatthe ensemble of ligandgated ion channels haveevolved to maximize mutual information between theinput (c) and the output (Nopen) within biophysicalconstraints. Such a measurement was made for theBicoid/Hunchback system in the early Drosophilaembryo, and the predicted probability distribution ofHunchback gene expression was strikingly close tothe empirical probability distribution.50 Sometimes,an organism cannot control the probability distribution of ligand concentration, but this does notpreclude use of information theory. For instance,the chemotactic receptors described in Case Studiesin MWC Thinking must use information aboutchemoattractant concentration to decide on whetherthe cell runs or tumbles. There are differentinformationtheoretic optimal ways to move in

1451Review: Statistical Mechanics of MWC Models
different chemoattractant gradients, and the optimalmovements can be compared to the observedmovements of the organism, as was done in Ref. 72.The idea that biological systems might have
evolved to maximize the mutual information betweentheir “input” and “output” is certainly not new. Thisoptimality principle has previously been applied to avariety of “information bottlenecks” in the brain, inwhich some physical barrier prevents information inone region from being copied directly into anotherregion, but where the second region needs theinformation encoded by the first region. For example,the light intensities hitting photoreceptors must beencoded as binary spikes sent by retinal ganglionneurons to the lateral geniculate nucleus, and thedistribution of spike times chosen should conveymaximal information about the incident lightintensities.57,59 A few investigators have begun tocalculate the mutual information between the inputand output of molecular systems,60,73,74 in particular, for genetic regulatory circuits and signal transduction pathways.It is important to note that mutual information
calculations are suggestive and potentially helpfulfor understanding organism behavior but neverdefinitive. Mutual information between the inputand output could be well correlated with anotherquantity that the system has evolved to maximize, inthe same way that cooperativity is correlated withdynamic range and channel capacity in the calculation above. Additionally, in these calculations, wemight have incorrectly identified the input and outputof the system.Finally, mutual information calculations can be
incredibly difficult. In particular, it is often difficult toanalyze nonlinear systems subject to timevaryingenvironmental stimuli and systems with feedback.Many biological systems, however, respond nonlinearly to timevarying environmental stimuli and arepart of a feedback loop. Analyzing these systemsusing information theory will likely require theoreticaladvances in communication theory.
Dynamical MWC
Molecules that are described in terms of the MWCconcept are not necessarily optimal informationencoders as described in the previous section.Nevertheless, there are many processes in the lifeof an organism where signaling molecules shouldrespond quickly and definitively to changes in ligandconcentration. Another feature that might beexpected of such molecules is that they block outhighfrequency molecular noise so that they onlyrespond to changes that occur over “long” timescales. To explore such timedependent properties,we need to go beyond the equilibrium statisticalmechanics description exploited thus far.
Though the original formulation of the MWC modelwas primarily an equilibrium concept, generalizationto the dynamical situation is relatively straightforwardand has been undertaken by many workers in themeantime,8,75–81 sometimes under the heading ofthe “kinetic allosteric model”.82–84 The chemicalreactions underlying the dynamical MWC modelwere outlined in one of the original papers,4 and anapplication of transition state theory85–87 to thosechemical reactions yields the kinetic allosteric model.The dynamical MWC model that we will present inthis review article uses only the law of mass actionand does not come close to using the full power oftransition state theory, which can calculate the valuesof rate constants from first principles.85–87
A smaller set of researchers have turned tothe language of control theory88,89 to describethe kinetics of MWC molecules using transferfunctions.8,80,90,91 In doing so, these researchershave described the dynamics of an MWC moleculeusing frequency instead of time. The two descriptionsare mathematically related; highfrequency signalsoscillate quickly and change on small timescales,whereas lowfrequency signals oscillate slowly andchange on long timescales. Transfer functions andfrequency response functions are often economicalways of describing the response of a system to achange in the inputs. These functions are often usedin electrical engineering and signal processing todesign filters that block noise and not signal. Morerecent articles have calculated the transfer functionsthat describe how MWC molecules respond tochanges in ligand concentration8,80 and showedthat a general MWC molecule does not respondstrongly to quickly changing ligand concentrations.8
In this section, we will describe a general approachto the dynamics of MWC molecules, once againillustrated through the special case of ligandgatedion channels. Specifically, we will examine theresponse of such a channel to changes in theconcentration of the relevant gating ligand.
Transition matrices and master equations
The general MWC molecule can be active orinactive and can have anywhere from 0 to n ligandsbound to it in each of these states of activity, giving atotal of 2 × 2n states. For even small numbers ofbinding sites,models of the transitions between these2 × 2n states can become unwieldy. However, if all ofthe binding sites are identical, then this description
can be greatly simplified. There arenk
� �different
ways for the receptor with n sites to bind k ligands, butall of these configurations have the same energy andtherefore the sameBoltzmannweight. Hence,we candescribe the state of an nsite MWC molecule interms of only two variables: the configurational stateof the receptor and how many ligands are bound to

O0
(b)
(a)O1 O2
C0 C1 C2
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (ms)
Pro
babi
litie
s of
eac
h st
ate
P(O2,t)
P(O1,t)
P(O0,t)
P(C0,t)
P(C1,t)
P(C2,t)
fL
2fC
bL
bC
2fO
bO
fC
2bC
fO
2bO
Fig. 11. Dynamics of an MWC ligandgated ion channelwith two binding sites. (a) Schematic showing states andrates of transition between MWC states in a simplifiedkinetic allostery model. (b) Probabilities of being in eachstate as a function of time, starting from a nonequilibriumconfiguration, as calculated using Eq. (33) with c = 0.5 μMand rate constants fO = 1 μM
−1 ms−1, bO = 170 ms−1,
fC = 1 μM−1 ms−1, bC = 0.04 ms
−1, fL = 1 μM−1 ms−1
and bL = 8 × 10−4 ms−1. The dotted lines show the
equilibrium values of each of these probabilities.
1452 Review: Statistical Mechanics of MWC Models
the receptor in total. In this simplified description,there are 2 × (n = 1) different states for an MWCmoleculewith n binding sites. For example, for the toymodel of the nACh receptor considered here, thepossible ligand–receptor configurations are {O2,O1,O0,C0,C1,C2} where Oi denotes the open nACh ionchannel with i bound ACh molecules and Ci denotesthe closed nACh ion channel with i bound AChmolecules, as shown in Fig. 11. The state of thesystem x(t) can be described as a list of theconcentrations of each of these configurations,
x tð Þ ¼
O2½ �O1½ �O0½ �C0½ �C1½ �C2½ �
1CCCCCCA
ð25Þ
although the ordering of the states in x is completelyarbitrary. A dynamical MWC model will describe how
x evolves with time. If the system is Markovian(“memoryless”), then x obeys a firstorder ordinarydifferential equation,
dxdt
¼ Mx ð26Þ
where M is a socalled “transition matrix” of size2(n + 1) × 2(n + 1). For those unfamiliar with matrices and vectors, row i of Eq. (26) is
dxidt
¼X2 nþ1ð Þj¼1
Mijxj ð27Þ
and this formulation is mathematically equivalent tothat given by Eq. (26). Deriving a dynamical MWCmodel is therefore equivalent to specifying theelements of the transition matrix, Mij.The statistical mechanics of chemical reactions
constrains the form of the matrix elements Mij. Tosee this, consider the situation shown in Fig. 11: howmight we change the concentration of open ionchannels with two ligands bound, [O2]? Based on thearrows in the diagram, there are two elementaryreactions that can change this concentration. First, aligand could bind to the open site of an open ionchannel with one ligand bound, O1 + L → O2. Second, an open ion channel with two bound ligandscould lose one ligand to the solution, O2 → O1 + L.The law of mass action implies that
d O2½ �dt
¼ kO1→O2 L½ � O1½ �−2kO2→O1 O2½ � ð28Þ
The kinetic rateskO2→O1 andkO1→O2 are linked togetherby the requirement that the ratio of kinetic rates yieldsthe equilibrium constant, kO2→O1kO1→O2 ¼ K
cð Þd . The factor of 2 in
the reaction rate for O1 + L → O2 arises because of adegeneracy in state space: there are two “types” ofO1molecules, one in which the ligand is bound at the leftsite and one in which the ligand is bound at the rightsite. An alternative and equivalent viewpoint is thatthere are two ligands on O2 that can unbind from thereceptor, but there is only one site on O1 to which aligand can bind.8 The kinetic rates kO1→O2 and kO2→O1reflect the height of activationenergy barriers betweenthe states, and the concentration dependence of thereaction rates encodes the frequency with which thereactants will meet. Those readers interested incalculating these rates more exactly using transitionstate theory should consult other references, forexample, Refs. 85,86. Note that Eq. (27) when i = 1 is
d O2½ �dt
¼ M11 O2½ � þM12 O1½ � þM13 O0½ �þM14 C0½ � þM15 C1½ � þM16 C2½ � ð29Þ
Comparison of Eqs. (28) and (29) indicates that
M11 ¼ −2kO2→O1 ; M12 ¼ kO1→O2c; M13 ¼ M14¼ M15 ¼ M16 ¼ 0 ð30Þ

1453Review: Statistical Mechanics of MWC Models
Where c = [L], the ligand concentration. The rateskO1→O2 and kO2→O1 are denoted as the forward andbackward rates fO and bO, respectively. Similar logiccan be used to determine the rest of the transitionmatrix M, which we list in full here as
M ¼
−2bO fOc 0 0 0 02bO − bOþ fOcð Þ 2fOc 0 0 00 bO − fLþ2fOcð Þ bL 0 00 0 fL − bLþ2fCcð Þ bC 00 0 0 2fCc − bCþ fCcð Þ 2bC0 0 0 0 fCc −2bC
1CCCCCCA
ð31ÞThe rates fO, bO, fC, bC, fL and bL correspond to theforward and backward rates for binding a ligand tothe receptor in its open state, the forward andbackward rates for binding a ligand to the receptorin its closed state and the forward and backwardrates for switching from the closed state to the openstate, respectively, as shown in Fig. 11. SeeSupplemental Material, Appendix 4. The ratio offorward and backward kinetic rates is the equilibriumconstant, yielding
e−βε ¼ fLbL
; K oð Þd ¼bOfO
; K cð Þd ¼bCfC
ð32Þ
The transition matrix M in Eq. (31) includes only asubset of the elementary reactions that could affectconcentrations of the various ligand–receptor configurations. For example, another elementary reaction that is often included in dynamical MWCmodels is O2 → C2, which describes the phenomenon of the ion channel opening and closing whileligands are bound. It is also possible to allow forrate constants for each binding site to be different orto allow for rate constants that change as a functionof the total number of ligands. However, thetransition matrix in Eq. (31) has sufficient complexityto give the correct qualitative behavior of thetransfer function.8 Our aim is to illustrate thegeneral principles underlying dynamical MWCmodels, not to find the exact kinetic rates or transferfunction for a particular receptor.Equations (26) and (31) give us a simple frame
work with which to analyze the dynamics of an MWCmolecule. In fact, if the concentration is a function oftime c(t) and the rate parameters are constant, thenthere is an exact analytic solution for the state vectorx(t),89 namely,
x tð Þ ¼ e∫t0M c t ′ð Þð Þdt ′x 0ð Þ ð33Þ
If M is a function of x(t) because the rate parametersare being continuously altered by some feedbackmechanism, then solving Eq. (26) becomes a muchmore challenging proposition unless there is aseparation of timescales. For instance, if theequilibration time between MWC states is muchsmaller than the timescale on which rate constants
are altered, then x(t) ≃ xeq. This assumption can bea useful approximation for simulating systems withfeedback, for example, the precise adaptation ofbacterial chemotactic receptors.24
Responses to changes in ligand concentration:The frequency response of an MWC molecule
One of the most inte