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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.
Table of contents
- Thermodynamics of Biological Processes
- The laws of thermodynamics (article) | Khan Academy
- 1. Introduction: Thermodynamics is Not Just for Dead Stuff
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. 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 most interesting biological systems, the concentration of ligand will change over time e. 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 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, 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, ; Gunasekaran et al.
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. Regulation is one of the great themes of biology. The roots of regulatory biology are largely to be found in the study of prokaryotes Ackers et al. 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. 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. In the time since, these ideas have been used even more aggressively for an ever-increasing set of regulatory architectures Dodd et al.
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. 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, 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.
States and weights for simple repression. A promoter has a binding site for a repressor molecule which excludes the binding of RNA polymerase. To derive these weights, we use the same approach as that described for ligand—receptor binding, except now we assume that both polymerases and repressors when not bound to the promoter are distributed among N NS sites on the bacterial genome this is essentially the size of the genome.
The energies in the Boltzmann factors are computed as the difference between the energy when repressor or polymerase is bound specifically to the promoter region of the DNA and when they are bound nonspecifically somewhere else on the genome Bintu et al. Details about how this formula is obtained in analogy to the probability of the ligand binding to a receptor from Eq.
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.
For repression, the fold-change is always less than one, while for activation, the fold-change is greater than one. 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.
Fold-change in gene expression. Two different promoters have been characterized as a function of the number of repressors Oehler et al. The thermodynamic models predict a precise dependence of the fold-change in gene expression on the concentration of repressors. The theoretical predictions are given by the curves, and the experimental results for two different promoters are given by the data points.
These predictions were obtained using the reasoning outlined in Fig. 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. 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.
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. 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. Transcriptional time series for several different classes of rate constants. The schematic emphasizes two scenarios in which thermodynamic models of gene regulation are valid.
In case 1, the promoter switches fast on the time scale defined by transcription initiation, while in case 2, the opposite limit is illustrated. In both limits, the steady-state number of transcripts mRNA degrades due to the action of RNases and by dilution after cell division is proportional to the fraction of time the promoter is in the active state state 1 , which can be computed using equilibrium techniques.
We are grateful to Alvaro Sanchez for his articulation of the ideas embodied in this figure. 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 Monod et al. 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.
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.
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: One of the most important experimental findings about these binding probabilities is the existence of cooperativity: In this situation, the simple binding curves such as those shown in Fig.
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. 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 specifically as a way to explain the nontrivial shape of the observed binding curve Pauling, 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. 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, Incorporation of 2,3-BPG into a Pauling-style model for hemoglobin or, similarly, incorporation of the Bohr effect, etc.
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: 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.
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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. 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: 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, 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. MWC model of ligand-gated ion channel.
Thermodynamics of Biological Processes
The channel is presumed to exist in one of two states, closed and open. The binding affinities of the ligands for the two binding sites on the channel are the same and they depend upon whether the channel is closed or open. This dependence leads to cooperative binding of the ligands.
A States and weights for a toy model of a ligand-gated ion channel with two binding sites for the gating ligand. B Probability of the two states of the channel open and closed as a function of the gating ligand. Notice how in the absence of ligand the probability of the channel being open is the same as that calculated in Eq. 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.
The outcome of this model is that the open probability as a function of ligand concentration has the simple but subtle form. 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, 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, 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.
The laws of thermodynamics (article) | Khan Academy
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, 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, Snapshots from the Rogers video showing the directed motion of a neutrophil.
The motion of a bacterium such as E. 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. 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. To illustrate how equilibrium statistical mechanics has been used to study chemotaxis, we consider the simplified scenario shown in Fig. 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.
1. Introduction: Thermodynamics is Not Just for Dead Stuff
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. States of a single chemoreceptor. A The chemoreceptor can exist in four distinct states, characterized by whether or not it is occupied by a ligand and by whether it is in the active state or not.
B The probability of being in the active on state is obtained by summing over the statistical weights of the states where the receptor is active and normalizing by summing over the statistical weights of all states. 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. In this case, the states and weights are shown in the figure and reflect the four eventualities that can be realized: 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. The states and weights corresponding to this model are shown in Fig. States and weights for chemoreceptors in the chemotaxis process. This figure shows the states and weights for a single receptor. 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, In this case, as shown in Fig.
MWC model for bacterial chemotaxis. The chemotactic receptors are modeled to exist in clusters with N distinct receptors. Collectively, these receptors can either be in the off or on state, where when on they are able to phosphorylate their downstream response regulator.
A States and weights diagram for the receptors. B Plot of the probability of the active state as a function of the concentration of chemoattractant for different numbers of receptors within a cluster. 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.
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. 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 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.
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. 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.
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, ; Raveh-Sadka et al. 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. 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.
MWC model for eukaryotic action at a distance. The regulatory region of the DNA is pictured here to have seven distinct binding sites. The organization of the DNA itself is further posited to exist in two different states. A States and weights corresponding to the model. B Data for the normalized level of Hunchback protein as a function of the level of Bicoid protein Gregor et al. 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, 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. The data for the relationship between bicoid binding and hunchback expression has been explored in a recent paper Gregor et al. 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. 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.
Of the nearly citations at the time of this writing of the original paper by Monod, Wyman, and Changeux Monod et al. 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.
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 from evolution to economics, from materials science to biology Rudnick and Gaspari, 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.
There is a vast literature on the use of models from equilibrium statistical mechanics to explore the properties of biological polymers de Gennes, ; Grosberg and Khokhlov, 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.
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. 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. 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. 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. Random-walk models of tethered ligand—receptor pairs. A Schematic of the tethered ligand—receptor pair and the associated statistical weights.
The multiplicities for the polymer are computed using a one-dimensional toy model of the tether. The rest of the parameters are defined as in Fig. For more realistic calculations, see Van Valen et al. B Concentration dependence of the probabilities of the different states that can be realized by the tethered ligand—receptor pair.
The volume of the elementary box v has been chosen to be approximately 1. 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. What this means really is that we evaluate the entropic cost of loop 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.
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