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Clustering and Signalling of Cell Receptors

As a response to ligand binding, transmembrane cell receptors often enhance their clustering, or oligomerization, during the signalling process. Here we present a statistical mechanical model which combines the aspects of clustering and signalling. In this model, receptors float on the surface, while for two neighboring receptors, there is an interaction energy dependent on their conformational states. On the other hand, ligand binding of a receptor shifts the energy difference between the two conformational states. Due to thermal fluctuation, the effects of clustering and signalling are statistical average quantities. This model reduces to a floating Ising model with a random field. We calculate the signalling in a grand canonical ensemble mean field approach, using Hubbard-Stratonovich transformation and replica method. Monte Carlo simulations are also performed. Essential biological features are obtained in our model.


INTRODUCTION
On the surface of a cell, there are many transmembrane receptor molecules. They are "sensory machines" of the cell. Binding extracellular ligands changes the conformational states or, in our point of view, the probability distribution of the conformational states. Consequently a cascade of responses occurs inside the cell. This is called signalling, which is often quite sensitive. An example is bacterial chemotaxis [1]: through the detection by its receptors, a bacterium swims towards higher concentration of a chemical it likes or a lower concentration of a chemical it dislikes. In this case, thousands of receptors cluster together at a pole of the cell. Recently, based on detailed analyses on the experimental findings, I proposed that cooperative receptor-receptor interaction enhances the signalling, and constructed a statistical mechanical theory for this process [2][3][4]. My idea has since received considerable attention 1 . In [4], good consistency was found in comparing this theory with experimental results [5][6][7][8][9][10], and it was also suggested what kinds of experiment are needed.
In the case of many other cell receptors, however, usually they float on the cell membrane. A ubiquitous phenomenon is that ligand binding enhances the clustering of receptors during the signalling process. Investigations on this issue are widely reported for various receptors, for example, integins, which attach cells to the extracellular matrix, tyrosine kinase receptors, cytoskine receptors, growth factor receptors, tumor necrosis factor receptors, including death receptors, antigen receptors such as T-cell receptors, bone morphogenetic protein receptor, G-protein-coupled receptors, etc [11][12][13][14][15][16][17][18][19]. In our previous works [2][3][4], the position of each receptor is fixed, and there is an a priori cluster. In the present article, we shall consider statistical mechanics of floating receptors. Based on our model, it is proposed that signalling sensitivity and clustering enhancement, in certain cases, are likely two consequences of the same mechanism based on conformation-dependent local receptor-receptor interaction.
In Sec. 2, we give a detailed description of the model. A mean field calculation of the signalling, as a statistical average quantity quantity, is given in Sec. 3. In Sec. 4, we discuss receptor clustering. Results of Monte Carlo simulations are given in Sec. 5. Finally a summary and discussion is made in Sec. 6.

THE MODEL
Consider a collection of receptors on a lattice. Fig. 1 is an illustrative configuration. Let the coordinate number be ν, which is 6 for a honeycomb lattice and is 4 for a square lattice. Suppose that the receptors can move around on the lattice. Hence at an instant, a receptor has ≤ ν nearest-neighboring receptors. The conformational state of the receptor at lattice point i is represented as V i , the value of which is either V 0 or V 1 , if we follow the two-state convention. V i can be understood as the position of the receptor molecule. 1 The reference was mostly made to a publication which popularizes some rudiments of my idea Obviously, V i is influenced by whether this receptor is bound to ligand. We suggest that V i is also influenced by the conformational states of its nearest-neighboring receptors, due to receptor-receptor interaction which is dependent on their conformational states. We use H i to represent whether there is a ligand binding, that is, H i = H = 0 if the receptor is bound to ligand, otherwise H i = 0. H can be understood as the force or torque generated by the ligand binding. HV i is the energy due to ligand binding. Given the ambient chemical concentration [L], the occupancy, i.e. the fraction of receptors with ligand bound, is where K d is the dissociation constant. Therefore each H i is randomly distributed between H and 0, with probabilities c and 1 − c, respectively. Formally, the probability distribution can be written as where , it is reasonable to regard the randomness of H i as quenched [3,4]. We associate each site i with a variable n i , which is 1 if there is a receptor on this site, and is 0 otherwise. In reality, of course V i and H i are only defined when there is a receptor on site i. However, because the presence or not of a receptor at site i has been determined by n i , for the convenience of treatment, equivalently we may associate V i and H i to each site i, no matter whether there is a receptor on site i. Since the receptors are allowed to diffuse, n i is time-dependent. One may constrain that the total number of receptors, N = i n i , is conserved. It is also possible that the receptors may get out of the relevant region, hence N is not conserved. In any case, we may use grand canonical ensemble approach characterized by the chemical potential, which, in the first case, is given by the constraint on the conservation of receptor number.
Therefore, in absence of thermal fluctuation, we have where the summation is over the nearest neighbors j of site i, U i is a threshold value, T ij is a coefficient for the coupling between nearest neighbors. j t ij n j V j is the influence, i.e. force or torque, on the receptor at site i from the nearest-neighboring sites. H i is the influence, i.e. force or torque, generated by the ligand binding of the receptor at i itself. Eq. (3) means that if the total influence on a receptor, as the sum of that due to its ligand binding and that due to the interaction with the nearest-neighbors, is larger than a certain value U i , the receptors is in state V 1 . Otherwise it is V 0 . Equivalently, with t ij = t ji and t ii = 0, the dynamics is determined in the following Hamiltonian, where ij represents pairs of nearest neighbors, t ij and U i have been set to be constants. Eq. (4) gives the total energy of the system, with the kinetic energy neglected. The first term is the total interaction energy of nearest-neighboring pairs. The second term is the energy due to ligand binding. The third term is the original energy, which, together with the first term, determines the probability distribution of the conformational states of all the receptors in the absence of ligand binding of any receptor. Without ligand binding and coupling with others, the energy of a receptor is UV 0 for state V 0 , and is UV 1 for state V 1 .
With ligand binding but without coupling, it is (U − H)V 0 for state V 0 , and is (U − H)V 1 for state V 1 . The interaction energy of the pair ij , if sites i and j are both occupied by receptors, is −tV i V j , the value of which is dependent on both V i and V j . Since ligand binding can cause more Precise form of H is subject to future experiments. However, (4) captures the essential elements while allows a theoretical treatment.
For convenience, we transform to a spin language, with the spin defined as In order to simplify the mathematics without loss of the essence, suppose V 1 = −V 0 . Therefore we obtain the following simplified Hamiltonian with We call this model, defined by (6) and (7), floating Ising model with a quenched random field at each site. Note that it is not a diluted Ising model, since n i is not a quenched disorder, but is a dynamical variable on the same footing as S i . It should be mentioned that a cell signalling process often involves different and largely separated time scales. On a much longer time scale the degrees of signalling and clustering evolve after the stimulus, till they return to the pre-stimulus levels. So there is a piece of short-time scale physics, and there is a piece of long-time scale physics. Here it suffices to discuss the the short-time scale physics, i.e. the quasi-equilibrium part of the theory. With an incorporation of the feedback, the long-time scale behavior is similar to that described in Ref. [3,4]. With the adaptation process taken into account, the ligand concentration above should be replaced as the change of the ligand concentration. However, as studied in [3,4], perfect adaptation is equivalent to no ligand binding.
Guo and Levine studied the clustering of receptors under the assumption that the interaction between neighbors depends on whether the receptors are liganded [20], while the signalling problem was not considered. As described in our model, We think that it is more likely that the interaction depends on the conformations of the receptors, rather than whether the receptors are liganded. In addition to the consistency with the situation of chemotaxis [2][3][4], this is also supported by the strong experimental evidence that oligomerization is stabilized by receptor-receptor interaction promoted by conformational changes of the receptors [11][12][13][14]18,19].

MEAN-FIELD CALCULATION OF SIGNALLING
The activity of the system is i n i V i , where the summation is over all the lattice sites, . . . represents ensemble average, i.e. average over the probability distribution of all possible configurations of the system. It is straightforward to write it as ( where m = n i S i is the average spin per site, I is the number of lattice sites. The signalling can be characterized by the change of W when the occupancy is changed from Therefore we set out to calculate m as a function of c, using grand canonical ensemble approach. The grand partition function is where µ is the chemical potential, H is as given in (6). Using Hubbard-Stratonovich transformation, we obtain Based on ln Z = lim n→0 (Z n − 1)/n, the replica method yields where α = 1, · · · , n denote the replicas. Averaging ln Z over the B i distribution (7), one obtains where, for notation convenience, we have set c 1 = c, and c 2 = 1 − c.
The mean field value of φ α i can be obtained by deciding the saddle point of the integrand in Eq. (11), for which one can obtain where Φ n is the saddle point value of φ α i , corresponding to n replicas, f k = 2+e βµ+(Φn+βB k ) + e βµ−(Φn+βB k ) , g k = e βµ+(Φn+βB k ) − e βµ−(Φn+βB k ) , (k = 1, 2). Taking n → 0 limit, we have is the mean-field value of the Hubbard-Stratonovich field. The saddle point approximation implies that m, as a measure of the activity, is calculated in the following way. For given where Φ is the mean-field value of the Hubbard-Stratonovich field, as given in (13). Therefore Φ = βνJm. Eq. (13) can thus be written as The average receptor number is N = ∂ln Z/∂(βµ), and can be evaluated as c k e βµ+βνJm+βB k + e βµ−βνJm−βB k 2 + e βµ+βνJm+βB k + e βµ−βνJm−βB k .
(15) and (16) can also be obtained in a simple mean field approach regarding H − µN as an effective Hamiltonian, and considering four possibilities of the state (n i , S i ) at site i, subject to an effective field νJm + B i .
Hence we have obtained an analytical expression for the signalling, as a statistical average quantity. If the receptor number is conserved in the system, (16) gives µ as a function of the receptor number.
Sensitivity of the signalling to ligand concentration is given by ∂m/∂c, which can be arbitrarily large.

CLUSTERING
The conformation-dependent interaction not only enhances signalling sensitivity, it is also responsible for clustering. The first term of the Hamiltonian (6) implies that, in order to minimize the Hamiltonian, the receptors tend to aggregate together to maximize the number of nonzero n i n j with S i = S j for neighboring ij . Therefore receptors with the same conformational state tend to cluster. However, because of thermal fluctuation, they cannot all cluster together, since there is a probability distribution for various configurations. As an illustration of the situation, a snapshot obtained in a Monte Carlo simulation is shown in Fig. 1.
The second term in (6) determines how ligand binding affects the clustering situation. It can be seen that U determines the bias of the distribution of the receptor state {S i } between 1 and −1 when there is no ligand binding (H i = 0). That is, if U = 0, for not-too-strong interaction compared with the temperature (βJ) 2 , there is an equal number of receptors with spin 1 and those with −1. If U > 0, there are more receptors with spin −1. U also determines whether ligand binding enhances or suppresses the clustering. For example, we may consider two typical cases: Case (i), U = H. According to (3), this means that merely a ligand binding of a receptor is enough to change its conformational state. In this case, Case (ii), U = 0. According to (4), this means that in the absence of ligand binding (H i = 0), there is no bias in the state distribution. In this case, These two cases can map to each other with the transformation H → −H and c → 1 − c.
In case (i), since B 2 < 0, in the absence of ligand, all S i tend to be close to −1. Ligand binding causes more S i to be 1. Because receptors with the same value of S i tend to cluster, those with different S i tend to be disconnected, the conclusion is that ligand binding suppresses clustering.
In case (ii), if there is no ligand binding, there is no biasing field, so there are equal probabilities for S i to be 1 and −1, consequently the clustering is minimized. When there is ligand binding, the clustering is enhanced.
For other values of U, i.e. U = 0 and U = H, we have the following general picture. Without ligand binding, there is a uniform field B 2 at every site. In the presence of ligand binding, there is a random field as given by (7). With a rough mean field estimation, one may see that whether the clustering is enhanced depends on whether (1 − c)B 2 + cB 1 has larger absolute value than |B 2 |. Thus only if H > 2U/c, is the clustering enhanced by ligand binding. Case (ii) belongs to this region.
The clustering of receptors can be studied quantitatively by defining a clustering correlation function as where the average is over different sites i and different directions of r first, and then over the thermodynamic ensemble, i.e. different possible configurations at the same temperature. n = N /I is the density of receptors on the lattice. If there is no clustering correlation, n i n i+r = n 2 , consequently C(r) = 0. This correlation function measures the deviation from the non-correlating case and allows comparison of situations for different receptor densities. This definition is similar to the two-point correlation function in the study of galaxy clustering. One may also define higher-order correlation functions. We leave for future investigations the analytical calculation of the clustering correlation function, and turn to numerical simulations here.

RESULTS OF MONTE CARLO SIMULATIONS
To investigate the extent of clustering and to calculate the activity, we have done Monte Carlo simulations using the Metropolis algorithm on a square lattice. In the simulations, we conserve the receptor numbers. We have specifically studied Case (ii), from which one may obtain the results for case (i) simply by changing c to 1 − c. The results show strong correlations for small values of r and weak correlations for large values of r. The decay looks like exponential, as expected for a non-critical system. We studied correlation functions for different values of the ligand binding fraction c, with a same receptor density n . See Figs. 2(a) and 2(b) for results under different values of the coupling-noise-ratio βJ. From the correlation function for small values of r, it is clear that the larger the ligand binding fraction c, the larger the correlation. This confirms the above analyses.
Comparing Figs. 2(a) and 2(b), it can be seen that with larger coupling-noise ratio βJ, as in Fig 2(a), the correlation function C(r) is larger at small values of r, while smaller at large values of r, indicating that clustering is stronger. On the other hand, for larger βJ, the decay of C(r) with r is faster, because it is further from criticality.
We also studied the correlation functions for different values of receptor densities n with a same ligand binding fraction c (see Fig. 2(c)). It is shown that the smaller the receptor density, the larger the correlation function. This can be understood, since the larger the density, the less freedom two receptors can approach each other.
In the simulations, we also obtained the activity. The activity W can also be written as ′ i S i , where ′ i represents summation over the receptors instead of the lattice sites. Hence it equals NM, where M is the average activity per receptor. Fig. 3 gives the relation between the average activity per receptor M and the ligand binding occupancy c, for different values of coupling-noise ratio βJ. From the plots, we see that the activity increases with the ligand binding fraction c, with βJ, and with the receptor density n , as consistent with the mean field solution.

SUMMARY AND DISCUSSION
In this paper, we propose a statistical mechanical theory which accounts for clustering and signalling of receptors in a same framework. Clustering and signalling of a network of receptors are treated as statistical average quantities. In our model, which can be reduced to a so-called floating Ising model in a random field, the interaction energy between neighboring receptors depends on their conformational states, therefore through this interaction, the conformational state of one receptor influences those of its neighbors. On the other hand, since the receptors are allowed to move on the membrane, the receptors with a same conformational state tend to cluster together, in order to decrease the total energy of the system. Therefore clustering and signalling are unified as two consequences of the same coupling between receptors.
According to our theory, clustering exists even in absence of the ligand binding. This is consistent with a recent experimental finding [18]. We have studied clustering, based on an appropriate definition of clustering correlation function. Monte Carlo simulations were made. To obtain the situation that ligand binding enhances the clustering, a parameter, namely the threshold value U for the change of the conformational state, which determines the state distribution in the absence of ligand binding, must be within a certain range. If the values of the variable characterizing the two conformational states are symmetric, i.e. with a same magnitude and with opposite signs, then a simple possibility is that this threshold value is zero. We note that it was found experimentally that, in the absence of ligands, the receptors are hindered to cluster by certain inhibitors, which are squeezed out when ligands bind [21]. In such a case, it may be constrained that clustering is always enhanced by ligand binding.
By using a mean field theory based on Hubbard-Stratonovich transformation and replica method, as well as by using Monte Carlo simulations, we also studied the activity as a function of ligand binding fraction c.
This model is also be interesting from the point of the view of statistical mechanics. On the other hand, it is straightforward to make appropriate extensions of this model, for example, to put in more details of the realistic systems. The adaptation can be studied by straightforwardly generalizing our previous approach based on a counteracting field as a feedback from the signal to the field [3]. In this case, both signalling and clustering are adapted through a feedback, on a long time scale.
A simple and direct experimental test of our theory is to examine whether conformational state of a receptor can be changed by ligand binding of its nearest-neighboring receptors. It is interesting to study the forces generated by ligand bindings of a receptor and its neighbors. Parameters in our model need to be measured. The techniques used in studying single molecules may be useful for this subject.  Monte Carlo steps after an initial random configuration. An empty circle represents a receptor with S i = −1 and B i = 0, a filled circle represents a receptor with S i = −1 and B i = B, an empty square represents a receptor with S i = 1 and B i = 0, a filled square represents a receptor with S i = 1 and B i = B. The probability for B i to be B is 0.8. βB = 2, βJ = 0.8. Obviously, increasing βJ enhances clustering. (iv) N = 300, βJ = 1.6 (⊙). In this case, the deviation of M(c = 0) from 0 is spontaneous "magnetization" due to strong coupling, but it is always brought to 0 by the negative feedback on a long time scale [3,4]