Influence of temperature and diffusive entropy on the capture radius of fly-casting binding

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1 . Research Paper. SCIENCE CHINA Physics, Mechanics & Astronomy December 11 Vol. 54 No. 1: 7 4 doi: 1.17/s Influence of temperature and diffusive entropy on the capture radius of fly-casting binding CHANG Le, GUO XinLu, ZHANG Jian, WANG Jun * & WANG Wei * National Laboratory of Solid State Microstructure and Department of Physics, Nanjing University, Nanjing 19, China Received May 19, 11; accepted July 6, 11; published online October 6, 11 The recognition and binding of proteins through the fly-casting mechanism are important biological processes. In this paper, a physical model for fly-casting binding is described based on the capillarity theory for protein chains. It is found that the capture radius for the fly-casting binding process is maximized at the transition temperature at which the free energy of the monomeric extended state of the protein equals that of the folded state. The factors related to the folding barrier or binding affinity do not change the condition needed to realize the optimization for fly-casting processes. These results will aid in the comprehensive understanding of binding processes. protein binding, fly-casting mechanism, protein folding PACS: Ee, Aa, He 1 Introduction The binding between proteins and various substrates is a basic process in biological systems to build functional modules and to transmit various molecular signals [1]. There are diverse binding behaviors and kinetics between various ingredients. The research which explores the universal physical principles has attracted much attention in protein studies [ 7]. Physically, the binding is a dynamic process related to the folding/unfolding of the monomers and the association of the related partners. Such kinds of processes are largely affected by various interactions (such as the monomeric stability and the binding affinity) to stabilize certain conformations and by the environmental conditions (such as the concentration and the temperature) to modulate the balances between various thermodynamic states. The interplay of these factors introduces many mechanisms for the binding process. For example, by emphasizing the importance of the interfacial interaction, the lock-and-key mechanism was established to describe the complementariness of interfaces between the partners after binding [1,]. With more consideration for the conformational entropy, some other mechanisms (such as *Corresponding author ( wangj@nju.edu.cn; wangwei@nju.edu.cn) ContributedbyWANGWei(EditorialBoardMember) induced fit [] and population shift []) are introduced for the cases with conformational changes during the binding process. More recently, some studies suggest that the protein would experience an apparent conformational deformation to accelerate the binding process, known as the fly-casting mechanism [4 7], in which the dynamic features of the binding process are focused on. These mechanisms outline the physical picture for the binding process. As a kinetic theory of protein binding, the fly-casting mechanism is believed to be universal in the recognition of the protein to certain targets, and has been extended to explain the DNA searching processes of DNA-binding proteins [8]. A series of experiments, simulations and theories were set up to illustrate the unfolding process during binding processes [7,9 1]. It is observed that the capture radius (which is defined as the distance beyond which the binding would occur irreversibly) is apparently larger than the gyration radius of a protein monomer. Many factors could influence the capture radius, such as the kinetic barrier for protein unfolding, and the affinity of binding. Recently, the translational entropy and the monomeric stability received more attention in the studies on binding processes. These effects could also strongly perturb the dynamics of protein binding processes. It has been found that the translational entropy could greatly c Science China Press and Springer-Verlag Berlin Heidelberg 11 phys.scichina.com

2 8 Chang L, et al. Sci China Phys Mech Astron December (11) Vol. 54 No. 1 accelerate the binding kinetics [1,14] and the stability of monomers were also essential in determining the kinetic features of binding processes [5,6]. These physical interactions and factors could be easily modulated by the environmental conditions (such as the protein concentration and temperature), and thus new dimensions could be introduced to describe and control the binding process. They are definitely necessary inclusions in a better theory for the binding processes of proteins. However, the effects of these two kinds of physical ingredients in fly-casting processes are not well discussed. Our work focuses on the effects of these factors on the fly-casting process. The understanding of the contributions of these factors would be helpful in establishing a comprehensive picture of the diverse binding phenomena. In this work, a free energy functional model for the binding process is employed. In this model, the capillarity interaction [7,15,16], the temperature and the translational entropy of monomers are considered. The free energies of the protein system in the bound and dissociated states are given. The potential of the mean force for binding is obtained as a function of the distance between the protein and the target through the minimization over all the other degrees of freedom. The capture radius is obtained as the maximum of the binding free energy profile. Comparing with previous models, our capture radii are apparently large and the free energy decays in a logarithmic manner for large protein-to-target distances. These are more consistent with the simulation results. More interestingly, it is found that there is a non-monotonic variation of the capture radii for different temperatures. A maximal capture radius can be reached when there is a balance between the monomeric folded state and the unfolded state (namely, around the monomeric folding temperature). Such a behavior results from the competitions between the monomeric stability and the binding affinity. As a conclusion, this model offers a simple way to understand how the environmental factors affect the fly-casting process. Model and theory During the binding between a protein and its target, the flycasting mechanism describes a process with partially unfolded intermediates. Similar to that which is modeled in previous study [7], the partially bound conformation of the protein is described by three groups (as shown in Figure 1(a)), namely, the bound group, the unbound globule group, and the extended linker group. These three groups are indexed as 1, and, respectively. A certain conformational state can be described with the sizes of these three groups, i.e., N 1, N, and N. Here, the protein size N gives a constraint for these sizes as N 1 + N + N = N. Alternatively, when the protein is dissociated from the target, the protein will fluctuate between its folded state with globule shape and unfolded state which is extended (as shown in Figure 1(b)). Similar to the description for the bound state, the dissociated state can be depicted with the sizes of the globule conformation and of the extended (a) Bound Target (b) Dissociated N 1 N N R Protein Figure 1 The sketch of the typical states, (a) the bound and (b) the dissociated states. The relations between proteins and targets are exemplified. Some quantities describing the sizes of groups and the geometry of system are marked. coils, N and N. Further, as a measurement of the progress of the binding process, the distance between the target and the center of mass of the protein, R,isalsodefined as another coordinate to characterize the geometry of the system. All these quantities establish a simple phase space of the protein system related to binding. Similar to previous studies [7,16], the capillarity approximation is used for the free energy of the protein system. That is, the free energy is proportional to the surface of the system. For the globule conformation, the energy related to the surface can be expressed as F glb (n) = γn, (1) where n is the number of residues in the globule. The factor γ is the penalty for the residue to be exposed in solvent and resembles the effect of surface extension, and n measures the surface size of a compact globule with n residues. For extended conformations the residues are largely solvated. The conformational entropy of the residues is another key determinant for the free energy in the extended state. With the approximation of the free rotation model, the free energy of the extended state can be expressed as F ext (n) = n(γ τts ), () in which n is the number of residues in the extended state, s is the average entropic gain of each residue when becoming coiled, and τ is a factor measuring the contribution of the entropy. For simplicity, this term can be reduced to F ext = γ n with γ = γ τts <γ. Note that the strength-like quantity γ is temperature dependent. Under physiological conditions, proteins can jump between the compact globule state and the loose extended state (known as protein folding/unfolding) thermodynamically. At the folding transition temperature T f of a protein with n residues, we have the relationship F glb (n) = F ext (n). It is not difficult to get the barrier

3 Chang L, et al. Sci China Phys Mech Astron December (11) Vol. 54 No. 1 9 B = 4/7γn [15], which gives the height of the free energy barrier to folding/unfolding at the temperature T = T f. Besides the interactions inside proteins, the binding interaction is another necessary ingredient for binding processes. Following a previous study [7], the binding free energy is defined to be proportional to the size of the bound part, F bind (n) = δn, () here δ measures the strength of binding. With these kinds of free energies, the free energy for protein systems can be easily evaluated. For the bound state, we have the condition N 1 >. The corresponding free energy would take the form F b = F bind (N 1 ) + F glb (N 1 ) + F glb (N ) + F ext (N ). (4) This is a summation of the free energies of three groups. Considering the case that the linker part is very flexible and almost fully stretched, the geometry of this partially bound state can determined. That is, the distance R between the center of mass and the target could be found to be R = 1 ( an N N + a N + 1 ) N 1 N N, (5) where the length a represents the distance between residues and is used as the unit of length. This gives another relationship for the lengths N 1 N for a certain coordinate R. Combining this with the constraint N = N 1 + N + N, the length N 1 can be represented as a function of N and R as N 1 = N N N 4 + NR a. (6) Therefore, the free energy can be derived as [ ] F b (N, R) = γ N 1 (N, R) + N (γ + δ)n 1 (N, R) δn, (7) here, < N < (RN/a) 4 since all the lengths are positive, namely, N 1 > andn >. Assuming that the conformational change is faster than the diffusive search for a target, the free energy can be minimized with respect to the quantity N to obtain the potential of mean force (PMF) along the coordinate R. Since the differentiation F b / N >, the free energy minimum is reached at N =. As a result, the PMF in the bound state is F b (R) = γ N NR a + ( γ + δ ) NR a δn. (8) Similar to that of the bound state, the free energy of the dissociated state can also be described as a summation of the terms in eqs. (1) and (). Before binding, the protein is free to move without geometric constraints. The translation entropy S t should be considered for this case. For a target molecule, this entropy can be expressed as S t k B ln ( 4πR ). The touching point between the protein and target R = R = 1 N 1 gives the boundary of this entropy, where the entropy would cease to be zero. Thus, the translational entropy would be S t = k B (ln R ln R ). With these considerations, the free energy can be written as F d = F glb (N ) + F ext (N N ) TS t = γn + γ (N N ) lnr + lnr. (9) Physically, there are generally two kinds of states, folded globule and extended coil. The protein fluctuates between these two states. These two states can be identifiedas the two minima for a certain R. These minima can be determined by the condition F d N = or the boundary (such as N = ) which satisfies the condition of a local minimum. Typically, these two minima are located at N = (unfolded) and N = N (folded), respectively. The corresponding free energy would be F U = γ N and F F = γn, respectively. The global minimum of these two states is an appropriate representative for the dissociated state. That is F d = min [ ] γ N,γN lnr + lnr, (1) here, the function min[a, b] gives the smaller value of the quantities a and b. During the binding process, the protein will generally adopt the state with minimal free energy thermodynamically. Therefore, the observable free energy of the protein system will be the minimum of the free energies F b and F d. If F b (R) < F d (R), the system is in the bound state; otherwise, it is in the dissociated state. The free energy profile is a segmented function composed of the F b and F d. To ease the comparison between the free energy profiles in various conditions (namely with various parameters), all the free energy profiles, F b and F d, are shifted by a value corresponding to their own folded state, Δ= F F [15,16]. This kind of shifting operation does not affect the shapes of the free energy profiles, and will not introduce any non-physical biases. Obviously, at the transition temperature T = T f,wehave the condition γ N = γn. The free energy difference between the binding dissociated states can be deduced as F b F d (R ) = γ ( N 1 N 1 N1 ) + γ ( N N 1 N ). This reproduces the case of ɛ = in the previous work [7]. This validates our model for the description of binding processes. Furthermore, through the introduction of some new factors, our model goes beyond the previous model and can be used to describe the effects of translational entropy and temperature. Results and discussion.1 The PMF profiles at various temperatures The profiles of potential of mean force F(R) for various val-

4 4 Chang L, et al. Sci China Phys Mech Astron December (11) Vol. 54 No. 1 ues of γ are calculated based on the above free energy functionals (as shown in Figure ). There are generally two parts for each PMF. For example, for the case with γ = γn 1, the PMF grows with increasing R when R < R c,andthe PMF decreases for larger R. The free energies of these two parts are determined by the terms F b and F d, respectively. Thus, these two parts are related to the bound and dissociated states, respectively. That is, when the protein is far away from the target, the kinetics of the protein are driven by the free energy F d. For this case, the free energy decays in a logarithmic manner, and the translational entropy related to the diffusive search takes the most important role. On the other hand, when the protein goes close enough to the target, the attraction between protein and target could overcome the translational entropy and realize the binding behavior. The edge between these two parts has the maximum of PMF, which marks the transitional state of the process. The corresponding distance R = R c acts as the boundary between binding and dissociation, and is defined here as the capture radius of the binding process. Clearly, we have the relation that F b (R) = F d (R) at the distance R = R c. This provides a criterion to estimate the capture radius under various situations including temperatures, binding affinities and so on, which are generally modulated by various parameters in our model (such as γ,δ). Moreover, the size of the critical distance R c is also larger comparing with previous study [7]. This demonstrates the necessity of the translation entropy for a better description of the binding process. This kind of behavior is more consistent with the simulation results [17] (as shown in the inset of Figure ). All the PMFs have an uphill behavior when approaching the target. This comes from the reduction of F/k B T f R c γ =.8 γn 1/ γ =.9 γn 1/ γ =1. γn 1/ γ =1.5 γn 1/ γ =. γnt 1/ PMF (kcal. mol 1 )..1.. R/Na..1.. R/Na Figure (Color online) The PMF along the coordinate R. Several cases with different ratios between γ and γ are given. The barrier is fixed to B = 5k B T f. The affinity is set to 1k B T f. The length of the protein chain is N = 1. The inset gives free energy curves from molecular dynamic simulations adapted from our submitted work [17]. The model system is the dimer of arc repressor (with PDB code 1ARR). The simulations are carried out with a Go-like model as well as umbrella sampling. The PMF is calculated using the WHAM algorithm [18 ]. 4 4 the translational entropy. The PMF from the simulation results also has a peak, and on the two sides of the peak, the model protein behaves differently, either binding or dissociating. This is consistent with our assignment for the capture radius. The balance between attractive energy and entropy in our model is also observed in the simulations [17]. This further supports our model. It is found that the growing part of the PMF in the region with small R becomes steeper for larger γ and the decreasing phase keep a similar shape except that the conformational free energy F dc = min [ ] γ,γn +Δvaries for different γ. When the temperature-related factor γ >γn 1 (namely at temperatures lower than T f ), the decreasing phases have the same F dc.differently, the F dc decreases gradually as the factor γ becomes smaller (namely at higher temperatures than T f ). The different behaviors of these two phases of the PMF influences the sizes of the capture radius R c. For the cases where T < T f, the capture radius R c becomes smaller for larger γ. This is a direct result of the condition that the increasing phase becomes steeper, which indicates that the attractive basin in the free energy landscape related to binding becomes steeper, while the dissociated basin is not seriously modified. Considering the detailed relation between the folded and extended states of a dissociated protein, the deepening of the binding basin mainly comes from the stabilization of the folded structures by the capillarity interaction. In other words, following the increase in γ, the protein is constrained by the stronger capillarity interaction. The protein is clearly not easy to deform and the fly-casting process is thus suppressed. On the other hand, for the cases where T > T f, the capture radius R c becomes smaller for lower γ. The shifting of the increasing phase with smaller F dc is the main reason. This corresponds to the stabilization of the extended structures for smaller γ. This kind of variation in free energy makes the protein prefer to be in the dissociated state. Only when the attraction is strong enough, binding occurs. That is, the transition state to binding is rather close to the target. This produces small values of R c, which behaves differently from the case with T > T f. It is interesting to point out that the intrinsically disordered proteins (IDP) (See the reviews [1 4] and references therein) generally correspond to the cases with T < T f, namely, they have small folding temperatures and are extended at room temperature. Our studies suggest that lower temperatures would enlarge the capture radius and accelerate the binding process. This is consistent with experimental observations [1,5]. Together with these two kinds of trends for R c, there is a maximum of R c when T = T f. This marks the competition between the stability of binding and possible flexibility of protein monomers. Around the monomeric folding transition, the largest capture radius of binding would make the fly-casting processes easier to be observed in experiments. Based on the above analyses, it can be concluded that the capture of a protein can be modulated by the basic features of the protein through environmental conditions. The competition between folded and

5 Chang L, et al. Sci China Phys Mech Astron December (11) Vol. 54 No extended structures would produce complex behavior for the binding process.. The interplay of various factors on capture radius As mentioned above, a maximum of the capture radius for the binding process can be reached at the folding temperature. In that situation, the binding is optimized with the largest range for the recognition. Is this kind of optimization for the binding a universal feature of the binding process? What would be the interplay between the environmental factors (such as the temperature) and the intrinsic interactions of the proteintarget system? Answers to these questions would be valuable for understanding and controlling the binding processes, and will be discussed in this section. As shown in the previous theory [7], the capture radius can be affected by some factors related to the interaction properties of the protein and of the complex, such as the folding barrier and binding affinity, besides the environmental factors mentioned above. The interplay between these factors and the temperature on the capture radius have been evaluated (as shown in Figures and 4). In Figure, the dependence of the capture radius on both the ratio of the free energies of the extended and folded states, r = γ N/γN, and the binding affinity δ is given. Similar to the previous studies, the capture radius will increase with an increase of the affinity δ for any temperature. The increase rates of R c vary for various temperatures. Around the ratio r = 1, the capture radius increases more prominently. This implies that the binding is more sensitive in the situation with r = 1 (namely at the transition temperature). While for a certain affinity δ, the maximization of the capture radius around γ N/γN = 1 is always observed. This indicates that the binding affinity does not qualitatively affect the optimization of the capture radius by the temperature. At the same time, the variations of the capture radii for different r and folding barrier B are also calculated as shown in Figure 4. A similar relationship. 1.5 R c /Na. γ N/ γn / δn/k B T f R c /Na. Figure 4 (Color online) The dependence of the capture radius R c on the ratio r and the barrier height B. Inthisfigure, the binding affinity is fixed at 1k B T. The length of the protein chain is N = 1. as Figure is also observed, except that large capture radii can be obtained only for small B of barriers. This is also consistent with the previous study [7]. The sensitivity of the capture radius around r = 1 is also observed. These results imply that the binding strength seems to be an orthogonal factor with respect to the temperature. This suggests the possibility to modulate the binding dynamics through only the environmental factors such as temperature. 4 Conclusions The fly-casting binding mechanism is a dynamic process which can be modulated by various interaction and environmental factors. In this work, based on a theoretical model including the temperature and the translational entropy, it is found that there is an optimal capture radius for fly-casting binding at the folding transition temperature. The maximal value of the capture radius does not depend on the factors related to interactions (such as the folding barrier and binding affinity). This reflects the different effects of the factors related to the interactions or environments. These results will be helpful to further model and control the binding process of proteins..1 γ N/ γn / This work was supported by the National Basic Research Program of China (Grant No. 7CB81486), the National Natural Science Foundation of China (Grant Nos , 184 and ) and the Natural Science Foundation of Jiangsu Province (Grant No. BK98) B/k B T f Figure (Color online) The dependence of the capture radius R c on the ratio r = γ N/γN and the binding affinity δn. Inthisfigure, the height of the folding barrier is fixed at 5k B T. The length of the protein chain is N = 1. 1 Lodish H, Berk A, Matsudaira P, et al. Molecular Cell Biology. 5th ed. New York: W H Freeman, Koshland D E. The key-lock theory and the induced fit theory. Angew Chem Int Ed, 1994, : Bosshard H R. Molecular recognition by induced fit: How fit is the concept? Physiol, 1, 16:

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