On the dynamics of molecular conformation

Abstract

Understanding the mechanism of fast transitions between conformed states of large biomolecules is central to reconciling the dichotomy between the relatively high speed of metabolic processes and slow (random-walk based) estimates on the speed of biomolecular processes. Here we use the dynamical systems approach to suggest that the reduced time of transition between different conformations is due to features of the dynamics of molecules that are a consequence of their structural features. Long-range and local effects both play a role. Long-range molecular forces account for the robustness of final states and nonlinear processes that channel localized, bounded disturbances into collective, modal motions. Local interconnections provide fast transition dynamics. These properties are shared by a class of networked systems with strong local interconnections and long-range nonlinear forces that thus exhibit flexibility and robustness at the same time.

Biomolecules typically have a large number of degrees of freedom. This fact would imply that the dynamics of such a biomolecule is chaotic (1) and in turn that transition times between different states can be estimated by using the random-walk model and are thus enormously long (2). However, many biomolecules have the ability to move rapidly and coherently between different conformations (3, 4). There are a number of approaches to this problem that take into account the large-scale vibrational dynamics of biomolecules, e.g., the normal mode analysis (NMA; or elastic network models) (4–7) and the protein quake concept described by Ansari et al. (8). Although dynamical systems theory contributed to the understanding of various aspects of molecular and chemical motion (9–13), it has not been used to provide a coherent mathematical explanation that encompasses all of the phenomena observed in refs. 4–8. Taking the phase–space perspective of dynamical systems theory, we suggest here that several well characterized dynamical processes govern fast transitions between conformed states.

Model

Consider a simple model of a class of macromolecules that exhibit a strong circular backbone structure. Attached to the backbone are side chains that are represented as a single mass on a pendulum attached to the backbone (see Fig. 1). These side chains are able to interact with other molecules or other side chains of the same molecule by forming hydrogen bonds. The model that we study is a simplified representation of a macromolecule, where only torsional degrees of freedom (those degrees that contribute to rotations around the backbone) are taken into account. Such models have been used, for example, for modeling of the coarse-grained dynamics of the DNA molecule (11, 14) and minimalist models of protein folding (15, 16). Consider a situation in which there are two backbones with side chains facing each other (see Fig. 1 for a graphical description), but one of the strands with its side chains is held immobilized, a choice sometimes made in molecular dynamics simulations. The other strand's backbone is also immobilized, but its side chains are allowed to move in the plane orthogonal to the backbone and feel force due to the Morse potential interaction with the side chains of the other strand. The side chains are coupled to nearest neighbors by torsional spring forces. The attractive–repulsive force is derived from the Morse potential, and it does not depend on the position of the side chain with respect to its nearest neighbors. In this sense, the effect of the attractive–repulsive forces and the effect of dihedral angle dynamics are clearly distinguished in this model.

Fig. 1.

A schematic representation of the example system. It consists of two backbones with side chains facing each other. One of the strands with its side chains is held immobilized. The neighboring side chains interact with a torsional potential. The top side chain interacts with the bottom (immobilized) side chain through the attractive–repulsive Morse potential. Note that in the simulations, the ends of the backbone are connected, making it circular.

Equations of Motion

The equations of motion for the kth side chain of the top strand, describing changes in angle θ around the backbone, read

where k = 1, …, N, θ_N+1 = θ₁, θ₀ = θ_N (we assume the backbone is circular), m is the mass of the side chain, h is the pendulum (side chain) length, x₀ is the equilibrium distance for hydrogen bonds, a_d is the decay coefficient for the hydrogen bond force, and D_b is the Morse potential amplitude. Changing the dihedral angle potential from harmonic to various other forms used in the literature does not affect the qualitative nature of the results.

By introducing the new time scale τ = $\sqrt{\raisebox{1ex}{$\left(m{h}^{2}t\right)$}\!\left/ \!\raisebox{-1ex}{$S$}\right.}$ and keeping the same “double dot” notation for the second derivative with respect to τ, we get

The number L defined by

measures the ratio between the strength of nearest-neighbor interactions and local (Morse potential) dynamics of side chains. If this number is large, the nearest-neighbor interaction is much stronger than local dynamics. However, we note that if all of the elements of the lattice are in the same state, then it is the repulsive–attractive term that determines the motion. The Hamiltonian for the above system reads

All of the simulations in the paper are done with N = 30, L = 10, h = 10, a_d = 0.7, and x₀ = 3, except for the contour plot of the single-component Hamiltonian, described next. All of the simulations were done with a matlab (MathWorks, Natick, MA) Runge-Kutta routine, which conserved energy very well for the time scale of simulation presented here.

Behavior of the Model

The system described above has two global energy minima. In fact, the phase space of a single component of the system, when a side chain is isolated from its neighbors on the same strand, is shown in Fig. 2 in the form of the contour plot of its Hamiltonian

In Fig. 2, a_d = 0.3 for clearer presentation (contours of the Hamiltonian get very dense in the repulsive region for larger decay coefficients a_d). This phase space, which is parameterized by the angle θ, and the associated angular velocity v, has two stable equilibria, positioned symmetrically with respect to θ = 0. There are also two unstable equilibria, at θ = 0 and θ = π. Upon coupling of N such components, the state that occurs when all of them reside in the left equilibrium state is one of the two global energy minima, the other one occurring when all of the side chains reside in the right equilibrium state. We will call these equilibria “exact conformed states.”

Fig. 2.

The phase space of a single degree of freedom oscillator with the Morse potential, whose Hamiltonian is shown in Eq. 4. In this figure, a_d = 0.3 for clearer presentation. This phase space, which is parameterized by the angle θ and the associated angular velocity v, has two stable equilibria, positioned symmetrically with respect to θ = 0. There are also two unstable equilibria, at θ = 0 and θ = π. Each of the stable equilibria is surrounded by a family of periodic orbits. We call these two zones “conformational sacks.” There is another, high-energy family of periodic orbits surrounding the conformational sacks and separated from them by the separatrices that connect the fixed point at π to itself.

A remarkable feature of this model system is its capability of converting localized disturbances into large-scale coherent motion. Consider the sequence of snapshots in Fig. 3 obtained by simulating the system of 30 side chains.

Fig. 3.

Sequence of system configurations in transition between the two global energy minima obtained by simulation of the equations of motion (2). (a) At the initial time, one of the free side chains is forced into the repulsive region of the hydrogen bond. (b–d) The energy so acquired first gets distributed to other side chains sequentially on both sides of the localized perturbation. (e–h) However, when the number of side chains in motion is large enough (e), they self-organize and perform a synchronized rotational motion around the backbone of the strand (f). Finally, they perform a flip to the other side of the strand (g), changing the molecule's conformation. They then oscillate in the potential well on the other side of the strand (h) until another sufficiently strong disturbance induces a new change of conformation.

At the initial time (Fig. 3a), one of the free side chains is forced into the repulsive region of the Morse potential. The energy so acquired first gets distributed to other side chains sequentially on both sides of the localized perturbation (Fig. 3 b–d). However, when the number of side chains in motion is large enough (Fig. 3e), they self-organize and perform synchronized rotational motion around the backbone of the strand (Fig. 3f). Finally, they perform a flip to the other side of the strand (Fig. 3g), thus changing the molecule's conformation. They then oscillate in the potential well on the other side of the strand (Fig. 3h) until another sufficiently strong disturbance induces a new change of conformation. In some cases it takes a couple of back-and-forth motions between the two conformed regions before the system is “captured” to stay in one of them. The initial disturbance needs to exceed an energy threshold for the transition to occur (see details in the next section). The addition of small dissipation and noise terms (Langevin dynamics) does not change the qualitative nature of the phenomenon, only the energy threshold of the initial perturbation needed for transition. Using symplectic integrators does not qualitatively change the results either, and quantitative differences are small.

It is worth pointing out several features of the phenomenon presented in Fig. 3. (i) If the neighbor-to-neighbor torsional coupling is not present, the synchronized motion does not occur. In that case, the side chains execute dynamics corresponding to the Morse potential interaction with the side chain of the other, immobilized strand and move along the contours shown in the phase–space portrait in Fig. 2. (ii) For pure torsional coupling, when no Morse potential interaction with the other strand is included, the synchronized motion also does not occur, except only for a special set of initial conditions that are already “synchronized”: setting all of the side chains to the same angular position initially, and giving them the same initial velocity leads to spin of the molecule around the backbone. Thus, both the dihedral angle dynamics and Morse potential interactions are needed for the observed transition from one state to another. The essentials of the mechanism of transition can be understood by using the dynamical systems theory of motion close to internal resonance (17). In particular, consider the average angle of the system defined by

We denote the average angular velocity of the side chains by v. It can be shown that, provided at time 0 all of the angles and velocities of the individual side chains are the same, the angles and velocities of side chains remain synchronized (i.e., the same) for all time. In fact, every energy surface of the system has at least one such synchronized trajectory. These trajectories execute the average angle and average velocity dynamics that is the same as that represented in the single-component phase portrait shown in Fig. 2. Given this “synchronized by initial conditions” picture, it is instructive to compare that single side-chain phase space with the trajectory of the whole system starting from the “nonsynchronized” initial condition shown in Fig. 3a projected onto the (θ, v) plane shown in Fig. 4. The system shows very interesting behavior in this projection. It essentially follows the contours of the constant Hamiltonian around the global energy minima, with small “kicks” taking it off one of the contours onto another.

Fig. 4.

A trajectory of the whole system starting from initial conditions depicted in Fig. 3a, projected onto the (θ, v) plane. The system essentially follows the contours of the constant Hamiltonian around the global energy minima, with small kicks taking it off one of the contours onto another. These kicks happen in the so-called resonance zone, close to the surface v = 0 in the phase space. Once the system exits one of the conformational sacks, it follows the trajectory of constant v and uniform θ, until it enters the other conformational sack.

These kicks happen in the so-called resonance zone (17), close to the surface v = 0 in the phase space. There are two essentially different parts of the phase space depicted in Fig. 2: the resonance zone that includes trajectories close to the each of the two exact conformed state and the high-energy zone, outside of the resonance zone. When the system is in one of the exact conformed states and a localized perturbation, shown in Fig. 3a, is added, it starts oscillating around the exact conformed states and performs almost integrable dynamics within the resonance zone. This almost-integrable dynamics along the contours of the projected Hamiltonian is interrupted by kicks that happen during passage through the thin zone in which the Morse potential force is close to zero, and linear torsional coupling dominates. At a certain moment, release from resonance happens, and the molecule transitions by crossing the separatrix indicated in Fig. 2 to the collective dynamics mode described previously. In this collective dynamics mode, the system follows a trajectory whose mean is one of the synchronized trajectories in which all of the angles and velocities are equal. Note that this mode of motion is essentially a rigid mode. This finding links the theory described here with the theory of normal mode analysis (NMA; or elastic network models) (4–7) in the sense of the observation of Delarue and Sanejouand (4) that “biologically relevant large amplitude motions are essentially rigid body motions of such domains.” The added ingredients introduced here are the necessity of repulsive–attractive interactions for the transformation of the localized initial disturbances to collective modes and the necessity of pathways in the phase space that correspond to collective modes of NMA and connect the conformation zones. These collective modes serve as “conformational superhighways” of molecular dynamics.

The last phase in the process of transition is that of the molecule getting entrained in the resonance zone, around the other exact conformed state. We could say that the whole transition process is in fact “resonance-enabled.”

Remarkably, the synchronization described here occurs because of a resonance phenomenon in a conservative, locally coupled system and does not depend on the dissipativity of the dynamics (18) or global coupling in conservative systems (19). In interesting studies of Morita and Kaneko (20, 21) of relaxation processes in both globally and locally coupled conservative systems, collective behavior (in the sense of energy accumulation in a particular mode) was observed, leading to large departures from equilibrium before the final relaxation process. The phenomena that we describe here also exhibit collective motion before relaxation. The class of Hamiltonians that exhibits the particular relaxation process described here is different, however, with the requirement of strong local interactions and weaker global interactions leading to the specific phenomenon that we describe.

Quantitative Analysis of the Model

We performed a suite of numerical simulations to better understand properties of the conformational transition described in the previous section. In Fig. 5, we show results of an investigation into the amount of initial energy necessary for conformational change and its dependence on the structure of the perturbations and the number of side chains involved in motion. Note that the energy level of the saddle located at θ = π in the case of 30 side chains is 0.0214. In Fig. 5a we plot time to reach that saddle vs. initial perturbation energy, which indicates the amount of time it takes to transition between the two conformational equilibria. All of the data are obtained with N = 30 side chains. Four different initial configurations are considered. In all of them, initial velocity of all side chains is zero. Open shapes indicate cases in which transition was not achieved. Black squares are data for initial perturbation for which half of the side chains are displaced from the equilibrium on the repulsive side of the potential, and half are in equilibrium position, corresponding roughly to the first mode of oscillation of the linear part of the system. Among all of the nonuniform perturbations studied here, this initial perturbation requires the least initial energy to perform conformational transition. This energy is an order of magnitude smaller than that of a random perturbation, the data for which are in green diamonds on the same plot. This feature is easily seen from Fig. 5b where minimum initial energy levels for various perturbations are shown. The black bar corresponds to the first-mode perturbation, the light blue to random energy perturbation. It is interesting to note that even displacing a single side chain with all of the others in equilibrium (the data for which are given in red circles in Fig. 5a and the dynamics of which was shown in Fig. 3) does better, at the same initial energy, than the random perturbation in terms of how much time it takes to reach the saddle point. The initial energy necessary to induce conformational transition with a single side chain perturbed is shown as a dark blue bar in Fig. 5b and is close to that required by a random perturbation. It is also interesting that structured perturbations that do not correspond to low oscillation modes of the linear part of the system do not do very well in terms of minimum initial energy for conformational flip: the data for time-to-saddle vs. initial energy for a sinusoidal perturbation of the shortest spatial wavelength around the left conformed state are shown in Fig. 5a in blue squares, and its minimum initial energy is shown as a yellow bar in Fig. 5b. These results are quite robust over a range of N: in Fig. 5c we show time-to-saddle vs. initial energy for N = 4, 10, 30, 50, and 100 for the first-mode initial condition described above, and we see that the behavior is qualitatively very similar.

Fig. 5.

Numerical results for time and energy needed for conformational change. (a) Time to conformational saddle plotted against initial perturbation energy for both structured energy and random energy perturbations. Open shapes represent cases in which conformational change was not observed. Black squares represent first-mode perturbation, red represents single-site perturbation, green indicates random perturbation, and blue represents sinusoidal perturbation of high spatial frequency. (b) Minimum initial energy needed for conformational transition for structured and random energy perturbations. Blue bar, single site perturbation; light blue bar, random perturbation; yellow bar, sinusoidal perturbation; black bar, first mode perturbation. (c) Time to conformational change plotted against initial energy perturbation for the structured (first mode) perturbation for various numbers of side chains N. Data for N = 4 are in red; N = 10, green diamonds; N = 30, blue; N = 50, black; N = 100, green squares. (d) Time to conformational change plotted against initial energy perturbation for the structured (first mode) perturbation with parameter L randomly distributed over side chains. Variances for L are 0 (red), 6.27 (green), 10.85 (blue), and 16.20 (black).

The observed conformational behavior is quite robust to changes in parameter L. We varied individual values L_i, i = 1, …, N, for N = 30 of L for side chains to obtain time-to-saddle vs. initial energy data shown in Fig. 5d for variance in L being 0 (red circles), 6.27 (green diamonds), 10.85 (blue squares), and 16.20 (black squares). It is seen that the observed times-to-saddle begin deviating only at the large variance or >16 where the standard deviation is 40% of the mean value of L = 10.

The results described in this section show both that structured perturbations are better at inducing conformational dynamics in the present model than random perturbations are and that such a conclusion holds robustly for a range of side-chain numbers N and distributions of parameter values L_i, i = 1, …, N.

Generalization

There are three elements of this conformational transition picture that we consider universal. (i) The first is the existence of exact conformed states in the phase space and relatively stable zones of integrable motion around them, that we call conformational sacks. These sacks are governed by effects of attractive/repulsive molecular forces and can be detected by using dynamical systems techniques such as the methods for detection of almost-invariant sets (10, 12). (ii) Next is the existence of rapid collective motions between conformational sacks provided by the elastic normal modes of the molecule. Because these types of motion are preferred by the molecule, the volume of the phase space explored by a biomolecule is minute compared with the total available volume, thus showing that the assumption of equal phase–space probabilities that is at the heart of Levinthal argument (2) is not correct. (iii) Targeted localized perturbations can bring about conformational transition, by pushing the two adjacent side chains or sets of side chains into the repulsive domain of their interaction, thus supplying potential energy that ultimately gets converted, through the process described above, to kinetic energy of collective-mode-mediated transition. This result leads to the conclusion that biological macromolecules have “hot buttons,” sites that, when acted on in the right way, lead to global conformational change by means of the mechanism of resonance, that involves transition across separating manifolds in phase space. This finding is closely related and provides a mathematical description of the phenomenon of “protein quakes” described by Ansari et al. (8). The schematic of the overall process is shown in Fig. 6.

Fig. 6.

Schematic representation of conformational dynamics in phase space. Areas I and II are conformational sacks. The transition pathway between them is determined by the elastic normal modes of the molecule. Exits and entrances into conformational sacks are controlled by the resonance dynamics. The transition is initiated by an external, local perturbation at a hot button site on the molecule.

There is a large set of N degree-of-freedom Hamiltonians that are currently used in studies of protein dynamics that satisfy the requirements for the conformational transition that we have described here. Specifically, these Hamiltonians are of the following form:

where p, q are the generalized momentum vector p = (p₁, …, p_N) and generalized position vector q = (q₁, …, q_N). The term N is the nonlinear term coming from nonlocal van der Waals interactions whereas L(q, p) is the bonded interaction term that contains linear and Fourier terms (see, e.g., ref. 22). In addition to the model presented above, we have studied a variety of models of this form that exhibit similar phenomena to the one presented here. For example, a linear model with local harmonic interaction and nonlocal Lennard–Jones interaction that has many conformed equilibria but exhibits transition between an equilibrium in which the distances between all of the side chains are the same and large (denatured) to the equilibrium in which all of the distances are the same and small (conformed). The reason for this transition is the existence of a normal mode in L that supports it in the way described above. This type of system can serve as a rudimentary model of an α-helix secondary structure (22), and its detailed study will be presented elsewhere.

It is interesting to note that the possibility of using the mechanism of resonance for achieving large changes in dynamical systems using small perturbations has been discussed (23), and it was argued that existence of internal resonances of the type described above is a necessary condition for inducing large changes with small perturbations. Methods for computing rates of transition across separatrices in high-dimensional Hamiltonian systems have been developed recently (13) and could be used to further elucidate the motion across separating manifolds.

Conclusion

We argue here that there are specific structural features of biological macromolecules that lead to their ability to quickly switch between different conformations. The picture that arises is that of a structure with a reduced number of degrees of freedom, with viable conformations being only those that are accessible through fast collective motions of the molecule. These fast collective motions are enabled by the local structure, which favors certain modes of motion. The dynamical systems mechanism that allows for efficient use of such motions is that of resonance. The conformations are decided by long-range effects of attractive and repulsive forces. The switch between two different conformations can be induced by local perturbations at a specific site in the molecule. Another intriguing related fact is that the dynamics of the molecule is not chaotic (ergodic) but in fact is piecewise integrable (integrable within the resonance zone and with integrable average outside of it; chaotic in the transition). In the case when certain modes of oscillation are overdamped, it is only the underdamped modes that can provide the conformational transition mechanism presented here.

Biological macromolecules are oscillator networks with a large number of degrees of freedom. They are designed to have functionality that is robust to a wide variety of perturbations but also to easily transform between different configurations when necessary. The essential structural features that allow for this capability are strong local interconnections (neighbor to neighbor) and global interconnections that are weak compared with the local ones, except in the special “resonance zones.” Note that the system described here acts as a “switch”: if we denote the left equilibrium point by L and the right equilibrium point by R, the fast transition between the state LLL … LLL and RRR … RRR is achieved with the energy input much smaller than expended by moving every single L to R individually, thus offering a new advantage for molecular computation (24). The main dynamical feature is that strong local interconnections provide fast pathways between zones in the phase space in which long-range forces dominate. Thus, we contend that we can learn a lot about designing robust flexible networks by studying design of biological macromolecules.