A synthetic picture of intramolecular dynamics of proteins. Towards a contemporary statistical theory of biochemical processes

Output-input ratio in thermally fluctuating biomolecular machines

Biological molecular machines are proteins that operate under isothermal conditions and hence are referred to as free energy transducers. They can be formally considered as enzymes that simultaneously catalyze two chemical reactions: the free energy-donating (input) reaction and the free energy-accepting (output) one. Most if not all biologically active proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. This makes the description of the enzymatic reaction kinetics in terms of conventional rate constants insufficient. In the steady state, upon taking advantage of the assumption that each reaction proceeds through a single pair (the gate) of transition conformational substates of the enzyme-substrates complex, the degree of coupling between the output and the input reaction fluxes has been expressed in terms of the mean first-passage times on a conformational transition network between the distinguished substates. The theory is confronted with the results of random-walk simulations on the five-dimensional hypercube. The formal proof is given that, for single input and output gates, the output-input degree of coupling cannot exceed unity. As some experiments suggest such exceeding, looking for the conditions for increasing the degree of coupling value over unity challenges the theory. Performed simulations of random walks on several model networks involving more extended gates indicate that the case of the degree of coupling value higher than 1 is realized in a natural way on critical branching trees extended by long-range shortcuts. Such networks are scale-free and display the property of the small world. For short-range shortcuts, the networks are scale-free and fractal, representing a reasonable model for biomolecular machines displaying tight coupling, i.e., the degree of coupling equal exactly to unity. A hypothesis is stated that the protein conformational transition networks, as just as higher-level biological networks, the protein interaction network, and the metabolic network, have evolved in the process of self-organized criticality.

Temperature and detection-wavelength dependence of the electron transfer rates in initial stages of photosynthesis

Unusual temperature behavior, observed in the initial electron transfer stages in the photosynthetic reaction centers of the purple bacteria, and a strong probing pulse wavelength dependence of transfer rates, determined in transient absorption spectroscopy, can easily be explained on assuming that the transfer takes place from dynamically unrelaxed states of protein environment. The transitions from the primary special pair (P) to a single bacteriochlorophyll (B) and next to a bacteriopheophytin (H) are controlled by diffusion down the energy value of underdamped vibrational modes of frequency 200 K, probably determining distances between the succeeding cofactors. The subsequent transition to the quinone A (Q) is controlled by diffusion in the position value of an overdamped conformational mode, probably corresponding to the local polarization. From the fit of available experimental data to simple theoretical formulas, the important physical conclusion arises that the very electronic transitions are fast as compared to the relaxation processes and, in the first approximation, only the latter contribute to the overall times of the initial electron transfer stages in photosynthesis.

EVOLUTION OF A CATALYTICALLY EFFECTIVE MODEL ENZYME: THE IMPORTANCE OF TUNED CONFORMATIONAL FLUCTUATIONS

Possible causal connections between the dynamics of a thermally fluctuating model enzyme molecule and catalysis are explored. The model is motivated by observations from experiment and simulation that amino acid residues residing in different enzymatic domains may show markedly different degrees of conformational freedom. Consequently, we are interested in the catalytic efficacy of an enzyme as a function of long-range many-atom cooperative effects resulting from strong, moderate, and weak interactions between enzymatic residues. Here we show and quantify through molecular dynamics simulations how the number and distribution of these interactions affects an enzyme's conformational fluctuation dynamics and its effectiveness as a catalyst. For any given distribution of "stiff" and "loose" enzymatic domains, catalytic fitness is defined as the number of chemical events — specifically the number of times a catalytic residue and substrate surmount a chemical reaction barrier — during molecular dynamics simulation. Through mutation, recombination, and a selection procedure following the ideas of Darwinian evolution, a genetic algorithm drives a population of enzyme molecules to greater catalytic fitness by modifying the mix of stiff and loose interactions. Approximately 30,000 different enzyme molecules are generated by the genetic algorithm — each with a unique number and distribution of strong, moderate, and weak inter-residue interactions. While the catalytically least fit enzyme exhibits 16 chemical events, the fittest boasts 253. That point mutations far from the active-site chemistry in the fittest enzyme have a strong effect on the number of chemical events suggests that catalysis depends, in part, on long-range many-atom globally correlated dynamical fluctuations.

Statistical properties of the dichotomous noise generated in biochemical processes

Dichotomous noise detected with the help of various single-molecule techniques convincingly reveals the actual occurrence of a multitude of conformational substates composing the native state of proteins. The nature of the stochastic dynamics of transitions between these substates is determined by the particular statistical properties of the noise observed. These involve nonexponential and possibly oscillatory time decay of the second order autocorrelation function, its relation to the third order autocorrelation function, and a relationship to dwell-time distribution densities and their correlations. Processes gated by specific conformational substates are distinguished from those with fluctuating barriers. This study throws light on the intriguing matter of the possibility of multiple stepping of the myosin motor along the actin filament per ATP molecule hydrolyzed.

Random walks with non-Gaussian step-size distributions and the folding of random polymer chains

In this paper, we study a random walker whose step-size distribution is of non-Gaussian bimodal form due to the addition of a quartic term in the exponential. By the central limit theorem, we know that in the limit of a large number of steps, the probability distribution representing the distance the walker has traveled becomes Gaussian. We investigate the nature of this convergence both numerically and analytically. We obtain a scaling relation describing the number of steps required for convergence in terms of the width and separation of the peaks of the step-size distribution. We assume in the concluding section that our model is well suited for the application of the folding of a random polymer chain.

Essential motions in a fungal lipase with bound substrate, covalently attached inhibitor and product

As an aid to understanding the influence of dynamic fluctuations during esterolytic catalysis, we follow protein flexibility at three different steps along the catalytic pathway from substrate binding to product clearance via a covalently attached inhibitor, which represents a transition-state mimic. We have applied a classical approach, using molecular dynamics simulations to monitor protein dynamics in the nanosecond regime. We filter out small amplitude fluctuations and focus on the anharmonic contributions to the overall dynamics. This 'essential dynamics' analysis reveals different modes of response along the pathway suggesting that binding, catalysis and product clearance occur along different energy surfaces. Motions in the enzyme with a covalently attached ligand are more complex and occur along several eigenvectors. The magnitudes of the fluctuations in these individual subspaces are significantly smaller than those observed for the substrate and product molecules, indicating that the energy surface is shallow and that a relatively large number of conformational substates are accessible. On the other hand, substrate binding and product release occur at distinct modes of the protein flexibility suggesting that these processes occur along rough energy surfaces with only a few minima. Detailed energetic analyses along the trajectories indicated that in all cases binding is dominated by van der Waals interactions. The carboxylate form of the product is stabilized by a tight hydrogen bond network involving in particular Ser82, which may be a potential cause of product inhibition. Considerations such as these should aid the understanding of mechanisms of substrate, inhibitor or product recognition and could become of importance in the design of new substrates or inhibitors for enzymes.

Radially softening diffusive motions in a globular protein

Molecular dynamics simulation, quasielastic neutron scattering and analytical theory are combined to characterize diffusive motions in a hydrated protein, C-phycocyanin. The simulation-derived scattering function is in approximate agreement with experiment and is decomposed to determine the essential contributions. It is found that the geometry of the atomic motions can be modeled as diffusion in spheres with a distribution of radii. The time dependence of the dynamics follows stretched exponential behavior, reflecting a distribution of relaxation times. The average side chain and backbone dynamics are quantified and compared. The dynamical parameters are shown to present a smooth variation with distance from the core of the protein. Moving outward from the center of the protein there is a progressive increase of the mean sphere size, accompanied by a narrowing and shifting to shorter times of the relaxation time distribution. This smooth, "radially softening" dynamics may have important consequences for protein function. It also raises the possibility that the dynamical or "glass" transition with temperature observed experimentally in proteins might be depth dependent, involving, as the temperature decreases, progressive freezing out of the anharmonic dynamics with increasing distance from the center of the protein.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.

Conformational behavior of ionic self-complementary peptides

Several de novo designed ionic peptides that are able to undergo conformational change under the influence of temperature and pH were studied. These peptides have two distinct surfaces with regular repeats of alternating hydrophilic and hydrophobic side chains. This permits extensive ionic and hydrophobic interactions resulting in the formation of stable beta-sheet assemblies. The other defining characteristic of this type of peptide is a cluster of negatively charged aspartic or glutamic acid residues located toward the N-terminus and positively charged arginine or lysine residues located toward the C-terminus. This arrangement of charge balances the alpha-helical dipole moment (C --> N), resulting in a strong tendency to form stable alpha-helices as well. Therefore, these peptides can form both stable alpha-helices and beta-sheets. They are also able to undergo abrupt structural transformations between these structures induced by temperature and pH changes. The amino acid sequence of these peptides permits both stable beta-sheet and alpha-helix formation, resulting in a balance between these two forms as governed by the environment. Some segments in proteins may also undergo conformational changes in response to environmental changes. Analyzing the plasticity and dynamics of this type of peptide may provide insight into amyloid formation. Since these peptides have dynamic secondary structure, they will serve to refine our general understanding of protein structure.

Time course of reactions controlled and gated by intramolecular dynamics of proteins: predictions of the model of random walk on fractal lattices

Computer simulations of random walk on the Sierpinski gasket and percolation clusters demonstrate that the short, initial condition-dependent stage of protein involving reactions can dominate the progress of the reaction over the main stage described by the standard kinetics. This phenomenon takes place if the intramolecular conformational transition dynamics modeled by the stochastic process is slow enough and the initial conformational substate of the protein already belongs to the transition state of the reaction. Both conditions are realized in two kinds of experiments: small ligand rebinding to protein after laser flash photolysis and direct recording of single protein channel activity. The model considered suggests simple analytical formulae that can explain the time behavior of the processes observed and its variation with temperature. The initial condition-dependent stage, and not the stage described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.