Stochastic Dynamics of Proteins and the Action of Biological Molecular Machines

Abstract

It is now well established that most if not all enzymatic proteins display a slow stochastic dynamics of transitions between a variety of conformational substates composing their native state. 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. Here, the criticality means that all the three classes of networks are scale-free and, moreover, display a transition from the fractal organization on a small length-scale to the small-world organization on the large length-scale. Good mathematical models of such networks are stochastic critical branching trees extended by long-range shortcuts. 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. The far-from-equilibrium degree of coupling between the output and the input reaction fluxes have been studied both theoretically and by means of the Monte Carlo simulations on model networks. For single input and output gates the degree of coupling cannot exceed unity. Study simulations of random walks on model networks involving more extended gates indicate that the case of the degree of coupling value higher than one is realized on the mentioned above critical branching trees extended by long-range shortcuts.

Steady-state distributions of probability fluxes on complex networks

“…Here, the first reaction (the input gate) donates the free energy to the system due to the external thermodynamic force (the affinity) and the second one (the output gate) consumes it, but partially because of dissipation, to perform work on some external system. This problem was the subject of our previous papers, concerning the action of the biological molecular machine [4,5]. There, however, we considered the same type of stochastic dynamics as in the present paper.…”

“…In Fig. 1b, this oversimplified picture of the coarse-grained enzymatic kinetics is replaced by the more detailed 'mesoscopic' scheme, promoted in our previous papers [4,5]. The gray rectangle represents an arbitrary network of stochastic transitions between numerous conformational substates, composing either the enzyme or the enzyme-substrate native state E. All these internal transitions satisfy the detailed balance condition.…”

Tracking individual membrane proteins and their biochemistry: The power of direct observation

“…A stochastic model can be used to describe the operation of molecular machines, which links allosteric motions and chemical reactions to changes in free energy. (Elber and Karplus, 1987;Jülicher and Bruinsma, 1998;Keller and Bustamante, 2014;Kurzynski and Chelminiak, 2014;Lazaridis and Karplus, 1999;Tsai, 2008) Such descriptions include the following phenomena: the closing of ligand binding clefts, (McCammon et al, 1976) rotations of subdomains, (De Groot et al, 1998;Noji et al, 1997) twisting of subunits, binding and unbinding of ligands, (Ghanouni et al, 2001;Lau and Roux, 2007;Ma et al, 2000;Ravindranathan et al, 2005) the metabolism of substrates (enzymes), (Rout et al, 2014) and the movement of proteins through lipid domains. (Wang et al, 2014) Each machine has a small number of key conformational degrees of freedom linked to these motions and chemical reactions, which result in the observable functional states (discrete states) of the protein.…”

Do biological molecular machines act as Maxwell's demons?

“…The assumption that most biochemical reactions are controlled by the slow dynamics of the proteins has a very rich literature confirmations [26,44]. It is this, in fact, that led to the now widely accepted change of the fundamental paradigm of molecular biology, that not only structure but also dynamics determine the function of the proteins [54][55][56][57][58]76].…”