Output-input ratio in thermally fluctuating biomolecular machines

Abstract

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.

Steady-state distributions of probability fluxes on complex networks

“…However, it becomes considerably wider in the case of disordered networks such as the random graphs or the scale-free networks with the Poisson and the power-law degree distributions, respectively [49]. In the previous paper [4], we supposed, being primarily motivated by the systems biology, that the network of states, like the protein interaction network and the metabolic network, has evolved in the process of self-organized criticality [50]. Such networks display a transition from the fractal organization on the small length-scale to the small-world organization on the large-length scale [51].…”

“…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.…”

Stochastic Dynamics of Proteins and the Action of Biological Molecular Machines

“…There are grounds to suppose that the conformational transition networks, as just as the two networks of the systems biology, the protein interaction network, and the metabolic network, have evolved to reach a scale-free, thus critical structure [8]. A controversy emerges if this structure is simulaneously fractal or small-world.…”

“…The even more convincing proof of the conformational transition dynamics of simple native proteins has been afforded by early molecular dynamics simulations [6,7]. Research of biomolecular dynamics is being developed faster and faster and today, even in the case of small, water-soluble proteins, one speaks about the "native state ensemble", and for very small proteins or protein fragments trials to reconstruct the actual networks of conformational transitions are realized (see papers cited in Reference [8]). …”

“…In References [13] and [12], basing on approximations carried too far, we suggested that the far-from-equilibrium degree of coupling can exceed unity already for reactions proceeding through single pairs of transition substates. In Reference [8], we proved the theorem that the value of the degree of coupling (5) should be lower than, or at the most equal to, unity, but only when the input and output reactions proceed through single pairs of transition conformational substates. It is reasonable to suppose that the chance of a higher degree of coupling is possible if the output gate is extended to two or more pairs of transition substates (Figure 2d).…”