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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.

Proteins are linear polymers of amino acids arranged in a sequence determined by genes. Since the origin of molecular biology in the 1950s, a paradigm has been commonly accepted, expressed shortly in two successive implications:

It assumes implicitly that the dynamics of native proteins reduces to simple normal vibrations about a single conformational state referred to as the “tertiary structure” of the protein. For at least two decades, however, it becomes more and more clear that not only structure but also more complex dynamics determine the function of proteins thus the paradigm has to be extended to [

Two classes of experiments imply directly that besides fast vibrations enzymatic proteins display also a much slower stochastic dynamics of transitions between a variety of conformational substates composing their native state. The first class includes observations of the non-exponential initial stages of reactions after special preparation of an initial microscopic state in a statistical ensemble of biomolecules by, e.g., the laser pulse [

Because of the slow character of the conformational dynamics, both the chemical and conformational transitions in an enzymatic protein have to be treated on an equal footing [

determining the time variation of the occupation probabilities _{l}_{l}_{′}_{l}

Contrary to the transition state theory, the stochastic theory of reaction rates takes into account the very process of reaching the partial thermodynamic equilibrium in the non-chemical degrees of freedom of the system described. In the closed reactor, the possibility of a chemical transformation of an enzyme will proceed before the conformational equilibrium has been reached results in the presence of a transient non-exponential stage of the process and in an essential dynamical correction to the reaction rate constant describing the following exponential stage [

The primary purpose of thermodynamics, born in the first half of the 19th century, was to explain the action of the heat engines. The processes they are involved in are practically reversible and proceed in varying temperatures. As a consequence, thermodynamics being the subject of the school and academic teaching, still deals mainly with equilibrium processes and changes of temperature. However, biological machines as well as many other contemporary machines act irreversibly, with considerable dissipation, but at constant temperature. Machines that operate under the condition

From a theoretical point of view, it is convenient to treat all biomolecular machines, also pumps and motors, as chemo-chemical machines [

The principle of the action of the chemo-chemical machine is simple [

In formal terms, the chemo-chemical machine couples two unimolecular reactions: the free energy-donating reaction R_{1} ↔ P_{1} and the free energy-accepting reaction R_{2} ↔ P_{2}. Bimolecular reactions can be considered as effective unimolecular reactions upon assuming a constant concentration of one of the reagents, e.g., adenosine diphosphate (ADP) in the case of ATP hydrolysis. The input and output fluxes _{i}_{i}

and

Here, symbols of the chemical compounds in the square brackets denote the molar concentrations in the steady state (no superscript) or in the equilibrium (the superscript eq). [E]_{0} is the total concentration of the enzyme and _{B}^{−}^{1}, where _{B} is the Boltzmann constant. The flux-force dependence is one-to-one only if some constraints are put on the concentrations [R_{i}_{i}_{i}_{i}_{i}] + [P_{i}

The free energy transduction is realized if the product _{2}_{2}, representing the output power, is negative. The efficiency of the machine is the ratio

of the output power to the input power. In general, the degree of coupling of both fluxes,

being itself a function of the forces _{1} and _{2}, can be both positive or negative. To avoid a misconception, let us stress that the definition (_{ij}_{ji}

where

are the Kedem-Caplan degree of coupling and the stoichiometric force ratio [

Usually, the assumption of tight coupling between both reactions is made (_{1} = _{2}, thus

The multiconformational counterpart of the scheme in

The flux-force dependence thus obtained for the two coupled reactions has a general functional form

The parameters _{+}_{i}_{−i}_{0}_{i}_{i}_{2} = 0 and for
_{i}_{i}_{0}_{i}_{+}_{i}_{−i}_{+}_{i}_{−i}

In Reference [^{4} iteration steps impossible. Only by increasing the number of the iteration steps to 10^{9} can one determine the fluxes, thus their degree of coupling, with the error lower than 0.3%. Preliminary estimations indicate that the result is in a good agreement with the stationary fluctuation theorem [

which can be equivalently rewritten as

Above,

and 〈. . .〉 is the average over that ensemble.
_{i}_{i}

The essential motive of our studies is a trial to answer the intriguing question of whether is it possible for the degree of coupling

No conventional chemical kinetics approach is able to explain such behaviors. In References [

Since the formulation by Bak and Sneppen a cellular automaton model of the Eldredge and Gould punctuated equilibriums [^{−α}

^{2}, with negative

where _{l}

For the system of gates as shown in _{2} = 0, and found _{1} = _{2} = 40 computer steps, those times being one order of magnitude shorter than the internal relaxation time _{rx} = 400 which means that the input and output reactions were controlled, though not completely, by intramolecular dynamics. The case of multiple output gates needs more systematic studies. For the system of gates shown in _{1} = _{2} = 40 we found

To end of the paper, it is worth presenting the scale of fluctuations for the dynamics discussed. ^{3} computer steps, and determined for the input force _{1} = 0.3 and the output force _{2} = 0. For the zero output force, the fluctuation theorem (

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. This makes the possibility of chemical transformations to proceed before the conformational equilibrium has been reached in the actual chemical state. In the closed reactor, it results in the presence of transient, non-exponential stages of the reactions. In the open reactor, a consequence is the necessity of determining the steady-state reaction fluxes by the mean first-passage times between transition conformational substates of the reactions rather than by conventional reaction rate constants. A hypothesis is stated that, as just as higher level biological networks, the protein interaction network, and the metabolic network, the protein conformational transition networks have evolved in a process of self-organized criticality. All the three classes of networks are scale-free and, probably, display a transition from the fractal organization on a small length-scale to the small-world organization on a 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. 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 has been confronted with the results of random walk simulations on various model networks.

For single input and output gates, the degree of coupling cannot exceed unity. As some experiments for the myosin II motor suggest such exceeding, looking for the conditions for increasing the degree of coupling value over unity (realization of a “molecular gear”) challenges the theory. Probably it holds also for the G-proteins and transcription factors, mutations of which can result in the cancerogenesis. Study simulations of random walks on several model networks involving more extended gates indicate that the case of the degree of coupling with the value higher than one is realized in a natural way on critical branching trees extended by long-range shortcuts. For short-range shortcuts, the networks are scale-free and fractal, and represent reasonable models for biomolecular machines displying tight coupling,

The study has been supported in part by the Polish Ministry of Science and Higher Education (project N N202 180038).

The authors declare no conflicts of interest.

Both authors equally contributed to the paper. The general conception and the theory is mainly due to MK, who also wrote the manuscript. The application of the critical branching tree model and numerical simulations are mainly due to PC. The authors approved the final version of the manuscript.

(_{0} is distinguished.

Development of kinetic schemes of the chemo-chemical machine. (_{1} ↔ P_{1} drives reaction R_{2} ↔ P_{2} against its conjugate force determined by steady state concentrations of the reactant and the product; (

Character of the functional dependence of the output flux _{i}_{i}_{i}_{i}

Exemplary simulated time course of the net number of the input (R_{1} ↔ P_{1}) and the output (R_{2} ↔ P_{2}) external transitions for the 5-dimensional hypercube with constant transition probabilities and the most optimally chosen input and output gates. (^{5} steps. In this particular case, the determined value of the degree of coupling was 0.39.

(^{2} which makes the network a scale free small world. Four output gates are distinguished, chosen for the Monte Carlo simulations; the unlabeled largest hub is the fourfold degenerated complement gate 2″.

(^{3} computer steps, determined for the input force _{1} = 0.3 and the output force _{2} = 0. The circles represent data for the fractal tree network (