Digitality Induced Transition in a Small Autocatalytic System

Autocatalytic reaction system with a small number of molecules is studied numerically by stochastic particle simulations. A novel state due to fluctuation and discreteness in molecular numbers is found, characterized as extinction of molecule species alternately in the autocatalytic reaction loop. Phase transition to this state with the change of the system size and flow is studied, while a singlemolecule switch of the molecule distributions is reported. Relevance of the results to intracellular processes are briefly discussed. Cellular activities are supported by biochemical reactions in a cell. To study biochemical dynamic processes, rate equation for chemical reactions are often adopted for the change of chemical concentrations. However, the number of molecules in a cell is often rather small [1], and it is not trivial if the rate equation approach based on the continuum limit is always justified. For example, in cell transduction even a single molecule can switch the biochemical state of a cell [2]. In our visual system, a single photon in retina is amplified to a macroscopic level [3]. Of course, fluctuations due to a finite number of molecules is discussed by stochastic differential equation (SDE) adding a noise term to the rate equation for the concentration [4, 5]. This noise term sometimes introduces a non-trivial effect, as discussed as noise-induced phase transition [6], noise-induced order [7], stochastic resonance [8], and so forth. Still, these studies assume that the average dynamics are governed by the continuum limit, and the noise term is added as a perturbation to it. In a cell, often the number of some molecules is very small, and may go down very close to or equal to 0. In this case, the change of the number between zero and nonzero, together with the fluctuations may cause a drastic effect that cannot be treated by SDE. Possibility of some order different from macroscopic dissipative structure is also discussed by Mikhailov and Hess [9, 10] (see also Ref. [11]). Here we present a simple example with a phenomenon intrinsic to a system with a small number of molecules where both the fluctuations and digitality(‘0/1’) are essential. In nonlinear dynamics, drastic effect of a single molecule may be expected if a small change is amplified. Indeed, autocatalytic reaction widely seen in a cell, provides a candidate for such amplification [12, 13]. Here we consider the simplest example of autocatalytic reaction networks (loops) with a nontrivial finite-number effect. With a cell in mind, we consider reaction of molecules in a container, contacted with a reservoir of molecules. The autocatalytic reaction loop is Xi + Xi+1 → 2Xi+1; i = 1, · · · , k;Xk+1 ≡ X1 within a container. Through the contact with a reservoir, each molecule Xi diffuses in and out. Assuming that the chemicals are well stirred in the container, our system is characterized by the number of molecules Ni of the chemical Xi in the container with the volume V . In the continuum limit with a large number of molecules, the evolution of concentrations xi ≡ Ni/V is represented by dxi/dt = rixi−1xi − ri+1xixi+1 +Di(si − xi) (1) where ri is the reaction rate, Di the diffusion constant, and si is the concentration of the molecule in the reservoir. For simplicity, we consider the case ri = r, Di = D, and si = s for all i, while the phenomena to be presented here will persist by dropping this condition. With this homogeneous parameter case, the