Chemiosmosis maintains life in nonequilibrium through ion transport across membranes, yet the origin of this order remains unclear. We develop a minimal model in which ion pump orientation and the intracellular electrochemical potential mutually reinforce each other. This model shows that fluctuations can induce collective pump alignment and the formation of a membrane potential. The alignment undergoes a phase transition from disordered to ordered, analogous to the Ising model. Our results provide a self-organizing mechanism for the emergence of bioelectricity, with implications for the origin of life.
Coupling between directional ion transport across a membrane and the generation of an electrochemical potential difference is essential for living cells [1]. Mitchell's chemiosmotic theory established that electric currents operate at the source of life: nonequilibrium ion currents across membranes drive cellular energy transduction, including ATP synthesis [2][3][4]. Through this coupling, living systems are maintained far from equilibrium by sustained directional ion fluxes across membranes.
At the origin of life, the emergence of sustained nonequilibrium ion flows across membranes is thought to have been a critical step. Phylogenetic evidence indicates that early life already synthesized ATP through membrane-potential-driven processes [5,6], and chemical and geological studies suggest that early cells may have emerged by exploiting natural proton gradients associated with alkaline hydrothermal vents [7,8]. However, cells can autonomously generate membrane potential through ion transport, even in the absence of preexisting ion gradients, leaving the origin of such mechanisms unresolved. Therefore, to elucidate the origin of life or protocells, it is necessary to understand how ion pumps could collectively generate and sustain membrane potential.
Although studies on the origin of life have examined the necessity of autocatalytic reaction sets [9][10][11][12], the generation of bio-information [13][14][15][16][17][18], the role of lipid membranes [19][20][21], and required nonequilibrium conditions [22,23], how nonequilibrium flows and membrane potential mutually sustain each other remains an open physical problem. Here, we introduce a minimal model describing how ion pumps act collectively by exploiting electrochemical potentials that are generated self-consistently. We show that nonequilibrium ion transport by pumps and the membrane potential emerge through a phase transition analogous to that of the Ising model. We further discuss the conditions for this transition, as well as the roles of fluctuations and symmetrybreaking fields.
In our model, protocells generate a membrane potential ∆ϕ via the flow of ions Q into and out of the cell through membrane-embedded pumps P = Fig. 1 T P i = + 1 P i = -1
Pump-ion coupling model, consisting of three regions; internal, external, and membrane. Pumps, P = {Pi} = {P1, P2, …, PN P }, are embedded in a membrane and orient either inward (Pi = +1) or outward (Pi = -1). Ions are present in both the internal and external regions. Pumps and ions can interact with each other, resulting in ‘flip’ of pumps and ’transport’ of ions. The direction of the arrows indicates the direction of ion transport; influx (F in ) and outflux (F out ).
{P 1 , P 2 , …, P N P }, with the total number of pumps N P fixed (Fig. 1). The total ion number N Q is set to be constant and assumed to be sufficiently large for continuum approximation of concentrations c I/E . Each pump is structurally asymmetric, and the direction of ion transport is determined by whether its head (ion-uptake side) faces inward or outward [24][25][26]. We represent the direction of the i-th pump by a binary variable P i ∈ {-1, 1}, corresponding to inward and outward directions, respectively.
First, the ions are transported across the membrane by chemically driven pumps [35], in the presence of the electric potential barrier of the lipid bilayer [27][28][29][30]. The ion fluxes into and out of the cell are proportional to the number of inward and outward pumps, n in/out P , reflecting the structural asymmetry of the pumps. In addition, the ion flux depends on the electrochemical potential difference ∆µ = µ I -µ E across the membrane, and, within a single Eyring-type barrier approximation [27][28][29][30], it is proportional to exp ( µ I/E ) . Here, µ I/E = µ 0 + β -1 ln c I/E + zeϕ I/E [31], where ϕ I/E denotes the electrostatic potentials inside and outside the cell [36].
Summarizing the above, ion transport is expressed as follows[37]:
where γ is the rate including chemical contributions to the pump [38]. Since N P is fixed, n out P = N P -n P , where n P ≡ n in P hereafter. The external ion concentration, determined by ion-number conservation, is given by c E = r V (c 0 -c I )+c 0 , where c 0 = N Q /(V I +V E ) is the average ion concentration and r V = V I /V E is the volume ratio. In addition, the membrane potential ∆ϕ = ϕ I -ϕ E is expressed as
where C is the membrane capacitance, and we define the ion concentration ratio q = (c I -c 0 )/c 0 [39]. Using the above definitions, the ion transport equation, Eq. ( 1), is given by
where α = βc 0 V I z 2 e 2 /(2C) quantifies the electrostatic energy associated with q, relative to thermal energy β -1 (see End Matter).
Next, we incorporate the interaction between the pumps and the membrane potential. The pumps are embedded in the membrane and equipped with a charged head, so that its orientation can change depending on the electrostatic potential, i.e., when the
This content is AI-processed based on open access ArXiv data.