(11.1) |

where refers to the probability that more than one event (i.e. dissociation or binding) occurs during a time increment . If we bring the term to the left, divide by and let approach to zero, we arrive at

where denotes the derivative of with respect to the time. For we have

(11.3) |

(11.4) |

(11.5) |

(11.6) |

Starting with , and using (11.7) in the form

we find by induction that . This relation determines the probabilities up to an arbitrary factor. Obviously, we must have that . The series is well known as the modified Bessel function. So . We therefore arrive at

This probability distribution relates to the Poisson distribution by just squaring the Poisson probabilities and renormalizing to assure that the sum of the probabilities remains 1. The normalizing constant in (11.8) compares with in the Poisson distribution. Since , so , the expected number of protons equals . This is lower than the value , which should be expected on the basis of the deterministic model. This is obvious when we obtain the variance by summing (11.7) for It is found to be . Although it is less than the variance of a Poisson distribution with the same mean it is still considerable for small , as is illustrated in Fig 11.1, which shows that the 95% confidence interval of the pH in a cell of neutral pH ranges from 6.85 to 7.15. The lifetime of a randomly selected water molecule, a hydroxylion and a hydronium ion follow an exponential distribution with mean , i.e. some h, for water and , i.e. some s, for both ions at C. Diffusion causes these particles to displace in time, , over a mean distance of , where denotes the diffusion coefficient. For HO, OH and H, the latter is 2.26, 5.3 resp. at C. The mean total lifetime displacement in an unbounded body of pure water is thus 1.37cm, 3.26m and 4.32m, as a crow flies. This means that the limited size of a cell is likely to influence the transport, even apart from influences exercised by, e.g. the membrane. We made some simplifying assumptions. The first one is that the cell consists of pure water, which is obviously not true. Its cytoplasma is well buffered. Although a full analysis would certainly be immensely complex, it is by no means certain that the stochastic fluctuations are more restricted in buffered mixtures. Buffers primarily balance net fluxes, but we did not discuss that situation. The assumption that water molecules dissociate independently from each other is hard to test at the moment. The significance of the weak bonds certainly depends on temperature. The intention of the example is to draw attention to the odd behaviour of the concept proton 'concentration' in small bodies. It might be relevant e.g. when we measure enzyme performance in well mixed extracts and try to evaluate its consequence for the living cell.