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Footnotes
[1] Definitions : PDE stands for Partial Differential Equation and parabolic means equations of the form :
We only treat the last case.
[2] Hint(Schulman, 1981) : Write down the binomial distribution for the number of molecules at a fixed position after N=Nl+Nr steps. Then determine the proper conditions and apply the appropriate Limit Theorem of Formal Statistics to find the Normal (i.e. Gaussian) distribution.
[3] This is generally done by using boundary detectiontechniques based on the ray-tracing techniques of Computer Graphics (Foley et al., 1990). See the Numerical Methods section.
[4] See fig.1 for a depicted example of an initialcondition.
[7] In this scheme, reversible reactions are written as a pair of irreversiblereactions!
[8] The spherical cell had a diameter of 20 μm and was simulated using a Crank-Nicholson solution method (eq.(35)) with 0.1 μm thick shells. Resting calcium concentration was 50 nM and diffusionconstant for calcium was 2*10-6cm2s-1. Removal mechanismswere not implemented. Slow buffer (EGTA): concentration 50 μM ,K,d=0.20 μM, f=2.5*106 M-1s-1 , b=0.5 s-1. Fast buffer (BAPTA): concentration 50 μM ,Kd=0.20 μM , f=4.0*108 M-1s-1 , b=8 s-1. Diffusible buffers had a diffusion constant of 2*10-6cm2s-1. The current was 100ms long and had an amplitude of 50 pA.
[9] Dapp was computed as
and the calcium influx was
scaled toachieve the same final free calcium concentration as in the case of realbuffer.
[10] Concentrations of 50 μM for the slowbuffer and 10 μM for the fast buffer. The cell is also larger(diameter of 60 μm ) and additional traces are shown forconcentrations 20 μm and 30 μm below the plasmamembrane.
[11] Since computers are generally deterministicmachines, it is better to use the term 'pseudo-random' numbers. It is not therandomness of every individual number per se that's important but thestatistical properties of a sequence that have to match the statisticalproperties of a sequence of 'real' random numbers.
[12] Of course, nothing keeps you from reformulating aset of reaction-diffusion equations into a set of integral equations and usingthe first type to solve them. Good luck!
[13] That is, on nowadays workstations. More molecules (on the order of hundred thousand to a million) can be tracked using workstation clusters or supercomputers.
[14] Such a solution is called the Green's Function of the differential equation at hand. Hence the name Green's Function Monte Carlo.
[15] It can be proven that var
when no additional
information about the solution is used.
[16] An exact definition of detailed balance strongly depends on the type of stochastic process involved (see (Fishman,1996)).
[17]
is the integer part of x.
[18] A definition for 'implicit method' is given afterintroducing the equations.