Download Advances in Biochemical Engineering, Volume 11 by D. Ramkrishna (auth.) PDF

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108) for the c-dependent first order product density. Interestingly enough, it is not coupled to any other equation as is, for example, Eq. (15). Thus the expected value of the population density can be obtained if Eq. (108) can be solved, which must be done subject to an appropriate initial condition. If initially there are N O cells of physiological states, say zi, z2, . , - Z N 0 in an environment of concentration c o , then No FI(Z; C; 0) = t5 (C -- Co) ~ t~ (Z -- Zi) (111) i= 1 which is an example of how initial conditions are specified for Eq.

Thus F(C) = F(E[C]) + ~ /(C - EIC]). V}nF(E[C]) n=l (99) a E[F(C)] = F(E[CI) + ~ {E[C - E[C]]- VtnF(E[C]) (100) so that n=l From (100), we may infer that when the concentration fluctuations are negligible E[F(C)] ~ F(E[C]) (101) Eq. (101) is useful in establishing the connection between the stochastic model equations to be presented in the next section and those in the deterministic formalism. The circumstances under which the fluctuations in the environmental variables may be neglected will be considered at a later stage.

1 The Master Density Function Since we must be concerned with the distribution of physiological states of all the ceils in the population and the environmental variables, we define a master density function a Ju(zl, z2 .... zv; c; t)dol dD2 ... , v a n d the environmental vector C is in a volume dc located at e somewhere in the m-dimensional volume ~. Aside from the physiological state, cells are assumed to be indistinguishable. The multivariate probability density function for the concentration vector C, denoted fc(e, t) is given by fc(C; t) = Z ~1 1~ f doi Jv(Zl , z2, --.

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