By Pablo A. Iglesias, Brian P. Ingalls

A survey of ways engineering options from regulate and structures thought can be utilized to assist biologists comprehend the habit of mobile platforms.

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**Additional resources for Control Theory and Systems Biology**

**Example text**

On the other hand, if bifurcations are far away in parameter space, then the nominal behavior may be highly robust to changes in operating conditions. 7 Systems with Inputs and Outputs Thus far, we have considered systems that evolve autonomously, and we have treated the whole state vector in our analysis. In control engineering, it is more common to consider systems that respond to external inputs and provide speciﬁc output signals to their environment. We ﬁrst consider systems of the form d sðtÞ ¼ f ðsðtÞ; uðtÞÞ: dt ð1:9Þ In this equation, the vector 2 3 u1 ðtÞ 6 .

5). Let AJ be a principal submatrix of A and PJ be a subvector of P, both corresponding to the indexes in J. 6) provides a bound on the error between the exact solution PJ to the (inﬁnite) chemical master equation and the matrix exponential of the (ﬁnite) reduced system with generator AJ . This result is the basis for an algorithm to compute the probability density function with guaranteed accuracy. The FSP approach and various improvements on the main algorithm are described by Munsky and Khammash (2008).

In contrast, a discrete stochastic formulation of the same reaction describes the probability that the numbers of molecules of species A and B take certain integer values at a given time t. In this way, populations of the species within the network of interest are treated as random variables. In this description, reactions take place randomly according to certain probabilities determined by several factors including reaction rates and species populations. For example, given certain integer populations of A and B, say NA and NB , at time t, the probability that the above reaction takes place within the interval ½t; t þ dtÞ is proportional to ðNA Â NB =V Þ dt, where V is the volume of the space containing the molecules of A and B and dt is a small time increment.