EE 5393 UMN Circuits, Computation, and Biology Winter 2013 ⊕ ⊕ Homework # 2 Due March 29, 2013 [in class] 1. Analyzing Stochasticity in Molecular Reactions Randomness is inherent to all forms of biochemical computation: at any given instant, the choice of which reaction fires next is a matter of chance. Certain biochemical systems appear to exploit this randomness for evolutionary advan- tage, choosing between different outcomes with a probability distribution – in effect, hedging their bets with a portfolio of responses that is carefully tuned to the environmental conditions. For instance, the lambda bacteriophage, a virus that infects the E. coli bacteria, chooses one of two survival strategies: either it integrates its genetic material with that of its host and then replicates when the bacterium divides (call this “stealth” mode); or else it manipulates the molecular machinery of its host to make many copies of itself, killing the bacterium in the process, and thereby releasing its progeny into the environment (call this the “hijack” strategy). The choice of which strategy to pursue, while based on environmental inputs, is probabilistic: in some cases, the virus chooses the first strategy, say with probability p and the second with probability 1 - p . Clearly the virus is hedging its bets, an approach that provides significant advantages in an evolutionary context. A model for the biochemistry of lambda is at: A set of initial values for the molecular types is at: With this model, we can assume that lambda has entered “stealth” mode when cI 2 > 145; it has entered “hijack” mode when Cro 2 > 55.
EE 5393, Winter ’13 2 For the reactions and initial values given, compute the probability that lambda has entered “stealth” mode vs. “hijack” mode for a range of values of the molecular type MOI = 1 . . . 10. For this problem, you can either use your own code for simulating chemical reactions from Homework 1 or use my code: . 2. Computing with Molecular Reactions For this problem, use Phil Senum’s compiler: (a) Euclid’s Algorithm Euclid’s algorithm is an efficient method for computing the greatest com- mon divisor (GCD) of two integers, also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Ele- ments. In its simplest form, Euclid’s algorithm starts with a pair of positive integers and forms a new pair that consists of the smaller number and the difference between the larger and smaller numbers. The process repeats until the numbers are equal. That number then is the greatest common divisor of the original pair.
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