429-2009-Q9key - " ) = Pr( n )exp( x n ) n = N...

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580.429 SBE3 2009 Quiz 9 Name: Diffusion and generating functions. Here we will model the diffusion of a receptor on the surface of a cell in one dimension. Each time it hops, it moves a distance d roughly equivalent to its diameter. It hops to the left with probability p and it hops to the right with probability 1– p . We follow its motion for N total hops. 1. (2 pts) What is the probability of the specific pattern of hops L n R N n , i.e. n hops to the left followed by N n hops to the right? Probability: p n (1– p ) N–n 2. (2 pts) Now disregarding the order of the hops, (i) how many different paths are there with a cumulative number of n hops to the left and N n hops to the right, and (ii) what is the aggregate probability of all of these paths, defined as Pr( n )? Total number of paths: C( N , n )= N !/ n !( N n )!= N n ! " # $ % , all acceptable Pr( n ): C( N , n ) p n (1– p ) N–n , with other expressions for C( N , n ) acceptable. 3. (3 pts) When the particle hops n times to the left and N n times to the right, it lands at position x n = d ( N ! 2 n ) . Define the generating function ! (
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Unformatted text preview: " ) = Pr( n )exp( x n ) n = N # = Pr( n )exp[ d ( N $ 2 n )] n = N # . Using the formula for Pr( n ), you can simply the generating function to the closed form solution ( ) = e dN T N , where the term T depends on p , d , and . Provide the value of T and the value of ( ) evaluated at = . ( ) = exp[ dN ] C ( N , n ) p n (1 # p ) N # n e # 2 dn n = N $ = exp[ dN ] C ( N , n )( pe # 2 d ) n (1 # p ) N # n n = N $ = exp[ dN ][ pe # 2 d + 1 # p ] N T = pe ! 2 d + 1 ! p ( = 0) = 1 (normalization) 4. (3 pts). The mean position after N hops can be obtained as x = d ln ! ( " ) / d evaluated at = . What is ln ( ) and what is x in simplified form? ln ( ) = dN + N ln[ pe # 2 d + 1 # p ] d ln ( ) / d = dN + N # 2 dpe # 2 d pe # 2 d + 1 # p x = dN # 2 dpN = Nd (1 # 2 p ) which makes sense because the mean position is Nd for p = 1 (all hops to the left) and Nd for p = 0 (all hops to the right)....
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This note was uploaded on 03/30/2010 for the course SBE 580.429 taught by Professor Joelbader during the Fall '09 term at Johns Hopkins.

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429-2009-Q9key - " ) = Pr( n )exp( x n ) n = N...

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