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Pathway Modelling

Click here for probabilistic approach.

Let us consider the probability of CAI-1 interacting with our fusion kinase with a reaction rate k, diffusion rates D1 and D2 and they start at a distance d apart at the beginning of the time interval http://lh4.ggpht.com/coa07mhk/SQZQsubfzCI/AAAAAAAAANw/92s3DnMCXMo/s144/delta_t.PNG The reaction between two diffusing particles can be considered to take place in two steps. First the encounter of CAI-1 with Fusion kinase and second the actual reaction of passing protein to response regulatory receptor. If we let CAI-1 as entity A and fusion kinase as entity B then the rate of reaction can given as:

http://lh6.ggpht.com/coa07mhk/SQZQslkxAzI/AAAAAAAAAOA/j5Bd0dJ8vtg/s288/eq2.PNG

Where http://lh4.ggpht.com/coa07mhk/SQZQs58C4WI/AAAAAAAAAOI/b5qn3x5R07s/s144/eq3.PNG is the probability of interaction between CAI-1 and Fusion Kinase.

http://lh4.ggpht.com/coa07mhk/SQZQs9D-DyI/AAAAAAAAAOQ/FdveIv00mZU/s144/eq4.PNG is the reaction once CAI-1 has interacted with Fusion Kinase.

This interaction will follow the law of diffuion and if we let the process under consideration as Poisson process, the probability can be given as:

http://lh4.ggpht.com/coa07mhk/SQZQx-YyI9I/AAAAAAAAAOY/8wieBfvt0mI/s144/eq5.PNG

http://lh3.ggpht.com/coa07mhk/SQZQyavXd-I/AAAAAAAAAOg/qsc_qvPTep8/s144/eq6.PNG

To find the probability of rest of reaction i.e. from activation of UvrY till the shining of our protein we can use the Gaussian Distribution, which can be given as:

http://lh3.ggpht.com/coa07mhk/SQZQypqSCUI/AAAAAAAAAO4/cURij-XIVgc/s288/eq9.PNG

http://lh5.ggpht.com/coa07mhk/SQZQ0R3tEeI/AAAAAAAAAPA/N_x0O2WupfY/s800/G1.PNG

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