E6010_Lecture2_NetMotif_Autoregulation_NP

E6010_Lecture2_NetMotif_Autoregulation_NP - Autoregulation:...

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Autoregulation: A Network Motif E6010: Lecture 2 Prof. Predrag R. Jelenkovi´c Dept. of Electrical Engineering Columbia University , NY 10027, USA predrag@ee.columbia.edu Jelenkovi´c (Columbia University) Network Motifs 1 / 32
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Outline 1 Last Lecture 2 Introduction to Lecture 2 3 Patterns, Randomized Networks, and Network Motifs 4 Detecting Network Motifs by Comparison to Randomized Networks 5 Autoregulation: A Network Motif Jelenkovi´c (Columbia University) Network Motifs 2 / 32
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Last Lecture Transcription networks: basic concepts The cognitive problem of the cell Elements of transcription networks Separation of time scales; typical approximate time scales for E. coli (Table 2.2 from the book): Binding signaling molecule to a TF 1 msec. Binding active TF to its DNA site 1 sec. Transcription + translation of the gene 5 min. 50% change of protein concentration 1 h. Steady state approximation Modularity of transcription networks The signs on the edges: activators and repressors The numbers on the edges: the input function - Hill function Logical input functions: a simple approximation Multi-dimensional input functions Dynamics and response time of simple gene regulation Jelenkovi´c (Columbia University) Network Motifs 3 / 32
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Hill Function Most transcription factors are composed of repeated protein subunits (dimers, tetramers). Each of the subunits can bind inducer (signaling) molecule. Full activity of TF is reached when multiple subunits bind the molecule. Assume that n molecules of S x can bind X forming the active complex X * [ nS x X ] Let X T be the total concentration of bound and unbound X , then the conservation law gives us [ nS x X ] + X 0 = X T where X 0 stands for unbound (inactive) X. Then, collision rate = k on X 0 S n x , where k on denotes the on rate of complex formation. The complex [ nS x X ] disassociates with rate k off disassociationrate = k off [ nS x X ] . Jelenkovi´c (Columbia University) Network Motifs 4 / 32
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Hill Function Then d [ nS x X ] dt = k on X 0 S n x - k off [ nS x X ] . This equation reaches equilibrium within milliseconds; separation if time scales! Implying k off [ nS x X ] = k on X 0 S n x , which, when replaced in conservation law, yields k off k on [ nS x X ] = ( X T - [ nS x X ]) S n x . Finally, by solving for bound X [ nS x X ] X T = S n x K n x + S n x Hill function , where K n x = k off / k on ( n = 1 Michaelis-Menten equation). Hill equation can be considered as the probability of finding a bound (active) X . Jelenkovi´c (Columbia University) Network Motifs 5 / 32
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Binding a repressor to a promoter X and D (promoter region) bind to form [ XD ] . Conservation law yields D + [ XD ] = D T . Then, the rate of formation of [ XD ] is given by d [ XD ] dt = k on XD - k off [ XD ] . Again, due to the separation of time scales!, the preceding equation reaches its steady state within a second, implying, in conjunction with the conservation law, yields, D D T = 1 1 + X / K d , where K d = k off / k on .
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E6010_Lecture2_NetMotif_Autoregulation_NP - Autoregulation:...

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