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A binds independently from other TFs A activates = a/(1+a) A represses = 1/(1+a) A represses, B activates = b/(1+a+b+ab) A and B compete for the same site A and B activate = (a+b)/(1+a+b) A represses, B activates = b/(1+a+b) A and B repress = 1/(1+a+b) B binds to bound A (compulsory order binding) B activates activator A = ab/(1+a+ab) B represses activator A = a/(1+a+ab) B activates repressor A = (1+a)/(1+a+ab) A and B bind as a complex A and B form a hetero- oligomeric activator = ab/(1+ab) A forms a homo- oligomeric activator = a n /(1+a n ) A, B, and C, form a complex repressor = 1/(1+a n b m c k ) Active configurations Logical equivalent Expression for Weights 0ABAB W0W0 WAWA WBWB W AB --- X A BA B 0001 - XXX A B + (1- ) 011 - X -- A BA B (1- ) 010 The Logic of Gene Regulation Maria J. Schilstra & Hamid Bolouri Biocomputation Research Group, University of Hertfordshire, Hatfield, UK http://strc.herts.ac.uk/bio/maria/NetBuilder/index.html Introduction: bio-logic Informal descriptions of experimental observations on gene expression patterns often translate readily into the language of formal logic: Informal statement Proposed logical equivalent 1) “Transcription factors A and B are both necessary for the expression of gene G” G = A B “Either of the transcription factors A or B is sufficient for expression of gene G” G = A B “Expression of gene G is stimulated by A but repressed by B” G = A B 1) : logical AND, : logical OR, : logical NOT operators Questions 1. How do these logical operations relate to the biochemistry of gene expression? 2. Do the rules of Boolean algebra hold for ‘bio-logical’ operations? 3. Is the logical approach always justified? Transcription initiation: a minimal model Transcription factors (TFs) bind to their cis-regulatory binding sites, where they somehow affect the transcription initiation rate. A B A B RNApol The transcription initiation rate k initiation is determined by: 1.The maximum initiation rate (k iniMax ) 2.The occupancy of the TF binding sites (y 0, y A, y B, y AB ) 3.The extent to which each complex stimulates initiation (w 0, w A, w B, w AB ) A + B - A B ++ No A or B: no initiation Only A: weak stimulation Only B: repression of the effect of other factors A and B together: significant stimulation EXAMPLE: If A and B do not affect each other’s binding (i.e. K A /K A(B) = 1), then the expression for can be linearized to: = W 0 + W A + W B + W AB ( = [B]/(K B + [B]), = [B]/(K B + [B]), W 0 = w 0, W A = w A – W 0, w AB = w AB – (W 0 +W A +W B ) From TF binding to cis-regulatory logic The modulation factor = W 0 + W A + W B + W AB determines the transcription initiation rate. Special cases: 1) A B A B 0 ABAB The expressions for the modulation factor hold for Boolean and continuous values between 0 and 1 The expressions for obey the laws of Boolean algebra (associative, commutative, De Morgan’s, etc.) An expression for can substitute a variable in another expression for . Primitives In the expression for , and are primitive: they cannot be linearized further. If binding of A depends on the presence of bound B, it may be necessary to use a primitive that contains [A] and [B] terms. Below: various primitives. a = [A]/K A, b = [B]/K B Fraction of time that the binding sites for A and B are unoccupied (y 0 ), occupied only with A (y A ), or simultaneously with A and B (y AB ). Different TF combinations stimulate or repress transcription to different extents. Dependence of the dynamics of the concentration of the gene product P on the transcription initiation rate, k initiation : Answers 1. The biochemistry of gene expression relates to combinatorial logic as shown above 2. The rules of Boolean algebra hold for bio-logical operations, but... 3.... strictly speaking, the logical approach is only justified for independent cis-regulatory binding sites. In practice, its use (including the choice of primitives) will depend on the accuracy of the data to be modelled, and on prior knowledge of the system

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Laws of Boolean Algebra Commutative Law Associative Law Distributive Law Identity Law De Morgan's Theorem.

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