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Thermodynamic Models of Gene Regulation Xin He CS598SS 04/30/2009

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Thermodynamic Background: Micro-states Micro-states: a bio-molecular system can exist in a number of different “states”. Folded state Unfolded state A Protein: DNA: Unbound state Bound state

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Thermodynamic background: Boltzmann Distribution Probability of state s Boltzmann constant Temperature Energy of state s Intuition: if a state has lower energy, the additional energy (because the total energyis conserved) is used to increase the entropy of the environment, thus it is more likely. Partition function Boltzmann weight

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Thermodynamic Background: Gibbs Distribution Suppose the system exchanges, not just energy, but also molecules, with its environment, the probability of a state will also depend on the number of molecules in the state. Number of molecules in state s Chemical potential Concentration Standard condition: e.g. 1mol/l Chemical potential at the standard condition

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Application of Gibbs Distribution to Protein- DNA Interaction A B A A promoter/enhancer sequence can bind multiple protein molecules. Suppose in one state s, two types of molecules A and B are bound, the probability of the state is given by: Free energyNumber of bound molecules Chemical potentialConcentration [Shea & Ackers, JMB, 1985] ΔG s usually consists of two parts: protein-DNA interaction energy; and protein-protein interaction energy

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Transcription Factor-DNA Binding A Question: what is the probability that a site is bound by its corresponding TF? Boltzmann weight of the bound state Equilibrium binding constant of the consensus site Mismatch energy Log-likelihood ratio score Site occupancy

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Gene Expression and Promoter Occupation mRNA level: At steady state: Transcription factors activate or repress gene expression level by modifying the promoter occupancy by RNAP. Probability of promoter occupation by RNAP mRNA degradation rate

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Transcriptional Activation by Recruitment Strength of interaction between A and RNAP, in the range of 20~100 Promoter occupancy:

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Transcriptional Repression by Exclusion Promoter and O R cannot be simultaneously occupied

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Combinatorial Transcriptional Control (I) Weight of a state TF-DNA, RNAP-DNA interactions TF-TF, TF-RNAP interactions Indicator variable of the i-th site

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Combinatorial Transcriptional Control (II) Total weight of all states where the promoter is occupied by RNAP: Total weight of all states where the promoter is not occupied by RNAP: Probability that the promoter is occupied by RNAP:

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Synergistic Activation Assumption: RNAP can simultaneously contact two TFs, A and B.

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Competitive Activation Assumption: binding of A or B excludes the other factor.

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Computing Partition Functions Problem: the number of states is exponential to the number of sites. To compute the partition function, one needs to sum over all states. Assumption: each bound TF interacts only with its neighboring TF Define σ[i] as a state where the last bound site is i, and W(.) be the weight of a state: For a state σ[i], suppose the nearest bound site of i is j, then: Sum over all possible values of j, and all states: Interaction of TF with site i Interaction between TFs bound at site i and j

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Transcriptional Activation in Eukaryotic Cells Transcription involves assembly of many more proteins (GTFs, co-factors) Enhancer sequences are often located far from the transcription start site DNA looping for distant activators to interact with proteins in the transcriptional machinery

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Transcriptional Repression in Eukaryotic Cells (I) A.Competitive DNA binding B.Masking the activation surface C.Direct interaction with the general transcription factors

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Transcriptional Repression in Eukaryotic Cells (I) D.Recruitment of repressive chromatin remodeling complexes E.Recruitment of histone deacetylases

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References Terrence Hwa’s course of quantitative molecular biology http://matisse.ucsd.edu/~hwa/class/w07/ Biological background Alberts et al, Molecular Biology of the Cell Physical background Nelson, Biological Physics: Energy, Information, Life Thermodynamic Modeling of transcriptional regulation Buchler et al, On schemes of combinatorial transcription logic, PNAS, 2003 Berg and von Hippel, Selection of DNA binding sites by regulatory proteins, Trends Biochem Sci, 1998

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