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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
/1( ) s BE k TP s e
ZProbability
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.
/s BE k T
s
Z ePartition functionBoltzmann 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.
( )/1( ) s s BE N k TP s e
Z
Number of molecules in state s
Chemical potential
0
0
ln ( )B
ck T T
c
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:
( )/ /1( ) [ ] [ ]s A A B B B s BA BG n n k T G k Tn nP s e A B e
Z
Free energy Number of bound molecules
Chemical potential Concentration
[Shea & Ackers, JMB, 1985]
ΔGs usually consists of two parts: protein-DNA interaction energy; and protein-protein interaction energy
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Transcription Factor-DNA Binding
A
( )/ ( )/max[ ] [ ] ( )B BG S k T E S k Tq A e A K S e
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
max( ) / ( ) ( )BE S k T LLR S LLR S Mismatch energy
Log-likelihood ratio score
1
qP
q
Site occupancy
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Gene Expression and Promoter Occupation
mRNA level: [ ][ ]
d mP m
dt
At steady state: *[ ] /m P
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
/
/
( )/
(0,0) 1
(0,1) [ ]
(1,0) [ ]
(1,1) [ ][ ]
P B
A B
A P A P B
G k TP
G k TA
G G G k TA P
W
W P e q
W A e q
W A P e q q
Strength of interaction between A and RNAP, in the range of 20~100
Promoter occupancy:
(0,1) (1,1)
(0,0) (0,1) (1,0) (1,1) 1P A P
A P A P
q q qW WP
W W W W q q q q
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Transcriptional Repression by Exclusion
( 0, 0) 1
( 0, 1)
( 1, 0)
( 1, 1) 0
R P
R P P
R P R
R P
W
W q
W q
W
Promoter and OR cannot be
simultaneously occupied
(0,1) (1,1)
(0,0) (0,1) (1,0) (1,1) 1P
R P
qW WP
W W W W q q
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Combinatorial Transcriptional Control (I)
,( ) ji ii i j
i i j
W q
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)
: 1
( )P
ONZ W
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:
: 0
( )P
OFFZ W
Probability that the promoter is occupied by RNAP:
ON
ON OFF
ZP
Z Z
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Synergistic Activation
Assumption: RNAP can simultaneously contact two TFs, A and B.
(0,0,0) 1
(1,0,0)
(0,1,0)
(1,1,0)
AOFF
B
A B
W
W qZ
W q
W q q
(0,0,1)
(1,0,1)
(0,1,1)
(1,1,1)
P
A A PON
B B P
A B A B P
W q
W q qZ
W q q
W q q q
1 1
1 1A A B B
PA B
q qP q
q q
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Competitive Activation
(0,0,0) 1
(1,0,0)
(0,1,0)
(1,1,0) 0
AOFF
B
W
W qZ
W q
W
Assumption: binding of A or B excludes the other factor.
1
1A A B B
PA B
q qP q
q q
(0,0,1)
(1,0,1)
(0,1,1)
(1,1,1) 0
P
A A PON
B B P
W q
W q qZ
W q q
W
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Computing Partition FunctionsProblem: 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:
[ ]
( ) ( [ ])i
Z i W i
For a state σ[i], suppose the nearest bound site of i is j, then:
( [ ]) ( [ ]) ( , ) ( )W i W j i j q i
Sum over all possible values of j, and all states:
( ) ( ) ( ) ( ) 1j i
Z i q i i j Z j
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 biologyhttp://matisse.ucsd.edu/~hwa/class/w07/
• Biological backgroundAlberts et al, Molecular Biology of the Cell
• Physical backgroundNelson, Biological Physics: Energy, Information, Life
• Thermodynamic Modeling of transcriptional regulationBuchler et al, On schemes of combinatorial transcription logic, PNAS, 2003Berg and von Hippel, Selection of DNA binding sites by regulatory proteins, Trends Biochem Sci, 1998