Thermodynamic Models of Gene Regulation

Xin He

CS598SS

04/30/2009

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Transcriptional Repression in Eukaryotic Cells (I)

A. Competitive DNA binding

B. Masking the activation surface

C. Direct interaction with the general transcription factors

Transcriptional Repression in Eukaryotic Cells (I)

D. Recruitment of repressive chromatin remodeling complexes

E. Recruitment of histone deacetylases

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