Analysis Of Binding Interactions

Analysis of Binding Interactions: Obtaining Meaningful Values with Monte Carlo Simulations Simon Chiang

Thursday, April 16, 2009

Theoretical Studies of Protein-DNA Binding Interactions: Goal of studies:

Measurement of Energetic Parameters

Model Effect of Proteins on gene Expression

“Questionable” Resolution: -Poorly reproducible values -Large standard deviations

“Meaningful” Resolution: -Reproducible values -Small standard deviations

Identification

Rational Design

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Monte Carlo Simulations

Resolving Parameters from Data

Fractional Saturation

•

Results of a single site protein-DNA binding experiment

1.0 0.8 0.6 0.4 0.2 0.0 -12

-10

-8

Log [Protein]

Single Site Binding Isotherm Thursday, April 16, 2009

+

Keq ΔG = -RT ln Keq

Parameter to Resolve : ΔGbinding

Fitting Program Detail Fitting Program Evaluate Fit of Data to Curve (Least-Squares)

Calculate Curve Refine Guess Values

1.0 0.8 0.6

ΔGguess

0.4 0.2 0.0 -12

-10 Log [X]

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-8

Fit Value ± Uncertainty Fractional Saturation

Fractional Saturation

Model for Data 1.0 0.8 0.6

ΔGfit±σGfit

0.4 0.2 0.0 -12

-10 Log [X]

-8

Notes on fitting 

Fit Value only makes the fit curve optimized to a single, imperfect data set. Data Set 1

ΔGfit1 = -10.2 (kcal/mol)

Different Data Set, Different Curve, Different Fit Value



Data Set 2

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ΔGfit2 = -10.1 (kcal/mol)

x5000  

Fit Program ΔGfit1 ΔGfit2 ΔGfit3 ΔGfit4 ΔGfit5

Number of Occurences

Criteria for a Meaningfully Resolved Parameter

σ Fit ΔG Value

Single Peak, resolution of a single value

Fit Values over many data sets follow a distribution with a single peak in probability Distribution limited near this value (small uncertainty)

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Number of Occurences

Ideally Distributions are Gaussian

• •

σσ Fit Value

Fit programs generally report uncertainties in σ Standard deviations are only rigorously accurate for Gaussian Distributions, and can be wildly inaccurate for other distributions.

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Fit Program

x5000

Number of Occurences

Not all parameters follow Gaussian Distributions

Fit Value

No single peak in probability

Perilous to trust the results of a fit program: • No resolution of a single value •Reported σ could be very inaccurate (non-gaussian) Thursday, April 16, 2009

Results of 3 experiments given by dots

Number of Occurences

Triplicate is not enough.

Fit Value

Suggests a gaussian with low σ, resolving a single value 

Repeating an experiment many times is necessary to establish that a parameter is meaningfully resolved (ie reproducible).

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Monte Carlo Simulations:

Repeating experiments computationally

Simulated Data Sets Experimental Data Sets and Model for Data

Number of Occurences

Basic Procedure:

Fit Parameter Value

Parameter Distributions

•Simulated data points are made to resemble and have uncertainties equal in magnitude to the experimental data points Thursday, April 16, 2009

Generating Simulated Data A non-trivial task 

Several methods exist; subtle in approach, sometimes long in statistical acceptance.



A Universal Requirement: Uncertainties on experimental data points must be well-resolved 

Generally requires repeating an experiment several times. Well-resolved uncertainties on the data

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vs σ

Fit Parameter Value





Number of Occurences

Number of Occurences

Utility of Monte Carlo Simulations

Fit Parameter Value

Simulations allow an assessment of the parameter distributions from several rather than thousands of repetitions. Generally ~ few hrs CPU time for 10k data sets

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Overview of my program: •Able to study multi-site, arbitrarily configured systems binding multiple ligands undergoing linked rxns

+

Binding Simulator Program Binding Curves & Derivatives •

Experiments that monitor Fractional Saturation

Monte Carlo Simulations •Experimental/Computer generated Data

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Bacteriophage λ Right Operator (OR) λ repressor dimer cro gene (lytic state) OR1

OR2

(Ptashne, Mark A Genetic Switch 1986)

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OR3

cI gene (lysogenic state)

OR Statistical Mechanical Model •Complex set of equations: determined by five free energy parameters. (Ackers, G. et al PNAS (1982) 79, 1129-1133) 3 Intrinsic site binding free energies:

ΔG1 ΔG2 ΔG3

ΔG12

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ΔG23

Why not to blindly trust fit programs: A dramatic OR example Fractional Saturation

1.0 0.8 0.6

Site 1 Site 2

0.4

Site 3

0.2 0.0

-12

-11

-10

-9

-8

-7

-6

Log [ λ repressor dimer]

Fit Values for two data sets (kcal/mol): Set 1

ΔG1= -12.5 ΔG2= -10.5 ΔG3= -9.4 ΔG12= -2.9 ΔG23= -2.9

Thursday, April 16, 2009

ΔG Set1=2-12.5 ΔG2= -10.5 ΔG3= 0 ΔG12= -2.9 ΔG23= -12.9

Erroneous Conclusions: Resolved Parameters ΔG1, ΔG2*, ΔG12* Unresolved Parameters ΔG3, ΔG23

-12.0

-12.0

number of occurences

ΔG1 (σ = 0.1)

• •

-4.0

number of occurences

-12.4

-10.0

-11.0

ΔG2 (‘σ’= 0.5)

-3.0

-2.0

ΔG12 (‘σ’= 0.5)

-1.0

number of occurences

-12.8

number of occurences

number of occurences

Monte Carlo Analysis of OR

-11.0

-10.0

-9.0

-8.0

ΔG3 (‘σ’= 0.5) <σ> = kcal/mol -4

-2

0

ΔG23 (‘σ’= 0.5)

Fit values, uncertainties only meaningful for ΔG1 ‘standard deviations’ on other parameters are inaccurate

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Resolution problems endemic to cooperative systems Root of poor resolution is a signal to noise issue. Fractional Saturation





1.0 0.8 0.6

Site 1 Site 2

0.4

Site 3

0.2 0.0

-12

-11

-10

-9

-8

-7

Log [ λ repressor dimer]

-6

Example: Sites 1 and 2 fully saturated due to cooperativity at concentrations where site 3 begins to fill.  Therefore there is little direct data on ΔG23; the pairwise interaction between just site 2 and site 3.

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OR parameter resolution problem recognized/addressed in the literature 

Published solution uses empirically chosen mutant OR operators that emphasize poorly resolved parameters:

OR+

OR-1

OR-3

OR-1 -3

•

Global analysis of wild-type OR and 3 mutant operators required resolution. (Brenowitz, M., et al. PNAS (1986) 83, 8462-8466)

Thursday, April 16, 2009

-12.2

ΔG1 (σ = 0.1)

-3.2

-3.0

-10.7

-2.8

ΔG12 (σ = 0.1)

-10.5

-10.3

number of occurences

-12.4

ΔG2 (σ = 0.1)

-2.6

number of occurences

-12.6

number of occurences

-12.8

number of occurences

number of occurences

MC Analysis of published OR resolution technique:

-9.6

-9.2

-8.8

ΔG3 (‘σ’ = 0.2) <σ> = kcal/mol -3.6

-3.2

-2.8

-2.4

ΔG23 (‘σ’ = 0.2)

•Standard deviations for ΔG3, ΔG23 still somewhat inaccurate •Overall resolution is much better (more gaussian, lower σ) Thursday, April 16, 2009

Improved resolution with rational choice of mutants • Study of distributions clarifies how different mutants resolve specific parameters. OR+ OR-2

OR-1 -3

• Global analysis of wild-type OR and 2 mutant operators predicted to improve resolution

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-12.4

-10.7

number of occurences

ΔG1 (σ = 0.05)

-3.2

-10.5

number of occurences

-12.5

-10.3

-9.7

-3.0

-2.8

ΔG12 (σ = 0.1)

-2.6

-9.5

-9.3

ΔG3 (σ = 0.06)

ΔG2 (σ = 0.06) number of occurences

-12.6

number of occurences

number of occurences

MC Analysis of rationally designed resolution technique:

<σ> = kcal/mol -3.5

-3.0

-2.5

ΔG23 (σ = 0.2)

-2.0

•Rational design results in meaningful fit values and standard deviations for all parameters; resolution in fact improves. Thursday, April 16, 2009

Summary Monte Carlo Simulations examine the resolution of fit parameters without literally repeating experiments thousands of times.  Analysis of distributions can assist in the rational design of experiments. 



In the future my program and the insights obtained from analyzing OR will be applied to study the 4-site PRE system.

Thursday, April 16, 2009

Acknowledgements UCHSC Dept. of Pharmaceutical Sciences 

Dr. David Bain

Bain Laboratory   

Dr. Aaron Heneghan Nancy Berton Michael Miura

Thursday, April 16, 2009