The Structure, Function, and Evolution of Biological Systems

The Structure, Function, and Evolution of Biological Systems

Instructor: Van SavageSpring 2010 Quarter

4/13/2010

Recent papers using models of epistasis:Michel, Yeh, Chait, Moellering, Kishony

Measures of epistasisSince covariance is as fundamental as fitness, why notdefine relative covariance instead of relative fitness. Wedefine it relative to tri-modally binned covariance that itself varies, so relative to a shifting baseline.

€

˜ ε =Cov(wx,wy )

BinnedCov(wx,wy )=

wxy − wxwy

˜ w xy − wxwy

Absolute covariance

Relative covariance

€

ε =Cov(wx,wy ) = wxy − wxwy

Measures of epistasis—based onFBA predictions in yeast

Sort of unimodal distribution goes to trimodal distributionOpposite of Lenki et al. because synergy is enriched. Why?

….and some pathogens grow very quickly

a1-phm-gro.wmv

They can be killed by antibiotics…

a1-phm-kil.wmv

…but some bacteria can become resistant to the drug

Resistant Bacterium

Antibiotic

Sensitive Bacterium X X

Resistance confers a large fitness advantage in the presence of the drug

X X

Resistant bacteria, CFP

Sensitive bacteria, YFP

compDOX.mpg

Antibiotic resistance a growing public health threat

Years

I. How do drugs interact with each other, and how can we use their interactions to determine their mechanisms of action?

II. How do drug interactions affect the evolution of drug-resistant bacteria?

III. Future Directions: What role do birds play in the transmission of drug-resistant bacteria?

Main Questions

Multiple drugs combine to fight bacteria

Drug A Drug B

Two drugs can interact with each other to produce varying effects

Can we do reverse and cluster monochromatically to find functional groups?

Construct network for all pairwise interactions,Start with each gene in its own group. Cluster by pairs if they interact with other genes in same way.Require monochromaticity, each group must interact with allother groups in same wayWithin a group there is no requirement for monochromaticityMake cluster sizes as large as possible

Cluster Movie

How clusterable are networks?Is clustering unique?If not, which instantiation is chosen?

Drug-Drug Network Functional Classification

Cell Wall

Aminoglycosides

Folic Acid

30S50S

DNA

Protein Synthesis

Yeh, et al. – Nature Genetics 2006

Drug-Drug Network Functional Classification

Cell Wall

Aminoglycosides

Folic Acid

30S50S

DNA

Functional classification of a new drug

Protein Synthesis

Yeh, et al. – Nature Genetics 2006

Drug-Drug Network Functional Classification

Cell Wall

Aminoglycosides

Folic Acid

30S50S

DNA

Protein Synthesis

Yeh, et al. – Nature Genetics 2006

Drug-Drug Network Functional ClassificationCell Wall

Aminoglycosides

Folic Acid

30S50S

DNA

Protein Synthesis

Yeh, et al. – Nature Genetics 2006

Drug-Drug Network Functional Classification

Cell Wall

Aminoglycosides

Folic Acid

30S50S

DNA

Putative novel action mechanisms

Protein Synthesis

Yeh et al. – Nature Genetics 2006

Conclusions (part 1)

• Drugs can be classified by their underlying mechanism of action based only on properties of their interaction network.

• Drugs with novel mechanism of action can be identified as drugs that cannot be classified with any existing groups.

How do drug interactions affect the evolution of resistance?

Main result: Antagonism, typically avoided in clinical settings, better slows the emergence of resistant bacteria

Some drug concentrations select for resistance

MIC: Minimal Inhibitory Concentration

Freq

uenc

y of

resis

tanc

e

Drug concentrationMIC MPC

MutantSelectionWindow

0

1

0

10-8

10-4

wild type

MPC: Mutant Prevention ConcentrationThe Mutant Selection Window is one measureof the potential to evolve resistance

Dong et al. 1999, Drlica 2003

In two-drug treatments, the “Mutant Selection Window” becomes an “area” of drug concentrations.

Freq

uenc

y of

resis

tanc

e

Drug concentrationMIC MPC

MutantSelectionWindow

0

1

0

10-8

10-4

Single Drug

Concentration of drug X

Multi-drug

Conc

entr

ation

of d

rug

Y

Michel,Yeh, et al. – PNAS 2008Dong et al. 1999, Drlica 2003

Concentration of drug X

Conc

entr

ation

of d

rug

Y

We want to minimize the area that resistant mutants can grow. For distance, we choose straight lines drawn

through the origin. Why?

These lines imply constant ratio of drug concentrations. This is what would be designed in a single pill and the amount

prescribed would push you up and down this line. It would signal how much more of drug to prescribe to kill of

resistants and not just wild type. Could also look for lowest dosage that gives MPC.

Imaging platform delivers resistance frequencies on 2-D drug gradient

Michel,Yeh, et al. – PNAS 2008

Selection for resistance strongly depends on the drug combination

Michel,Yeh, et al. – PNAS 2008

1

103

MSW

Drug ratioERY FUSERY:FUS

102

10

Drug ratioAMI FUSAMI:FUS

1

103

MSW

102

10

Another view of antibiotic interactionsIsobolograms

MICA

MIC

B

MICA

MIC

B

MICA

MIC

B

Loewe additivity

Effect of drugs are independent, so all that matters is total concentration.Can imagine then that Cx+Cy=Cx,MIC or Cy,MIC.

Every drug is normalized to its MIC, so the combined MIC line is defined by

€

Cx

Cx,MIC

+Cy

Cy,MIC

=1

Loewe additivity and epistatic additivity

MICA

MIC

B

MICA

MIC

B

MICA

MIC

B

Loewe additivity

Fitness is scaled by MIC line for each drug independently. Combination is product of the two, and then just set Fxy equal to 0.

€

Fxy = FxFy = 1− Cx

Cx,MIC

⎛

⎝ ⎜

⎞

⎠ ⎟ 1−

Cy

Cy,MIC

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟~ 1− Cx

Cx,MIC

+Cy

Cy,MIC

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Suppression

Antagonism

The shape of equal inhibition lines in the dose-dose space defines the interaction between the drugs

Grow

th ra

te

Growth rate

[A]

[B]

MIC

SynergySynergy

Additivity

Minimal Inhibitory Concentration

MICA

MIC

B

Conc

entr

ation

of d

rug

B

A simple multiplicative model FAB = FA*FB

does not work

Synergy

Concentration of drug A

MICA

MIC

B

Antagonism

FAB<<1 FAB=1

wild-typegrowth

FA=1, FB=1 Multiplicative model predicts FAB=1

Michel,Yeh, et al. – PNAS 2008

There are many different resistance mechanisms

• efflux pump

• target affinity

• drug degradation

resistant mutants see lower levels of drug

Resistant mutants “see” lower effective drug concentrations

Concentration of drug A

Conc

entr

ation

of d

rug

B

wildtype

resistantmutant

Rescaling

Chait, Craney, Kishony – Nature 2007

Model for single drug

Can express frequency of bacteria at concentration Cx as

€

Fx (Cx ) = − dF(Cx )dCx

dCxCx, MIC

∞

∫ = −F(Cx )[ ]Cx, MIC

∞ ~ −F(∞) + F(Cx,MIC ) = F(Cx,MIC )

Recognize the probability density

€

p(Cx ) = − dF(Cx )dCx

Can also use theta/heaviside/step function or their eta function

€

Fx (Cx ) = ηCx,MIC

Cx

⎛ ⎝ ⎜

⎞ ⎠ ⎟p(Cx )dCx

Cx, MIC

∞

∫

Model for two drugs

By analogy,

€

Fxy (Cx,MIC ,Cy,MIC ) =Cy, MIC

∞

∫ η xy p(Cx,Cy )dCxdCyCx, MIC

∞

∫

Can directly measure and enforce MIC curve. Trying to use this and other information to predict MPC curve and thus mutant selection window. How do we approximate the joint probability distribution.Two extremes.

Independent probability distribution

€

pind (Cx,Cy ) = p(Cx )p(Cy )

If drugs are the same, this is extreme correlation in probability distribution. Does NOT imply additive epistasis at all.

€

pcorr (Cx,Cy ) = p(Cx )p(Cy )

Model for two drugs

Choose actual probability density to be linear combination of these two with free parameter ξ to tune model to data.

Measure px, py, and ηxy and all of these are experimentally tractable

Free parameter ξ is only part of model fit

Important to build simple models in terms of measurable parameters and only a few free parameters

€

pcorr (Cx,Cy ) = ξpxycorr + (1−ξ )pxy

ind

Single drug resistance and drug interactions predict multidrug resistance

Concentration of Drug A0

1

0

00

Conc

entr

ation

of D

rug

B

resistance to the drugcombination

single drug resistancedrug interactions

cross-resistance

measurements

1 parameter

MathematicalModel

Michel,Yeh, et al. – PNAS 2008

The mathematical model is in good agreement with the experimental data

EXPE

RIMEN

T

ERY

FUS

AMPCP

RAMI

FUS

MODE

L

ERY

FUS

AMI

FUS

AMP

CPR

Michel,Yeh, et al. – PNAS 2008

Synergistic drugs kill more effectively than antagonistic drugs. But how do they impact resistance? Consider simple example with only three populations: wild type, single type

resistant to drug A, and single type resistant to drug B.Independent probability distributions.

Resistant to AResistant to B

Some combinations of the two drugs better reduce the potential to evolve resistance

Resistant to AResistant to B

best “effective drug”A:B

Concentration of drug A

Conc

entr

ation

of d

rug

B

MSW “effective drug”2A:B

MSW

“effective drug”2A:3B

MSW

Antagonism

1

0

“effective drug” 2A:B0 1 (MIC) MPC

MSW1

0

“effective drug” 2A:3B0 1 (MIC) MPC

MSW1

0

“effective drug” A:B0 MIC MPC

MSW

Predicted resistance

Some combinations of the two drugs better reduce the potential to evolve resistance

best “effective drug”A:B

Concentration of drug A

Conc

entr

ation

of d

rug

B

MSW

Synergy Resistant to AResistant to B

1

0

“effective drug” A:B0 MIC MPC

MSW

Antagonistic combinations have smaller mutant selection windows: windows are scaled relative to MIC like everything else as an inset

Conc

entr

ation

of d

rug

B

Synergy

Concentration of drug A

Antagonism

1

0 0 MIC MPC

MSW1

0 0 MIC MPC

MSW

Michel,Yeh, et al. – PNAS 2008

Antagonistic combinations predicted to better reduce selection for resistance

Michel,Yeh, et al. – PNAS 2008

Suppression

Antagonism

The shape of equal inhibition lines in the dose-dose space defines the interaction between the drugs

Grow

th ra

te

Growth rate

[A]

[B]

MIC

SynergySynergy

Additivity

Minimal Inhibitory Concentration

Synergy

Bact

eria

l Fitn

ess

-+ Drug A

Drug B-

+

AR

BR

A simple model suggests profound impact of drug interactions on selection for resistance

-+ Drug A

Drug B-

+

BR

A R

Suppression

-+ Drug A

Drug B-

+

BR

AR

Directional Suppression

Hypothesis: suppressive combinations can select against resistance

There is very little fitness cost to resistance in a drug free environment

Resistant bacteria, CFP

Sensitive bacteria, YFP

compLB.mpg

Conclusions

• Synergistic combinations, currently preferred in clinical settings, may actually favor resistance

• Trade-off between immediate killing efficacy and future evolution of resistance

Next class we will move onto papers using networks motifsfor gene regulation

First Homework set is due in two weeks (April 20, 2010).