The Structure, Function, and
Evolution of Biological Systems
Instructor: Van Savage
Spring 2010 Quarter
4/13/2010
Recent papers using models of epistasis:
Michel, Yeh, Chait, Moellering, Kishony
Measures of epistasis
Since covariance is as fundamental as fitness, why not
define relative covariance instead of relative fitness. We
define it relative to tri-modally binned covariance that
itself varies, so relative to a shifting baseline.
Absolute covariance
  Cov ( w x , w y )  w xy  w x w y
Relative covariance

˜

Cov ( w x , w y )
BinnedCov ( w x , w y )

w xy  w x w y
w˜ xy  w x w y
Measures of epistasis—based onFBA
predictions in yeast
Sort of unimodal distribution goes to trimodal distribution
Opposite 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
Antibiotic
Resistant Bacterium
Sensitive Bacterium
X X
Resistance confers a large fitness advantage
in the presence of the drug
Resistant
bacteria, CFP
compDOX.mpg
X X
Sensitive bacteria,
YFP
Antibiotic resistance a growing public
health threat
Years
Main Questions
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 drugresistant bacteria?
III. Future Directions: What role do birds play in the transmission
of drug-resistant bacteria?
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 all
other groups in same way
Within a group there is no requirement for monochromaticity
Make 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
DNA
Aminoglycosides
Folic Acid
50S
30S
Protein Synthesis
Yeh, et al. – Nature Genetics 2006
Drug-Drug Network  Functional Classification
Cell Wall
DNA
Aminoglycosides
Folic Acid
Functional classification of
a new drug
50S
30S
Protein Synthesis
Yeh, et al. – Nature Genetics 2006
Drug-Drug Network  Functional Classification
Cell Wall
DNA
Aminoglycosides
Folic Acid
50S
30S
Protein Synthesis
Yeh, et al. – Nature Genetics 2006
Drug-Drug Network  Functional Classification
Cell Wall
DNA
Aminoglycosides
Folic Acid
50S
30S
Protein Synthesis
Yeh, et al. – Nature Genetics 2006
Drug-Drug Network  Functional Classification
Cell Wall
DNA
Aminoglycosides
Folic Acid
Putative novel action
mechanisms
50S
30S
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
Frequency of
resistance
wild type
Mutant
Selection
Window
1
10-4
10-8
0
0
MIC
Drug concentration
MPC
The Mutant Selection MPC:
Window
is one
measure
Mutant
Prevention
Concentration
MIC:
Minimal
Inhibitory Concentration
of 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.
1
Multi-drug
Mutant
Selection
Window
10-4
10-8
0
0
MIC
MPC
Drug concentration
Dong et al. 1999, Drlica 2003
Concentration of drug Y
Frequency of resistance
Single Drug
Concentration of drug X
Michel,Yeh, et al. – PNAS 2008
Concentration of drug Y
We want to minimize the area that resistant mutants
can grow. For distance, we choose straight lines drawn
through the origin. Why?
Concentration of drug X
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
102
102
MSW
103
MSW
103
10
1
10
1
ERY
ERY:FUS
Drug ratio
FUS
AMI
AMI:FUS
FUS
Drug ratio
Michel,Yeh, et al. – PNAS 2008
Another view of antibiotic interactions
MICB
MICB
MICB
Isobolograms
MICA
MICA
MICA
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
C x ,MIC

Cy
C y ,MIC
1
MICB
MICB
MICB
Loewe additivity and epistatic additivity
MICA
MICA
MICA
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.
F xy

Cx
 F x F y  1 
 C x ,MIC

Cy

1 
 C y ,MIC

 C
Cy
x


 ~ 1  

C y ,MIC

C x ,MIC




[B]
The shape of equal inhibition lines in the dose-dose space
defines the interaction between the drugs
Suppression
Antagonism
Synergy
Growth rate
Growth rate
[A]
MIC
Minimal Inhibitory Concentration
FAB<<1
wild-type
growth
Antagonism
MICB
Synergy
MICB
Concentration of drug B
A simple multiplicative model FAB = FA*FB
does not work
FAB=1
MICA
MICA
Concentration of drug A
FA=1, FB=1 Multiplicative model predicts FAB=1
Michel,Yeh, et al. – PNAS 2008
There are many different resistance
mechanisms
• efflux pump
resistant mutants see
lower levels of drug
• target affinity
• drug degradation
Concentration of drug B
Resistant mutants “see” lower effective drug
concentrations
Rescaling
wild
type
resistant
mutant
Concentration of drug A
Chait, Craney, Kishony – Nature 2007
Model for single drug
Can express frequency of bacteria at concentration Cx as

F x (C x ) 

C x , MIC

dF (C x )
dC x

dC x   F (C x ) C
x , MIC
~  F ( )  F (C x ,MIC )  F (C x ,MIC )
Recognize the probability density
p(C x )  
dF (C x )
dC x
Can also use theta/heaviside/step function or their eta function


F x (C x ) 

C x , MIC
C

x ,MIC
 
p(C x ) dC x
 C x 
Model for two drugs
By analogy,


F xy (C x ,MIC ,C y ,MIC ) 


xy
p(C x ,C y )dC x dC y
C y , MIC C x , 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
p
ind
(C x ,C y )  p(C x ) p(C y )
If drugs are the same, this is extreme correlation in probability distribution.
Does NOT imply additive epistasis at all.

p
corr
(C x ,C y )  p(C x ) p(C y )
Model for two drugs
Choose actual probability density to be linear combination of these
two with free parameter ξ to tune model to data.
p
corr
(C x ,C y )   p xy
corr
 (1   ) p xy
ind
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
Single drug resistance and drug interactions
predict multidrug resistance
measurements
Concentration of Drug B
single drug resistance
drug interactions
1 parameter
cross-resistance
Mathematical
Model
0 0
resistance to the drug
combination
1
0 0
Concentration of Drug A
Michel,Yeh, et al. – PNAS 2008
FUS
ERY
AMI
FUS
AMP
CPR
ERY
MODEL
FUS
CPR
FUS
EXPERIMENT
The mathematical model is in good agreement with
the experimental data
AMP
AMI
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 B
Resistant to A
Some combinations of the two drugs better
reduce the potential to evolve resistance
Concentration of drug B
Antagonism
Resistant to B
Resistant to A
“effective drug”
Predicted resistance
2A:3B
best “effective drug”
MSW
A:B
“effective drug”
MSW
2A:B
MSW
Concentration of drug A
1
0 0
MSW
MSW
MSW
1MIC
(MIC)MPCMPC MPC
“effective
“effective
“effectivedrug”
drug”
drug”2A:3B
2A:B
A:B
Some combinations of the two drugs better
reduce the potential to evolve resistance
Concentration of drug B
Synergy
Resistant to B
Resistant to A
best “effective drug”
A:B
1
MSW
MSW
Concentration of drug A
0 0
MIC
MPC
“effective drug” A:B
Antagonistic combinations have smaller mutant
selection windows: windows are scaled relative
to MIC like everything else as an inset
Concentration of drug B
Synergy
Antagonism
MSW
MSW
1
00
1
00
MIC MPC
MIC MPC
Concentration of drug A
Michel,Yeh, et al. – PNAS 2008
Antagonistic combinations predicted to better
reduce selection for resistance
Michel,Yeh, et al. – PNAS 2008
[B]
The shape of equal inhibition lines in the dose-dose space
defines the interaction between the drugs
Suppression
Antagonism
Synergy
Growth rate
Growth rate
[A]
MIC
Minimal Inhibitory Concentration
A simple model suggests profound impact of drug interactions
on selection for resistance
Bacterial Fitness
Synergy
-
Suppression
+
+
-
+
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
compLB.mpg
Sensitive bacteria,
YFP
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 motifs
for gene regulation
First Homework set is due in two weeks (April 20, 2010).