An Introduction to Systems Biology
Chapter10: Optimal Gene Circuit Design
Protein-level optimality studies
Protein-level optimality – Agenda
1.  Fitness-functions
2.  Optimal expression levels of a protein under constant conditions
–  Case: Optimality and tuning of the expression level of betagalactosidase (LacZ).
3.  Optimal protein regulation under variable conditions
–  To regulate, or not to regulate
4.  Selection of regulatory circuitries
–  No regulation
–  Simple regulation
–  Feed-forward loops
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27041, Introduction to Systems Biology
1. Fitness functions
The fitness function (f):
•  Mathematic formulation of the ability to achieve a given objective
involved in competition i.e.:
–  Growth (Most common for microorganisms)
–  Production of antibiotics/toxic compounds
–  Making nutrients unavailable to competitors
•  Under the assumption of actual fitness, higher fitness means evolutionary
selection.
•  The fitness function is the foundation of optimality theory.
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27041, Introduction to Systems Biology
1. Fitness functions
The general formulation of the fitness function (f):
f =b−c
f: Fitness
b: Benefit in terms of the fitness
c: Cost in terms of the fitness
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Protein-level optimality – Agenda
1.  Fitness-functions
2.  Optimal expression levels of a protein under constant conditions
–  Case: Optimality and tuning of the expression level of betagalactosidase (LacZ).
3.  Optimal protein regulation under variable conditions
–  To regulate, or not to regulate
4.  Selection of regulatory circuitries
–  No regulation
–  Simple regulation
–  Feed-forward loops
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
2. Optimal expression levels of a protein
under constant conditions
Theory: Evolutionary selection will favour cells with the optimal expression
level of a given protein at a specific set of conditions.
Case example: The lac operon
(Dekel and Alon, 2005, Nature, 436, 588-592)
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Growth of E. coli on glucose + lactose
Glucose
Lag
Lactose
Inada, T.; Kimata, K. & Aiba, H.
Genes Cells, 1996, 1, 293-301
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Growth of E. coli on glucose and lactose
Glucose:
•  Fast uptake
•  Fast conversion
Lactose:
•  Slower uptake
•  Slower conversion
Faster growth = Higher fitness = Evolutionary selection!
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Growth of E. coli on glucose + lactose
Growth is faster on glucose!
Inada, T.; Kimata, K. & Aiba, H.
Genes Cells, 1996, 1, 293-301
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Regulation of the lac operon
Inada, T.; Kimata, K. & Aiba, H.
Genes Cells, 1996, 1, 293-301
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Investigating the regulation of the lacoperon
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Investigating the regulation of the lacoperon
Establishing the fitness function:
f =b−c
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Benefit of the lac operon
Benefit (b):
The relative increase in growth rate due to
LacZ
1. Each molecule of degraded lactose must
bring a benefit.
L
b( Z , L ) = δ ⋅ Z
K+L
b( Z ) = δ [ZLin ]
2. It is known that transport (by LacY) is
slower than the breakdown rate i.e. the
limiting factor.
L
VZ [ZLin ] ≈ VY [YL] = VY Y
KY + L
3. LacY and LacZ are on the same operon
Z =Y
4. Substituting out gives
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CBS, Department of Systems Biology
[ZLin ] ≈ VY
L
Z
VZ K + L
Y
27041, Introduction to Systems Biology
Benefit of the lac operon
Strategy:
Check the benefit of
wildtype levels of Z for
different levels of L
Experimental design:
Fully induce the Lac
operon with IPTG
saturation and measure
growth rates at
increasing levels of L
(up to saturation)
Results:
The benefit coefficient is
determined.
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CBS, Department of Systems Biology
b( Z , L ) =
δ ⋅Z ⋅L
K+L
27041, Introduction to Systems Biology
Cost evaluation
Strategy:
Induce different levels of
the lac-operon without
the benefits and measure
the reduction in growth.
Experimental design:
Add different levels of
IPTG during growth on
glycerol and measure the
reduction in growth and
the Lac expression.
Results:
The cost is a non-linear
function of the
expression.
Each new LacZ protein costs more than the previous one!
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Cost is non-linear!
Reasons:
•  Resource-supply is limited and shared
•  The proteins losing the resources for LacZ may
also be growth-limiting.
Mathematical derivation:
g = g max
R
KR + R
g ( Z ) = g max
R − εZ
K R + R − εZ
εK R
Z
g − g (Z )
=
c( Z ) =
R( K R − R) 1 − Z /(( K R + R) / ε )
g
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CBS, Department of Systems Biology
c( Z ) =
η ⋅Z
1− Z M
27041, Introduction to Systems Biology
Cost equation
c( Z ) =
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η ⋅Z
1− Z M
27041, Introduction to Systems Biology
Cost equation
Adding this equation gives
the red line here.
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CBS, Department of Systems Biology
c( Z ) =
η ⋅Z
1− Z M
27041, Introduction to Systems Biology
Calculation of fitness
Determining the fitness function:
# δZL & # ηZ &
f L (Z) = b(Z,L) − c(Z) = %
( −%
(
$ K + L ' $1− Z / M '
Observations:
•  Fitness is dependent on both Z and L
•  One specific maximum level of Z is not always optimal!
€
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Calculation of fitness for different values of L
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Calculation of fitness for different values of L
Observations:
•  Linear increase in benefit, non-linear
in cost gives sudden drop in fitness.
•  Wild-type levels of Z are optimal
when L=0.6 mM.
Further calculations:
•  L > 0.05 mM for the lactose operon
to increase fitness.
# δZL & # ηZ &
f L (Z) = %
( −%
(
$ K + L ' $1− Z / M '
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Calculation of optimal level of Z for different
values of L
The fitness function:
# δZL & # ηZ &
f L (Z) = %
( −%
(
$ K + L ' $1− Z / M '
Determining optimum:
" df L %
$
'=0
# dZ &
Optimum function:
Z opt
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⎡
η ( K + L) ⎤
= M ⎢1 −
⎥
δL ⎦
⎣
CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Calculation of optimal level of Z for different
values of L
Optimum function:
Z opt
⎡
η ( K + L) ⎤
= M ⎢1 −
⎥
δ
L
⎣
⎦
Experimental setup:
•  Evolution for 530 generations at a
specific lactose concentration
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Evolution study of protein-level optimality
Experimental setup:
•  Serial dilutions in 10 mL tubes
every 6.6 generations.
•  Fixed starting concentrations of
lactose
Observations:
•  After 2-500 generations, the level
of LacZ adjusts.
•  Larger changes take longer.
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Evolution study of protein-level optimality
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Evolution study of protein-level optimality
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Evolution study of protein-level optimality
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Evolution study of protein-level optimality
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Protein-level optimality – Agenda
1.  Fitness-functions
2.  Optimal expression levels of a protein under constant conditions
–  Case: Optimality and tuning of the expression level of betagalactosidase (LacZ).
3.  Optimal protein regulation under variable conditions
–  To regulate, or not to regulate
4.  Selection of regulatory circuitries
–  No regulation
–  Simple regulation
–  Feed-forward loops
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
3. Optimal protein regulation under variable
conditions
•  Optimization in a constant environment:
•  Regulate level of expression to optimize fitness.
•  Optimization in a variable environment:
–  When does regulation increase the fitness?
•  To regulate or not to regulate?
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
3. Optimal protein regulation under variable
conditions
Theoretical consideration using fitness functions:
An environment displays condition C with probability p. Protein Z is only
needed under condition C.
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Organism 1: Z always on
f1 = pb − c
Organism 2: Z is regulated
f 2 = pb − pc − r
Organism 3: Z is not present
f3 = 0
CBS, Department of Systems Biology
27041, Introduction to Systems Biology
3. Optimal protein regulation under variable
conditions
Organism 1: Z always on
f1 = pb − c
Organism 2: Z is regulated
f 2 = pb − pc − r
Organism 3: Z is not present
f3 = 0
Z always on:
f1 > f 2 ∧ f1 > f 3 ⇒ p > 1 − r / c ∧ p > c / b
Regulation selected:
f 2 > f1 ∧ f 2 > f 3 ⇒ p < 1− r /c ∧ p > r /(b − c)
Z not present:
f 3 > f1 ∧ f 3 > f 2 ⇒ p < c /b ∧ p < r /(b − c)
€
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CBS, Department of Systems Biology
€
27041, Introduction to Systems Biology
3. Optimal protein regulation under variable
conditions
Organism 1: Z always on
Organism 2: Z is regulated
(c/b, 1-c/b)
Organism 3: Z is not present
Relative cost of
regulatory system
r/c
p
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
r/c
1
Relative
cost of
System
System
always
always
OFF
ON
regulatory
system
Regulation
0
0
p
1
Demand for X in the environment
Fig 10.6: Selection phase diagram, showing regions where gene regulation, genes always ON or genes always OFF are optimal. The x-axis is the
fraction of time p that the environment shows conditions in which the protein X is needed, and brings benefit (p is called the demand for X). The
ratio of protein cost to regulation system cost is r/c.
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Protein-level optimality – Agenda
1.  Fitness-functions
2.  Optimal expression levels of a protein under constant conditions
–  Case: Optimality and tuning of the expression level of betagalactosidase (LacZ).
3.  Optimal protein regulation under variable conditions
–  To regulate, or not to regulate
4.  Selection of regulatory circuit
–  No regulation
–  Simple regulation
–  Feed-forward loops
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
4. Regulatory circuit selection
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
4. Regulatory circuit selection
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
4. Regulatory circuit selection
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Regulatory circuitries
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27041, Introduction to Systems Biology
4. Regulatory circuit selection
When is it feasible to have a delay?
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Pulse influence on fitness
Evaluating a pulse of input Sx of
duration D in the presence of Sy
inducing Z:
•  Cost is constant for the duration D
•  Benefit increases with Z
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Pulse influence on fitness
Evaluating a pulse of input Sx
of duration D in the presence
of Sy inducing Z:
•  Cost is constant for the duration
D
•  Benefit increases with Z
•  Fitness (f = b – c) is negative
initially but grows over time
•  The integrated fitness function
D
ϕ ( D) = ∫ f (t )dt
0
can show the effect of the pulse
length
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Influence of pulse duration on fitness
Plotting the integrated
fitness function
•  Pulses of duration D<Dc are
infeasible to react to.
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CBS, Department of Systems Biology
D
ϕ ( D) = ∫ f (t )dt
0
27041, Introduction to Systems Biology
Pulse influence on fitness
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
a)
Simple regulation
b)
"
Sx
"
"
"
"
"Z
"
"
FFL
τ
"
"
cost
"
benefit"
"
"
fitness
Time
Time
10.9 Dynamics of gene expression and growth rate in a short, non-beneficial pulse and a long pulse of Sx and Sy
(a)  Simple regulation shows a growth deficit for both pulses, (b) FFL filters out the short pulse, but has reduced
benefit during the long pulse. The figure shows (top to bottom): (1) Pulse of Sx and Sy. (2) Dynamics of Z
expression. Z is turned on after a delay t (t=0 in the case of simple regulation), and approaches its steady state
level Zm. (3) Normalized production cost (reduction in growth rate) due to the production load of Z.
Cost begins after the delay t. (4) Normalized growth rate advantage (benefit) from the action of gene product Z
(5) Net normalized growth rate.
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Dynamic influences on regulatory circuit
selection
Application of cost-benefit to
modeling the choice of
regulatory circuits:
Plotting shows the areas where
the shown regulation has the
maximum fitness of the three.
•  If c > b, no regulation will be
selected.
•  The higher value of p, the larger
benefit of FFL.
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology
Summary
•  Cost-benefit analysis can be used for modeling of gene regulation at a
protein level.
•  Costs and benefits can be measured directly in a well-known system.
•  Optimal expression levels may be calculated
•  Optimal regulatory circuits may be modeled in both static and dynamic
environments
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CBS, Department of Systems Biology
27041, Introduction to Systems Biology