Optimality theory Optimal expression level of a protein under constant conditions Cost of the LacZ Protein Mathematical description of cost function The Benefit of the LacZ Protein Fitness Function and the Optimal Expression Level Cells Reach Optimal LacZ Levels in a Few Hundred Generations in Laboratory Evolution Experiments Environmental selection of the feedforward loop network motif References

1. Optimal Gene Circuit Design
Presented by:
Gaurav
19mslsbf03
Department of Computational
sciences
Submitted to:
Dr. Devinder Kaur
Department of Computational
sciences
https://www.the-scientist.com/infographics/designing-genetic-circuits-41838

2. Content
• Optimality theory
• Optimal expression level of a protein under constant
conditions
• Cost of the LacZ Protein
• Mathematical description of cost function
• The Benefit of the LacZ Protein
• Fitness Function and the Optimal Expression Level
• Cells Reach Optimal LacZ Levels in a Few Hundred
Generations in Laboratory Evolution Experiments
• Environmental selection of the feedforward loop network
motif
• References

3. Optimality theory
• Optimality theory aims to predict and
understand which circuit and parameters will
arise in each situation.
• Optimality theory is based on the ability of
natural selection to maximize fitness, and
hence to converge on particular circuit designs
and parameter values in a given environment.

4. Optimal expression level of a protein under
constant conditions
What sets the expression level of a protein? Why are
some proteins produced at a few copies per cell, others
at thousands and yet others at tens or hundreds of
thousands?
To address this question, we begin with a situation in
which fitness can be precisely defined:
Bacteria that grow in a constant environment that is
continually replenished. In this case, the fitness F is the
growth rate of the cells. The number of cells, N, grows
exponentially with time at rate F until they get too
dense:

5. Contd…
• Optimality theory
predicts that the protein
expression level that is
selected maximizes the
fitness function. The
fitness function in this
case is growth rate F as a
function of the number
of copies of the protein
expressed.

6. Contd…
• In principle, F(Z) can have
local maxima and minima. A
journey on this fitness
function can get stuck on
the local maxima or blocked
by impassable valleys.
Random mutations can
cause noise along this
journey, as can sampling
noise when population sizes
are small, an effect called
genetic drift.

7. Cost of the LacZ Protein
• Experimental measurement of the
cost function.
• To measure the cost of the lac
system, IPTG was used to induce the
lac system to various levels in the
absence of lactose.
• The cells grew on another carbon
source, glycerol.
• Expression of LacZ reduced the
growth rate of the cells.
• The cost c(Z) is defined as the relative
reduction in the growth rate when Z
units of the protein are made.
The cost rises linearly with Z at first, so that each unit of protein reduces growth by a fixed
amount. At higher protein levels, the cost starts to accelerate. The reason is that
production and maintenance of the lac proteins not only requires the use of the cells’
resources, but also reduces the resources available to other useful proteins.

8. Mathematical description of cost
function
• Assume that the growth rate of the cell
depends on an internal resource R. The
growth rate is typically a saturating function of
resources such as R, following a Michaelis–
Menten function.
• Each unit of protein Z reduces the resource by
a small amount, R → R − εZ. The cost begins
to diverge as Z approaches a cutoff level M, in
which so much Z is produced that R is
depleted .
• When only a few copies of the protein are
made, the cost goes as c(Z) ∼ c0Z. The cost of
a single protein is about 10−6, which makes
sense because there are about 106 proteins in
the bacterial cell.
• They show that the relative reduction in
growth rate due to the fully induced lac system
is about 4.5%.

9. The Benefit of the LacZ Protein
• Benefit: defined as the relative
increase in growth rate due to the
action of the protein.
• In the case of LacZ, the benefit is
proportional to the rate at which LacZ
breaks down its substrate, lactose.
• LacZ breaks down lactose at a rate
that is proportional to the number of
copies of the protein, Z, times a
saturating function of the
concentration of lactose, L.
• K = Michaelis CONSTANT2 and
b0 =maximal growth rate advantage
per LacZ protein at saturating lactose.
The relative increase in growth rate due to the fully induced level of LacZ with
saturating amounts of lactose is about 17% under the conditions of the experiment

10. Fitness Function and the Optimal
Expression Level
• Fitness function, equal to the
difference between benefit and cost.
• The fitness function is the growth rate
of cells that produce Z copies of LacZ
in an environment with a lactose
concentration of L.
• The position of this maximum, namely
the optimal protein level Zopt,
depends on lactose level, L.
• The optimal protein level Zopt can be
found by the point at which the
derivative of the fitness function with
respect to Z equals zero

11. Contd…
• Zopt rises with L because lactose increases
the benefit per LacZ protein, and hence
increases the selection pressure to
produce more proteins.
• The fully induced wild-type expression
level, ZWT = 60,000/cell is predicted to be
optimal when L ∼ 0.6 mM under these
experimental conditions.
• When there is no lactose in the
environment, the optimal level is Zopt = 0,
because the lac proteins confer only costs
and no benefits.
• In fact, zero expression is optimal as long
as lactose L is lower than a threshold Lc,
because costs exceed benefits.
The threshold level of lactose needed for selection of the gene system is Lc ∼ 0.06 mM.
If lactose environments with L < Lc persist for many generations, the organism will tend
to lose the gene encoding LacZ.

12. Cells Reach Optimal LacZ Levels in a Few Hundred Generations
in Laboratory Evolution Experiments
• To test the predictions of this cost–benefit
analysis, Erez Dekel performed a
laboratory evolution experiment. The
evolution experiment used the technique
of serial dilution.
• The concentration of the LacZ protein was
monitored over time.
• The LacZ protein level reached the
predicted optimal level in each tube to a
good approximation.
• Cells growing with no lactose lost their lac
expression altogether.
• Analysis of the evolutionary dynamics
indicated that the cells reached their
optimal, adapted levels in each case by
means of a mutation that changed the
LacZ protein level.

13. Environmental selection of the feedforward loop network motif
• Cost–benefit analysis for the
FFL in a given dynamically
fluctuating environment.
• Term environment means the
time-dependent profiles of the
input signals.
• In E. coli, about 40% of the
genes regulated by two inputs
are regulated by a FFL,
whereas the remaining 60% are
regulated by a simple two-
input design which is not a FFL

14. Contd…
• Suppose that the system is presented with a
pulse of input SX of duration D.
• The impact of this pulse on the cell’s growth is
given by the integrated fitness over the pulse
duration, φ()()DFtdtD=0.
• This integrated fitness shows that brief pulses
of input signals have a detrimental effect on
growth.
• They lead to a reduction in growth rate.
• The reason for this reduction is that when the
input pulses are shorter than a critical pulse
duration, Dc, protein Z does not have time to
build up to levels in which the accumulated
benefit exceeds the costs.

15. Contd…
• Fitness is reduced by brief input pulses, a circuit that
can avoid responding to brief pulses, but still respond
to persistent pulses, can be advantageous.
• The coherent FFL can perform precisely this type of
filtering task.
• In the coherent type-1 FFL with an AND input function,
Z is expressed only at a delay after the signals appear.
• Therefore, only pulses of input signals longer than the
delay time of the FFL will lead to Z expression.
• The delay in the FFL, TON, results from the time it takes
for transcription factor Y to accumulate and cross its
activation threshold for gene Z.

16. Contd..
• The delay in the FFL filters out
pulses that are shorter than
TON.
• This avoids the reduction in
growth caused by short pulses.
• The delay also has a
disadvantage, because during
long pulses, Z is produced only
at a delay and misses some of
the potential benefit of the
pulse.
• In some situations the FFL does
more harm than good

17. Contd…
• This diagram shows that the
FFL is more fit than simple
regulation when deleterious
brief pulses are common and
the benefit-to-cost ratio is not
too high.
• Simple regulation is superior
when brief pulses are rare.
• When costs exceed benefits,
neither circuit is selected

18. References
• Uri Alon - An Introduction to Systems Biology_
Design Principles of Biological Circuits-CRC
Press (2020)
• https://www.youtube.com/watch?v=PfPWpWj
vRPU&list=PLLbr-
B8cNbo6v4kc68JowzUeAYdh6gdQH&index=18

19. Thanks!