Design and performance of in vitro transcription rate regulatory circuits

Design and performance of

in vitro

transcription rate regulatory circuits

Elisa Franco and Richard M. Murray

Abstract

— This paper proposes a synthetic

in vitro

circuit

that aims at regulating the rate of RNA transcription through

positive feedback interactions. This design is dual to a pre-

viously synthesized transcriptional rate regulator based on

self-repression. Two DNA templates are designed to interact

through their transcripts, creating cross activating feedback

loops that will equate their transcription rates at steady state.

A mathematical model is developed for this circuit, consisting of

a set of ODEs derived from the mass action laws and Michaelis–

Menten kinetics involving all the present chemical species. This

circuit is then compared to its regulatory counterpart based

on negative feedback. A global sensitivity analysis reveals the

fundamental features of the two designs by evaluating their

equilibrium response to changes in the most crucial parameters

of the system.

I. I

NTRODUCTION AND

B

ACKGROUND

Building biological machinery out of known components

with the same confidence as one can build a silicon chip

is one of the crucial objectives of scientists who approach

synthetic biology with a quantitative mind. This will not only

allow to expand the pool of available molecular architectures

but also help to gain a better understanding of the char-

acteristics, modularity and evolvability of existing complex

biological networks still to be unraveled [1]. It is fundamental

to focus on basic functional motifs that are widely diffused in

nature, as they can be considered elementary building blocks

of large scale systems [2].

Building a circuit out of biological components is sim-

plified when operating

in vitro

: a higher control over the

environment and over unwanted reactions permits to moni-

tor more precisely the functional response of the designed

system. Utilizing few components is also beneficial to the

same purposes.

Recognizing the extraordinary importance of understand-

ing basic network motifs in a controllable environment, the

topic of this paper is the design and modeling of an

in

vitro

circuit that aims at regulating the transcription rate

of RNA through a positive feedback interconnection. This

work follows a previously proposed negative feedback-based

circuit having the same objective [4]. Transcription is a

fundamental part of the central dogma of molecular biology

and is naturally regulated in the cell: for instance it can be

turned on or off by binding of transcription factors, or by

secondary structure formation in the nascent RNA (see [5]

and references cited therein). The dynamics of several genes

Research supported in part by the Institute for Collaborative Biotech-

nologies through grant DAAD19-03-D-0004 from the U.S. Army Research

Office.

The authors are with the Division of Engineering and Applied

Sciences, California Institute of Technology, Pasadena, CA 91125.

elisa,murray

@cds.caltech.edu

.

can be coupled, and it is an interesting question whether

there exist mechanisms that match the transcription rates of

two or more genes.

The circuit described in this paper is composed of two

double stranded DNA (dsDNA) species coupled through their

transcripts with a mechanism of cross activation: if one of

the two transcripts is in stoichiometric excess, it is designed

to increase the production of the other RNA species by

releasing a single stranded DNA (ssDNA) activator molecule.

Thanks to this positive feedback loop, at equilibrium the

two transcription rates are equal. This feature is attractive

because it is a first step towards the design of concentration-

following circuits. The objective of our future work is in

fact to understand how positive and negative feedback motifs

can be used in order to keep a molecular species of interest

at a desired concentration level, or make it track a certain

concentration time profile.

The first

in vitro

transcriptional switches were designed

and synthesized by Kim [9], [8] as a possible biological

implementation of neural networks. More complex cell–free

environments for quantitative analysis have been proposed

in [11], where protein signaling patterns are considered.

However, the computational power of a simple setting

comprising only nucleic acids and few enzymes has been

theoretically proven to be superior [7] by virtue of its

simplicity. The same thermodynamics principles utilized to

build transcriptional switches are useful to construct several

other systems presenting a circuit–like behavior [12], [15]

or even to create nanomolecular devices [3], [13]. A further

motivation in focusing our attention on nucleic acids lies

in their important role in the control of gene expression,

which is being acknowledged and studied with increasing

interest [5].

This paper, building on previous work that analyzed a

negative feedback rate regulator [4], proposes its positive

feedback counterpart. The performance and features of the

two different designs are compared through a global sensi-

tivity analysis with respect to their feedback interconnection

and the environment enzymatic levels. These two architec-

tures based on transcriptional switches create a regulatory

mechanism not considered before. Following the procedure

in [4], the positive feedback rate regulator was designed

and mathematically modeled starting from the occurring

biochemical reactions; this new system is currently being

synthesized in laboratory. The employed pool of biological

machinery is of interest because it can be used to construct

a variety of molecular devices with different functionalities,

despite its simplicity and low number of components.

Proceedings of the

47th IEEE Conference on Decision and Control

Cancun, Mexico, Dec. 9-11, 2008

TuA05.3

161

II. C

IRCUIT DESCRIPTION AND MODELING

A. Circuit design

The first objective of this work is that of proposing a new

synthetic transcription rate regulatory network relying on

positive feedback, as an alternative to the negative feedback

based circuit described in [4]. The two design ideas are

schematically compared in Figure 1a and 1b.

Transcriptional circuits are composed of nucleic acids and

few enzyme species [9], [8]. The core of these circuits are

DNA templates (

100

–

120

nucleotides long) designed to be

transcribed into RNA (

80

–

100

nucleotides long) in the pres-

ence of the enzyme RNA polymerase (

R

p

). This process can

be switched on or off by displacement of part of the enzyme

binding area (the promoter). The ssDNA sequences allowing

completion of the promoter are called activators (

25

–

35

nucleotides long), which can be sequestered by ssDNA (or

RNA) sequences called inhibitors (25-35 nucleotides long).

In the positive feedback rate regulator, two templates

T

1

,T

2

are incomplete in their promoter region: activators

A

1

,A

2

can bind the templates completing the promoter and

allowing

R

p

to operate the transcription of RNA species

R

1

,R

2

. Transcription is normally off due to the presence

of two ssDNA inhibitor strands

S

1

,S

2

, that sequester the

activators. When transcription is initiated, the two RNAs are

designed to bind to each other, forming a double stranded

complex potentially available for further processing. By

construction, either product in excess with respect to the

other will promote the production of the other species by

binding to its inhibitor, thereby releasing the corresponding

activator. Since both transcripts have this cross-activation

feature, at steady state their production rates should be equal,

as demonstrated in Section III. RNase H (

R

h

), the other

enzyme species present, allows degradation of DNA-RNA

hybrids, introducing a further level of dynamic adaptation.

This architecture is schematically described in Figure 1 a).

Fig. 1.

Schematic representation of a) positive and b) negative feedback

rate regulators

The mechanism that allows turning on and off the tem-

plates is known as branch migration. Nucleic acid strands

provided with

toehold

regions [14] that remain exposed

in bound states can be displaced by specifically designed

RNA or DNA molecules. In our case, the RNA transcripts

‘encode’ for the inhibitor toehold and can initiate the strand

displacement. For instance, we can design

A

1

so that it binds

to

S

1

with free energy of

−

35

kcal/mol, and

R

1

so that it

binds to

S

1

with free energy of

−

40

kcal/mol: the second

will thus be a more favorable reaction.

The strand design method consists of finding the desired

complementarity regions and energetic constraints. This pro-

cedure can be automated by using Monte-Carlo optimization

of a user defined scoring function where the free energy

gain of unwanted secondary structures is suitably weighted

(in house software of the E. Winfree Lab at Caltech). Our

design for the strands is shown in Figure 2: following the idea

proposed in [4], the RNA transcripts will have ‘mirrored’

sequences in order to satisfy complementarity and cross

activation. As in the negative feedback rate regulatory circuit,

this design can produce unwanted interactions among the

strands. Specifically there will be a further off state of

the templates due to the binding of

R

i

to

T

j

. Moreover,

given that each RNA species also encodes for its activator

complementary sequence, there could be a self-inhibitory

action for each subsystem: this effect can be limited to

R

i

binding to free

A

i

if the activator toehold region is not

present in the RNA sequence (therefore

R

i

will not be able to

strip off

A

i

from the template

T

i

). The system behavior can

be monitored by labeling the DNA strands with fluorescent

dyes and quenchers [10], as detailed in Figure 2.

Fig. 2.

Design for the positive feedback regulator, sub–circuit of index

1

.

The arrow tick at the end of the strands indicates the

5

′

to

3

′

direction.

Starting from the

5

′

(left): fluorophore (cyan circle);

a

1

region (orange)

including part of the promoter region (blue box); initiation sequences

(cyan); complementary toehold

th

S

2

region (light purple); complementary

a

′

2

region (light blue);

a

1

region (orange); toehold

th

S

2

region (light green)

and at the

3

′

end

hairpin

region (brown). The hairpin is necessary to

avoid spurious elongation of the RNA strand during transcription [8]. The

sequence of the transcript

R

1

comprises all the regions of

T

1

right after the

promoter. Starting from the 3’ end (left) for

A

1

: quencher (black circle);

toehold

th

a

1

region (green); activator

a

′

1

region (red) comprising part of

the promoter.

B. Mathematical modeling

The chemical reactions occurring in the system are used

to derive a set of ordinary differential equations (ODEs).

Throughout this derivation, the dissociation constants are

omitted when assumed to be negligible. For enzymatic

reactions, we assume that the concentration of enzymes is

considerably lower than that of the DNA molecules, allowing

the classical steady state assumption for Michaelis-Menten

kinetics. The use of a deterministic continuous model is well

justified in this context: the experimental setting of transcrip-

tional circuits is such that the molecular concentrations are in

the order of

10

9

units per microliter. This case is dramatically

different from, for instance, certain transcription factors in

cellular environments, which could be

9

orders of magnitude

less concentrated; stochastic modeling is necessary in those

cases.

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

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162

The mass action reactions are, for

i

∈{

1

,

2

}

,

j

∈{

2

,

1

}

:

Activation

Inhibition

Release

Output formation

Unwanted interactions

T

i

+

A

i

k

T

i

A

i

→

T

i

A

i

T

i

A

i

+

S

i

k

T

i

A

i

S

i

→

T

i

+

S

i

A

i

A

i

+

S

i

k

A

i

S

i

→

A

i

S

i

R

i

+

A

j

S

j

k

R

i

A

j

S

j

→

R

i

S

j

+

A

i

R

i

+

S

j

k

R

i

S

j

→

R

i

S

j

R

i

+

R

j

k

R

i

R

j

→

R

i

R

j

R

i

+

A

i

k

R

i

A

i

→

R

i

A

i

R

i

+

T

j

k

R

i

T

j

→

R

i

T

j

(1)

The target enzymatic reactions are:

R

p

+

T

i

A

i

k

+

ON

ii

→

←

k

−

ON

ii

R

p

·

T

i

A

i

k

catON

ii

→

R

p

+

T

i

A

i

+

R

i

R

p

+

T

i

k

+

OF F

i

→

←

k

−

OF F

i

R

p

·

T i

k

catOF F

i

→

R

p

+

T

i

+

R

i

R

h

+

R

i

S

j

k

+

H

S

j

→

←

k

−

H

S

j

R

h

·

R

i

S

j

k

catH

S

j

→

R

h

+

S

j

(2)

The enzymatic processes involving unwanted complexes:

R

p

+

R

i

T

j

k

+

OF F

ij

→

←

k

−

OF F

ij

R

p

·

R

i

T

j

k

catOF F

ij

→

R

p

+

R

i

T

j

+

R

j

R

h

+

R

i

A

i

k

+

H

A

i

→

←

k

−

H

A

i

R

h

·

R

i

A

i

k

catH

A

i

→

R

h

+

A

i

R

h

+

R

i

T

j

k

+

H

T

j

→

←

k

−

H

T

j

R

h

·

R

i

T

j

k

catH

T

j

→

R

h

+

T

j

(3)

Given equations (1), (2), and (3) it is straightforward to

derive a set of ODEs as follows:

d

dt

[

T

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

]

−

k

R

j

T

i

[

R

j

] [

T

i

]

+

k

catH

T

i

[

Rh

·

R

j

T

i

]

d

dt

[

A

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

]

−

k

A

i

S

i

[

A

i

] [

S

i

]

−

k

R

i

A

i

[

R

i

] [

A

i

]

+

k

R

j

A

i

S

i

[

R

j

] [

A

i

S

i

] +

k

catH

A

i

[

R

h

·

R

i

A

i

]

d

dt

[

S

i

] =

−

k

A

i

S

i

[

A

i

] [

S

i

]

−

k

T

i

A

i

S

i

[

T

i

A

i

] [

S

i

]

−

k

R

j

S

i

[

R

j

] [

S

i

] +

k

catH

S

i

[

R

h

·

R

j

S

i

]

d

dt

[

R

i

] =

−

k

R

i

A

j

S

j

[

R

i

] [

A

j

S

j

]

−

k

R

i

R

j

[

R

i

] [

R

j

]

−

k

R

i

T

j

[

R

i

] [

T

j

]

−

k

R

i

S

j

[

R

i

] [

S

j

]

−

k

R

i

A

i

[

R

i

] [

A

i

]

+

k

catON

ii

[

R

p

·

T

i

A

i

] +

k

catOF F

i

[

R

p

·

T

i

]

+

k

catOF F

ji

[

R

p

·

R

j

T

i

]

d

dt

[

R

i

T

j

] = +

k

R

i

T

j

[

R

i

] [

T

j

]

−

k

catH

T

j

[

R

h

·

R

i

T

j

]

d

dt

[

R

i

R

j

] = +

k

R

i

R

j

[

R

i

] [

R

j

]

(4)

Where

[

T

i

]

represents the concentration of species

T

i

. The

molecular complexes that appear on the right hand side of the

above equation can be expressed as a function of the states

with some standard steps. Mass conservation immediately

yields the dynamics of

[

T

i

A

i

]

,

[

A

i

S

i

]

,

[

R

i

S

j

]

.

Assuming that binding of the enzyme is faster than

transcription or degradation in equations (2) and (3), and

defining the Michaelis–Menten coefficients (e.g. for the on

state of the template

k

MONii

=

k

−

ONii

+

k

catONii

k

+

ONii

), it is

possible to use mass conservation laws to obtain explicit

expressions for the enzyme concentrations. Due to space

limitations we only report the complete expression for the

term for

R

p

bound to

[

T

i

A

i

]

:

[

R

p

T

i

A

i

] =

[

R

p

tot

]

(1 +

P

i,j

[

T

i

A

i

]

k

MON

ii

+

[

T

i

]

k

MOF F

i

+

[

R

i

T

j

]

k

MOF F

ij

)

(5)

The nonlinear set of equations (4) was numerically an-

alyzed using the MATLAB ode23s solver. The parameter

values used in these simulations are reported in Table I.

These parameters are taken from the literature [8] and from

experimental data fitting on the negative feedback regulator

(data not shown here), and they are chosen so that the two

sub-circuits are identical. This is a simplifying assumption

that helps to provide intuition on the performance of the

circuit by just creating an imbalance in the concentration

of the strands. In particular, utilizing the parameters listed

in Table I, and initial conditions

T

tot

1

= 100

n

M,

A

tot

1

=

100

n

M,

S

tot

1

= 100

n

M,

T

tot

2

= 200

n

M,

A

tot

2

= 200

n

M and

S

tot

2

= 200

n

M. The enzymatic concentrations are

R

p

tot

=

20

n

M and

R

h

tot

= 2

n

M. These initial conditions are chosen

based on the amounts normally utilized in an experimental

setting and are targeted for reaction volumes of

70

μ

L. The

dynamics of

T

1

, T

2

on and of the total amount of RNA

produced are shown in Figure 3, simulated over a

6

hour

time window.

Fig. 3. a) Time profile of the templates in on state. b) Time profile of the

total amount of produced RNA transcripts

III. P

OSITIVE AND NEGATIVE FEEDBACK DESIGN

COMPARISON

A. Rate regulation

The models for the positive and negative feedback rate

regulator [4] can be simplified by eliminating negligible dy-

namics and unwanted interactions. The negligible dynamics

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

TuA05.3

163

are represented by transcription reactions that may occur

when the templates are off (not bound to their activator),

or when they are bound to an RNA species. The unwanted

interactions are those due to the specific design choice, but

may be eliminated with a more sophisticated design. The

binding reactions between RNA species and DNA templates

fall in this latter category; for the positive feedback-based

circuit, the binding between the RNA product and its own

activator is also considered an unwanted interaction. These

simplifications are be helpful in order to study the funda-

mental features of the two types of feedback.

Referring to the negative feedback circuit dynamics de-

rived in [4], we can write the following simplified equations,

where the off transcription reactions and the binding of

R

i

T

j

have been eliminated:

d

dt

[

T

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

] +

k

R

i

T

i

A

i

[

R

i

] [

T

i

A

i

]

d

dt

[

A

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

] +

k

catHii

[

R

h

·

R

i

A

i

]

d

dt

[

R

i

] =

−

k

R

i

R

j

[

R

i

] [

R

j

]

−

k

R

i

T

i

A

i

[

R

i

] [

T

i

A

i

]

−

+

k

catONii

[

R

p

·

T

i

A

i

]

d

dt

[

R

i

R

j

] = +

k

R

i

R

j

[

R

i

] [

R

j

]

(6)

For the positive feedback circuit just introduced, the equa-

tions can be simplified as follows, by neglecting the off

transcription reactions, the binding of

R

i

T

j

and of

R

i

A

i

:

d

dt

[

T

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

] +

k

R

i

T

i

A

i

[

R

i

] [

T

i

A

i

]

d

dt

[

A

i

] =

−

k

T

i

A

i

[

T

i

] [

A

i

]

−

k

A

i

S

i

[

A

i

] [

S

i

]

+

k

R

j

A

i

S

i

[

R

j

] [

A

i

S

i

]

d

dt

[

S

i

] =

−

k

A

i

S

i

[

A

i

] [

S

i

]

−

k

T

i

A

i

S

i

[

T

i

A

i

] [

S

i

]

−

k

R

j

S

i

[

R

j

] [

S

i

] +

k

catH

S

i

[

R

h

·

R

j

S

i

]

d

dt

[

R

i

] =

−

k

R

i

A

j

S

j

[

R

i

] [

A

j

S

j

] +

k

catON

ii

[

R

p

·

T

i

A

i

]

−

k

R

i

R

j

[

R

i

] [

R

j

]

−

k

R

i

S

j

[

R

i

] [

S

j

]

d

dt

[

R

i

R

j

] = +

k

R

i

R

j

[

R

i

] [

R

j

]

.

(7)

It is possible to prove that for both designs the steady

state transcription rate of

R

1

is equal to that of

R

2

. For the

negative feedback circuit, write

[

R

tot

i

] = [

R

i

] + [

R

i

A

i

] +

[

R

i

R

j

]

. Taking the derivative with respect to time, one can

immediately see that

d

dt

[

R

i

] = 0 =

d

dt

[

R

i

A

i

]

, provided

that the concentration of two species reaches an equilibrium.

Therefore one is left with

d

dt

[

R

tot

1

] =

d

dt

[

R

1

R

2

] =

d

dt

[

R

tot

2

]

.

Analogously for the positive feedback circuit, where

[

R

tot

i

] =

[

R

i

] + [

R

i

S

j

] + [

R

i

R

j

]

, one shows that when all the

other species in the solution have reached a steady state,

d

dt

[

R

tot

1

] =

d

dt

[

R

tot

2

] =

d

dt

[

R

1

R

2

]

. Note that the notation

[

R

i

]

indicates the concentration of the

i

th RNA species which is

not bound to other molecules.

The transient dynamics of

[

R

tot

i

]

can be evaluated by

explicitly writing their expression, where several terms will

cancel out. For the negative feedback regulatory circuit one

is left with the equation:

d

dt

[

R

tot

i

] =

k

catON

ii

[

R

p

·

T

i

A

i

]

−

k

catHii

[

R

h

·

R

i

A

i

]

. For the positive feedback regulator one

gets:

d

dt

[

R

tot

i

] =

k

catON

ii

[

R

p

·

T

i

A

i

]

−

k

catH

S

j

[

R

h

·

R

i

S

j

]

.

It

is clear from these expressions that, if one assumes identical

binding parameters for the two sub-circuits, in order to exper-

imentally verify

d

dt

[

R

tot

1

] =

d

dt

[

R

tot

2

]

one should be able to

measure both concentrations of

[

T

i

A

i

]

and

[

R

i

A

i

]

(negative

feedback circuit) or

[

R

i

S

j

]

(positive feedback circuit). While

by using fluorescent dyes one can easily estimate

[

T

i

A

i

]

and

use it as an indicator of the transcription rate; measuring

[

R

i

S

j

]

may not be as straightforward.

B. Sensitivity analysis

The basic architecture of the positive and negative feed-

back rate regulatory circuits is such that the objective of

equating the transcription rates of two sub-circuits will

always be fulfilled, when the initial conditions allow the

system to reach a steady state (excluding the dsRNA species

R

1

, R

2

which is not degraded). It is of interest to understand

what is the variability of performance due to changes of

certain critical parameters of the system.

Local sensitivity analysis restricted to a first order Taylor

series approximation was insufficient for this class of non-

linear systems. In fact, the sensitivity matrix technique [6]

did not yield meaningful results, due to the large integration

times and the type of nonlinearities. The analysis is therefore

explored numerically, by tuning the parameters in a certain

range and observing the variation in the solution of the

ODEs. Attention will be restricted to a limited number of

interesting parameters: the feedback strength (self inhibition

and cross activation binding rates), the concentration of

R

p

and of

R

h

. The feedback strength can be modulated

by changing the length of the toeholds. The enzymatic

concentration is often a source of experimental uncertainty,

as vendors only provide information about the activity of

the protein, defined as moles of substrate converted per unit

time.

Figures 4 and 5 show the simulation results for the steady

state amount of templates and the transcription rate versus

fold change in the parameter of interest. It is meaningful to

analyze the steady state behavior of the templates since it is

the most directly measurable concentration. The production

rate of RNA will also be evaluated, since the objective

of the design is that of equating such rate for the two

transcripts. The initial conditions were set to

T

1

= 100

n

M,

A

1

= 100

n

M,

T

2

= 200

n

M and

A

2

= 200

n

M for the

negative feedback circuit. For the positive feedback regulator:

T

1

= 100

n

M,

A

1

= 100

n

M,

S

1

= 100

n

M,

T

2

= 200

n

M,

A

2

= 200

n

M and

S

2

= 200

n

M. The nominal parameters

utilized in the simulations are reported in Tables I and II;

such parameters have been taken from the literature [8]

and from data fits performed on analogous circuits. The

nominal concentration of enzymes are

R

p

tot

= 20

n

M,

R

h

tot

= 2

n

M, based on experimental practice. This sen-

sitivity analysis was also performed on the full models of

the two systems, giving no relevant difference with respect

to the reported results (data not shown).

For the negative feedback circuit, Figures 4a and 4b),

one can see that both high feedback gain and high

R

p

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

TuA05.3

164

concentration will decrease the equilibrium amount of

T

i

A

i

on. For the positive feedback circuit, Figures 5a and 5b, the

effect is instead that of increasing the amount of

T

i

A

i

on. On

the other hand, the steady state rate of production of RNA

responds in a very different way to a variation of the two

parameters. In Figures 4d and 4e one observes that strong

self inhibition will obviously decrease the production of RNA

for the negative feedback circuit, while a high concentration

of

R

p

will boost it. As for the positive feedback case,

in Figures 5d and 5e show that a strong cross activation

will increase up to a certain saturation value the rate of

production, while the increment is almost linear with respect

to the

R

p

amount. Finally, the concentration of

R

h

has

similar effects on the concentration of templates on and

on the rate of production: for the negative feedback circuit,

Figures 4c and 4 f, a more significant degradation of RNA-

DNA hybrid

R

i

A

i

means more

A

i

capable of turning the

circuit on, and accordingly a higher yield of RNA. On the

other hand, the opposite effect is observed in the positive

feedback circuit, in Figures 5c and 5f, where degradation of

R

i

S

j

puts back into the circuit the inhibitor

S

i

.

Given the above analysis, the choice of a particular design

will depend on the performance specifications. Clearly the

superposition principle does not hold since the systems

are nonlinear. The two sub-circuits will reach the same

production rate of RNA using both designs: but feasibility

constraints should be taken into consideration. For instance,

operating at high feedback strength is better, since long toe-

holds would make the branch migration process faster [14].

Working with low amount of enzymes is also an advantage,

as they represent the most costly component of the circuits.

The negative feedback rate regulatory circuit has a clear

advantage over the positive feedback one in its lower number

of DNA species, which make it a simpler and cheaper circuit.

As an example, suppose an overall low amount of RNA

is desired. The negative feedback circuit should be utilized,

operating at ‘high feedback gain’ and low

R

p

, as shown in

Figures 6a and 6b (self inhibition binding rates ten times

higher than nominal,

R

p

= 10

n

). A low amount of RNA

could be obtained also using the positive feedback regulatory

circuit, at the expense of either using more

R

h

or designing

short toeholds for the cross–activation process, which would

make the reactions slower. If instead the objective is a

higher production of RNA, positive feedback loops should

be used, with a high cross activation interconnection and low

concentration of

R

h

, Figures 6c and 6d (cross activation ten

times higher,

R

h

= 1

nM

). A high amount of RNA could

be obtained also with the negative feedback loops, utilizing

though more

R

p

and short toeholds (more expensive and

slower reactions).

IV. C

ONCLUSIONS AND FUTURE WORK

A circuit aimed at matching the transcription rate of two

DNA templates has been presented in this paper. The circuit

design is based on positive feedback and is an alternative to a

previously described circuit based on negative feedback [4].

Fig. 4.

Steady state sensitivity analysis of the negative feedback rate

regulator: a), b) and c) show the equilibrium concentrations of

T

1

A

1

and

T

2

A

2

versus the fold change of self inhibition binding rates,

R

p

and

R

h

amounts; d), e) and f) show the corresponding production rate of RNA under

the same conditions. The simulation time is of

6

hours.

TABLE I

P

OSITIVE FEEDBACK REGULATORY CIRCUIT PARAMETERS

Units:

[

s/M

]

Units:

[1

/s

]

Units:

[

M

]

k

T

i

A

i

= 4

·

10

3

k

catON

ii

= 0

.

064

k

MON

ii

= 250

·

10

−

9

k

T

i

A

i

S

i

= 4

·

10

4

k

catOF F

i

= 1

·

10

−

3

k

MOF F

i

= 1

·

10

−

6

k

A

i

S

i

= 5

·

10

4

k

catOF F

ij

= 1

·

10

−

3

k

MOF F

ij

= 1

·

10

−

6

k

R

i

A

i

S

i

= 9

·

10

4

k

catH

S

i

=

.

106

k

MH

S

i

= 50

·

10

−

9

k

R

i

S

i

= 9

·

10

4

k

catH

T

i

=

.

106

k

MH

T

i

= 50

·

10

−

9

k

R

i

T

j

= 1

·

10

3

k

R

i

A

i

= 1

·

10

3

k

R

i

R

j

= 2

·

10

5

TABLE II

N

EGATIVE FEEDBACK REGULATORY CIRCUIT PARAMETERS

Units:

[

s/M

]

Units:

[1

/s

]

Units:

[

M

]

k

T

i

A

i

= 4

·

10

3

k

catON

ii

= 0

.

064

k

MON

ii

= 250

·

10

−

9

k

T

i

A

i

R

i

= 5

·

10

4

k

catH

ii

=

.

106

k

MH

ii

= 50

·

10

−

9

k

A

i

R

i

= 5

·

10

4

k

R

i

R

j

= 2

·

10

5

47th IEEE CDC, Cancun, Mexico, Dec. 9-11, 2008

TuA05.3

165

Fig. 5.

Steady state sensitivity analysis of the positive feedback rate

regulator: a), b) and c) show the equilibrium concentrations of

T

1

A

1

and

T

2

A

2

versus the fold change of cross activation binding rates,

R

p

and

R

h

amounts; d), e) and f) show the corresponding production rate of RNA

under the same conditions. The simulation time is of

6

hours.

A mathematical model of the positive feedback-based regu-

latory circuit has been derived along with the basic design

idea, showing that the theoretical properties of the circuit are

as anticipated by physical intuition. A sensitivity analysis has

been carried out, comparing the performance of both versions

of the rate regulatory architecture. Undergoing work is aimed

at experimentally verifying the two circuit properties and the

tradeoffs between positive and negative regulation. Future

work will focus on how to design concentration followers

with transcriptional circuits, starting from the rate regulatory

systems. The objective is that of understanding how to

match the concentration of two molecules with accuracy

through a specific feedback motif. This will allow us to gain

more insight on how living organisms perform this feature,

maintaining precise concentration levels of their molecular

components.

Acknowledgments

The authors would like to thank Erik

Winfree, Jongmin Kim, Per-Ola Forsberg and all the mem-

bers of the DNA and Natural Algorithms group at Caltech for

their helpful advise during the development of this project.

R

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Fig. 6.

Design choices to achieve different performances: a) and b) show

the time profile of templates on and total production of RNA for the negative

feedback circuit with high self inhibition and low amount of

R

p

; c) and d)

show the positive feedback circuit performance with high cross activation

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R

h

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