Simple mathematical models of gene regulatory dynamics

For a generic operon with a maximal level of transcription Nb din concentration units, the dynamics are given by Goodwin 1965, Griffith 1968a, Griffith 1968b, Othmer 1976, Selgrade1979 I

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in the Life Sciences

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in the Life Sciences

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Michael C Mackey • Moisés Santillán •

Marta Tyran-Kami´nska • Eduardo S Zeron

Simple Mathematical Models

of Gene Regulatory

Dynamics

123

Ciudad de MéxicoMexico

ISSN 2193-4789 ISSN 2193-4797 (electronic)

Lecture Notes on Mathematical Modelling in the Life Sciences

ISBN 978-3-319-45317-0 ISBN 978-3-319-45318-7 (eBook)

DOI 10.1007/978-3-319-45318-7

Library of Congress Control Number: 2016956565

© Springer International Publishing Switzerland 2016

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To students everywhere: past, present, and future.

We survey work that has been carried out in the attempts of biomathematicians

to understand the dynamic behavior of simple bacterial operons starting with theinitial work of the 1960s We concentrate on the simplest of situations, discussingboth repressible and inducible systems as well as the bistable switch and thenturning to a discussion of the role of both extrinsic noise and the so-called intrinsicnoise in the form of translational and/or transcriptional bursting We conclude with

a consideration of the messier concrete examples of the lactose and tryptophanoperons and the lysis-lysogeny switch of phage This survey has grown out ofour work over the past 20 years and is an enlarged version of our review paper(Mackey et al.2015)

June 2016

vii

We have benefited from the comments, suggestions, and criticisms of manycolleagues over the years (you will know who you are) and from the institutionalsupport of our home universities as well as the University of Oxford, the University

of Bremen, Bergischen Universität Wuppertal, and the International Centre forTheoretical Physics MCM is especially grateful to a comment from Dr JérômeLosson many years ago that directed attention to these fascinating problems.This work was supported by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada, the Polish NCN grant no 2014/13/B/ST1/00224, andthe Consejo Nacional de Ciencia y Tecnología (Conacyt) in México

ix

Part I Deterministic Modeling Techniques

1 Generic Deterministic Models of Prokaryotic Gene Regulation 3

1.1 Inducible Regulation 3

1.2 Repressible Regulation 5

2 General Dynamic Considerations 7

2.1 Operon Dynamics 7

2.1.1 No Control 9

2.1.2 Inducible Regulation 9

2.1.3 Repressible Regulation 13

2.1.4 Bistable Switches 13

2.2 The Appearance of Cell Growth Effects and Delays Due to Transcription and Translation 23

2.3 Fast and Slow Variables 26

Part II Dealing with Noise 3 Master Equation Modeling Approaches 31

3.1 The Chemical Master Equation 32

3.2 Relation to Deterministic Models 34

3.2.1 The Chemical Langevin Equation 36

3.3 Stability of the Chemical Master Equation 37

3.3.1 Algorithms to Find Steady State Density Functions 40

3.4 Application to a Simple Repressible Operon 43

4 Noise Effects in Gene Regulation: Intrinsic Versus Extrinsic 49

4.1 Dynamics with Bursting 50

4.1.1 Generalities 50

4.1.2 Distributions in the Presence of Bursting for Inducible and Repressible Systems 52

4.1.3 Bursting in a Switch 57

xi

xii Contents

4.1.4 Recovering the Deterministic Case 61

4.1.5 A Discrete Space Bursting Model 62

4.2 Gaussian Distributed Noise in the Molecular Degradation Rate 64

4.3 Two Dominant Slow Genes with Bursting 66

Part III Specific Examples 5 The Lactose Operon 73

5.1 The Lactose Operon Regulatory Pathway 73

5.2 Mathematical Modeling of the Lactose Operon 77

5.3 Quantitative Studies of the Lactose Operon Dynamics 83

6 The Tryptophan Operon 87

6.1 The Tryptophan Operon in E coli 87

6.2 Mathematical Modeling of the trp Operon 89

6.3 Quantitative Studies of the trp Operon 92

7 The Lysis-Lysogeny Switch 99

7.1 Phage Biology 102

7.2 The Lysis-Lysogeny Switch 104

7.3 Mathematical Modeling of the Phage Switch 107

7.4 Brief Review of Quantitative Studies on the Phage Switch 112

7.5 Closing Remarks 114

References 115

Index 123

The operon concept for the regulation of bacterial genes, first put forward by Jacob

et al (1960), has had an astonishing and revolutionary effect on the development

of understanding in molecular biology It is a testimony to the strength of thetheoretical and mathematical biology community that modeling efforts aimed atclarifying the implications of the operon concept appeared so rapidly after theconcept was embraced by biologists Thus, to the best of our knowledge, Goodwin(1965) gave the first analysis of operon dynamics which he had presented in his book(Goodwin1963) These first attempts were swiftly followed by Griffith’s analysis

of a simple repressible operon (Griffith1968a) and an inducible operon (Griffith1968b), and these and other results were beautifully summarized by Tyson andOthmer (1978)

Since these modeling efforts in the early days of development in molecularbiology, both our biological knowledge and level of sophistication in modeling haveproceeded apace to the point where new knowledge of the biology is actually drivingthe development of new mathematics This is an extremely exciting situation andone which many have expected—that biology would act as a driver for mathematics

in the twenty-first century much as physics was the driver for mathematics inthe nineteenth and twentieth centuries However, as this explosion of biologicalknowledge has proceeded hand in hand with the development of mathematicalmodeling efforts to understand and explain it, the difficulty in comprehending thenature of the field becomes ever more difficult due to the sheer volume of workbeing published

In this very short and highly idiosyncratic review, we discuss work from our

group over the past few years directed at the understanding of really simple operon

control dynamics We start this review in Chap.1 by discussing transcription andtranslation kinetics for both inducible and repressible operons In Chap.2we thenturn to general dynamics considerations which is largely a recap of earlier work withadditional insights derived from the field of nonlinear dynamics

The next two chapters deal with complementary approaches to the consideration

of the role of noise, with Chap.3 developing the theory of the chemical master

xiii

xiv Introduction

equation and Chap.4 considering the role of noise (in a variety of forms from avariety of sources) in shaping steady-state dynamic behavior for larger systems.Following this, we turn away from the realm of mathematical nicety to biological

reality by looking at realistic models for the lactose (Chap.5) and tryptophan(Chap.6) operons, respectively, and the lysis-lysogeny switch in phage (Chap.7).These three examples, probably the most extensively experimentally studied exam-ples in molecular biology and for which we have relatively large quantities of data,illustrate the reality of dealing with real biology and the difficulties of applyingrealistic modeling efforts to understand that biology

Deterministic Modeling Techniques

In this first part we treat very simple deterministic models for gene regulation.Models like these were the first that appeared, and are appropriate for situations inwhich one is looking at the behavior of a large number copies of the gene regulatorynetwork (e.g in a culture of many cells) where ‘large’ and ‘many’ mean something

on the order of Avagadro’s number ('6  1023).

produces messenger RNA (mRNA, denoted M here) Then through the process

of translation of mRNA, intermediate protein (I) is produced which is capable of controlling metabolite (E) levels that in turn can feedback and affect transcription

and/or translation A typical example would be in the lactose operon of Chap.5where the intermediate isˇ-galactosidase and the metabolite is allolactose Thesemetabolites are often referred to as effectors, and their effects can, in the simplestcase, be either stimulatory (so called inducible) or inhibitory (or repressible) to theentire process This scheme is often called the ‘operon concept’

We first outline the relatively simple molecular dynamics of both inducible andrepressible operons and how effector concentrations can modify transcription rates

If transcription rates are constant and unaffected by any effector, then this is called

a ‘no control’ situation

The lac operon considered below in Chap.5 is the paradigmatic example of

inducible regulation In an inducible operon when the effector (E) is present then the repressor (R) is inactive and unable to bind to the operator (O) region so DNA transcription can proceed unhindered E binds to the active form R of the repressor

and we assume that this binding reaction is

R C nE k

C 1

*

k 1

RE n;

© Springer International Publishing Switzerland 2016

M.C Mackey et al., Simple Mathematical Models of Gene Regulatory Dynamics,

Lecture Notes on Mathematical Modelling in the Life Sciences,

DOI 10.1007/978-3-319-45318-7_1

3

in which k1C and k1 are the forward and backward reaction rate constant, tively The equilibrium equation for the reaction above is

respec-K1D RE n

where K1 D kC

1 =k

1 is the reaction dissociation constant and n is the number of

effector molecules required to inactivate repressor R The operator O and repressor

R are also assumed to interact according to

O C R k

C 2

*

k 2

OR;which has the following equilibrium equation:

f E/ D 1 C K1E n

1 C K2R tot C K1E n D 1C K1E n

1.2 Repressible Regulation 5

where K D 1 C K2R tot Maximal repression occurs when E D 0 and even at that

point mRNA is produced (so-called leakage) at a basal level proportional to K1.Assume that the maximal transcription rate of DNA (in units of time1) is N'm.Assume further that transcription rate' in the entire population is proportional to

the fraction of unbound operators f Thus we expect that' as a function of theeffector level will be given by' D N'm f , or

effector binds the inactive form R of the repressor so it becomes active and take this

reaction to be the same as in Eq (1.1) However, we now assume that the operator

O and repressor R interaction is governed by

O C RE n

kC 2

*

k 2

ORE n;with the following equilibrium equation

K2D ORE n

O  RE n

; K2D k2C

The total operator is

O tot D O C ORE n D O C K1K2O  R  E n D O.1 C K1K2R  E n/;

so the fraction of operators not bound by repressor is

f E/ D O

O tot

1 C K1K2R  E n:

Assuming, as before, that the amount of R bound to O is small compared to the

amount of repressor gives

f E/ D 1 C K1E n

1 C K1C K1K2R tot /E n D 1C K1E n

1 C KE n ;

where K D K1.1 C K2R tot / In this case we have maximal repression when E is

large, and even when repression is maximal there is still a basal level of mRNA

production (again known as leakage) which is proportional to K1K1< 1 Variation

of the DNA transcription rate with effector level is given by' D N'm f or

The constants A; B  0 are defined in Table1.1

Table 1.1 The parameters A,

B,,  and  for the

inducible and repressible

we use that as a basis for discussion We let .M; I; E/ respectively denote the

mRNA, intermediate protein, and effector concentrations For a generic operon with

a maximal level of transcription Nb d(in concentration units), the dynamics are given

by (Goodwin (1965), Griffith (1968a), Griffith (1968b), Othmer (1976), Selgrade(1979))

It is assumed here that the rate of mRNA production is proportional to the fraction

of time the operator region is active, and that the rates of protein and metaboliteproduction are proportional to the amount of mRNA and intermediate proteinrespectively All three of the components.M; I; E/ are subject to degradation, and the function f is as determined in Chap.1above

To simplify things we formulate Eqs (2.1)–(2.3) using dimensionless tions To start we rewrite Eq (1.6) in the form

concentra-'.e/ D ' m f e/;

© Springer International Publishing Switzerland 2016

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Lecture Notes on Mathematical Modelling in the Life Sciences,

DOI 10.1007/978-3-319-45318-7_2

7

where'm(which is dimensionless) is defined by

classic operon model have been fully analyzed (Mackey et al.2011), the results

of which we simply summarize here We set X D x1; x2; x3/ and let S t X/ be the

2.1 Operon Dynamics 9

flow generated by the system (2.5)–(2.7), i.e., the function t 7! S t X/ is a solution

of (2.5)–(2.7) such that S0.X/ D X For both inducible and repressible operons, for all initial conditions X0D x0

No control simply means f x/  1, and in this case there is a single steady state

xD dthat is globally asymptotically stable

2.1.2 Inducible Regulation

2.1.2.1 Single Versus Multiple Steady States

For an inducible operon [with f given by Eq (1.2)] there may be one (X1or X3),

3/ corresponds to the induced state The steady

state values of x are easily obtained from (2.8) for given parameter values, and thedependence on d for n D 4 and a variety of values of K is shown in Fig.2.1.Figure2.2shows a graph of the steady states xversusd for various values of the

leakage parameter K.

Analytic conditions for the existence of one or more steady states come from

Eq (2.8) in conjunction with the observation that the delineation points are marked

by the values ofd at which x= d is tangent to f.x/ (see Fig.2.1) Differentiation

of (2.8) yields a second condition

1

 n K  1/D

x n1

of Eq ( 2.4), and the straight lines correspond to x= dfor (in a clockwise direction) d2 Œ0; d /,

d D d , d 2 d ; dC /, d D dC , and dC < d This figure was constructed with n D4

and K D10 for which d D 3:01 and dC D 5:91 as computed from ( 2.11 ) See the text for details Taken from Mackey et al ( 2011 ) with permission

1

2 3 4

et al ( 2011 ) with permission

2.1 Operon Dynamics 11From Eqs (2.8) and (2.9) the values of x at which tangency will occur are given by:

A necessary condition for the existence of two or more steady states is obtained

by requiring that the radical in (2.10) is non-negative:

leakage is appreciable (small K, e.g for n D 4, K < 5=3/2) then the possibility ofbistable behavior is lost

We can make some general comments on the influence of n, K, andd on theappearance of bistability from this analysis First, the degree of cooperativity.n/

in the binding of effector to the repressor plays a significant role and n > 1 is a

necessary condition for bistability If n > 1 then a second necessary condition for

bistability is that K satisfies Eq (2.12) so the fractional leakage.K1/ is sufficientlysmall Furthermore,dmust satisfy Eq (2.13) which is quite interesting For n ! 1

the limiting lower limit isd > 1 while for n ! 1 the minimal value of  dbecomesquite large This simply tells us that the ratio of the product of the production rates

to the product of the degradation rates must always be greater than 1 for bistability

to occur, and the lower the degree of cooperativity.n/ the larger the ratio must be.

If n, K andd satisfy these necessary conditions then bistability is only possible

ifd 2 Œd; dC (c.f Fig.2.3) The locations of the minimal.x/ and maximal

.xC/ values of x bounding the bistable region are independent of  And, finally,

0 5 10 15 20 0

2 4 6 8 10

.K;  d / in the interior of the cone there are two locally stable steady states X

1; X

3, while outside

there is only one The tip of the cone occurs at.K;  d/ D 5=3/ 2 ; 5=3/ p 4

5=3/ as given by Eqs ( 2.12 ) and ( 2.13) For K 2 Œ0; 5=3/ 2 / there is a single steady state Taken from Mackey

et al ( 2011 ) with permission

.xCx/ is a decreasing function of increasing n for constant  d ; K while xCx/

is an increasing function of increasing K for constant n;  d

2.1.2.2 Local and Global Stability

Although the local stability analysis of the inducible operon is possible (Mackey

et al.2011), the thing that is interesting is that the global stability is possible todetermine

Theorem 2.1 (Othmer 1976 ; Smith 1995, Proposition 2.1, Chap 4) For an

inducible operon with ' given by Eq (1.3), define I I D Œ1=K; 1 There is an

2.1 Operon Dynamics 13

2 If there are two locally stable nodes, i.e X1and X3ford2 d; dC /, then all

flows S t X0/ are attracted to one of them (See Selgrade ( 1979 ) for a delineation

of the basin of attraction of X1 and X3.)

2.1.3 Repressible Regulation

As is clear from a simple consideration of our dynamical equations the repressible

operon has a single steady state corresponding to the unique solution xof Eq (2.8).Again, rather remarkably, we can characterize the global stability of this singlesteady state through the following result from Smith (1995, Theorems 4.1 and 4.2,Chap 3)

Theorem 2.2 For a repressible operon with ' given by Eq (1.5), define I R D

ŒK1=K; 1 There is a globally attracting box B R RC

3 defined by

B R D f.x1; x2; x3/ W x i 2 I R ; i D 1; 2; 3g

such that the flow S t is directed inward everywhere on the surface of B R more there is a single steady state X 2 B R If X is locally stable it is globally stable, but if Xis unstable then a generalization of the Poincare-Bendixson theorem (Smith 1995 , Chap 3) implies the existence of a globally stable limit cycle in B R

of biological systems (In a gene regulatory framework we might term the doublepositive feedback switch an inducible switch, while the double negative feedbackswitch could be called a repressible switch.) Some laboratories have used this insight

to engineer in vitro systems to have bistable behavior and one of the first wasGardner et al (2000) who engineered repressible switch like behavior of the type

we study in this section Some especially well written surveys are to be found inFerrell (2002), Tyson et al (2003), and Angeli et al (2004)

Figure2.4gives a cartoon representation of the situation we are modeling here,which is a generalization of the work of Grigorov et al (1967) and Cherry and Adler(2000) The original postulate for the hypothetical regulatory network of Fig.2.4is

to be found in the lovely paper (Monod and Jacob1961) which treats a number ofdifferent molecular control scenarios, and the reader may find reference to that figurehelpful while following the model development below It should be noted that withthe advent of the power of synthetic biology it is now possible to construct molecularcontrol circuits with virtually any desired configuration and thereby experimentallyinvestigate their dynamics (Hasty et al.2001)

We consider two operons X and Y such that the ‘effector’ of X, denoted by E x,

inhibits the transcriptional production of mRNA from operon Y and vice versa Consider initially a single operon a where a 2 fx ; yg and denote by Na 2 fy; xg

the opposing operon For the mutually repressible systems we consider here, in

the presence of the effector molecule E a the repressor R Na is active (able to bind

to the operator region), and thus block DNA transcription The effector binds with

the inactive form R Naof the repressor, and when bound to the effector the repressor

Fig 2.4 A schematic depiction of the elements of a bistable genetic switch, following Monod and

Jacob ( 1961) There are two operons (X and Y) For each, the regulatory region (Reg x or Reg y)

produces a repressor molecule (R x or R y) that is inactive unless it is combined with the effector

produced by the opposing operon (E y or E x respectively) In the combined form (R x E y or R y E x)

the repressor-effector complex binds to the operator region (O x or O yrespectively) and blocks

transcription of the corresponding structural gene (SG x or SG y ) When the operator region is not

complexed with the active form of the repressor, transcription of the structural gene can take place

and mRNA (M x or M y) is produced Translation of the mRNA then produces an effector molecule

(E x or E y) These effector molecules then are capable of interacting with the repressor molecule of the opposing gene

2.1 Operon Dynamics 15becomes active We take this reaction to be in equilibrium and of the form

R Na C n Na E a • R Na E an Na: (2.14)

Here, R Na E an Na is a repressor-effector complex and n Na is the number of effector

molecules that inactivate the repressor R Na If we let the mRNA and effectorconcentrations be denoted by.M a ; E a/ then we assume that the dynamics for operon

The function f is calculated exactly as we have done in Sect.1.2 Explicitlyincluding the proper subscripts we have

where K a D K 1;a 1 C K 2;a R tot ;a/

We next rewrite Eqs (2.15) and (2.16) by defining dimensionless concentrations.Equation (1.5) becomes

a denotes the leakage and note that ifa goes to infinity then thetranscription goes to zero Similarly using a dimensionless mRNA concentration

Thus the equations governing the dynamics of this system are given by the fourdifferential equations

2.1.4.2 Steady States and Dynamics

The dynamics of this model for a bistable switch can be analyzed as follows Set

W D x1; x2; y1; y2/ so the system (2.17)–(2.20) generates a flow S t W/ The flow

S t W0/ 2 RC

4 for all initial conditions W0D x0; x0; ; y0; y0/ 2 RC

4 and t> 0.The steady states of the system (2.17)–(2.20) are given by x1 D x

2.1.4.3 Graphical Investigation of the Steady States

Figure2.5gives a graphical picture of the five qualitative possibilities for steadystate solutions of the pair of Eqs (2.17)–(2.20)

An alternative, but equivalent, way of examining the steady state of this model is

by examining the solution of either one of the pair of equations

x

d ;x D f x.d ;y f y x// WD F x x/; y

d ;y D f y.d ;x f x y// WD F y y/:

0 0.5 1 1.5 2 2.5 3

x2

0 0.2

0.4

0.6

0.8

1 1.2

y2

A B

C D

E

Fig 2.5 A graphical representation of the possible steady state solutions of Eqs (2.22 ) and ( 2.23 ).

We have plotted the y1and x1isoclines (y2 D d ;y f y x2/ and x2 D d ;x f x y2 / respectively), and

assumed that the y1isocline (the graph of y2 D d ;y f y x2/) is not changed but that x1 isocline (the

graph of x2 D d ;x f x y2 /) is varied as indicated by the labels A to E, e.g by decreasing d ;x (A)

There is a single steady state at a large value of x2and a correspondingly small value of y2 In this

case operon X of the bistable switch is in the “ON” state while operon Y is in the “OFF” state This

steady state is globally stable (B) A decrease in d ;xnow leads to a situation in which there are two

steady states, the largest (locally stable one) corresponding to the intersection of the two graphs, and the second smaller (half stable) one where the two graphs are tangent (C) Further decreases

in d ;x now result in three steady states For the largest (locally stable) one the operon X is in the

on state while Y is in the off state The smallest one (also locally stable) corresponds to operon Y

in the ON state and X is in the OFF state The intermediate steady state is unstable (D) This case

is like B in that there are two steady states, one (locally stable) defined by the intersection of the

two graphs in which Y is ON and the second at the tangency of the two graphs is again half stable.

(E) Finally, for sufficiently small d ;x there is a single globally stable steady state in which Y is ON and X is OFF

We choose to deal with the first Note that since both f x and f y are monotonedecreasing functions of their arguments, the composition of the two

2.1 Operon Dynamics 19

Fig 2.6 A graphical

representation of the possible

steady state solutions of the

equation x=d ;xD

f x.d ;y f y x// WD F x x/ The

smooth monotone increasing

graph is that ofF x x/ as

given in Eq ( 2.24 ), while the

straight line is that of x=d ;x

for different values of d ;x.

The five straight lines

correspond to the five

A B C D E

2.1.4.4 Analytic Investigation of the Steady States

Single Versus Multiple Steady States This model for a bistable genetic switch

may have one [W1( E of Fig.2.5or Fig.2.6) or W3(A)], two [W1; W

3, indicating that W1corresponds to operon X in the OFF state

and operon Y in the ON state while at W3X is ON and Y is OFF.

Analytic conditions for the existence of one or more steady states can be obtained

by first noting that we must have

x

d ;x D f x.d ;y f y x// WD F x x/ (2.25)satisfied In Fig.2.7we have illustrated Eq (2.25) for various values of parameters

In addition to this criteria, we have a second relation at our disposal at thedelineation points between the existence of two and three steady state These points

Fig 2.7 The plot ofd ;x

versus x obtained from

Eq ( 2.25 ) The figure was

constructed for the following

parameters: n x2 f1; 2; 3g,

n y2 f1; 2; 3; 4; 5g, yD 2,

xD 12, y D 10 The blue

and we increase n yfrom 1

(the lowest line) to 5 (the top

one) The red lines

correspond to n xD 2 and the

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x

0 0.5 1 1.5 2 2.5 3 3.5

are also determined by a second relation since x=dis tangent toF x x/ (see Fig.2.6

B, D) Thus we must also have

Indeed, from Eqs (2.22) and (2.23) we have

d ;xd ;yD xy

f x y/f y x/;

A necessary condition for there to be a solution to Eq (2.27), and thus a necessary

condition for bistability, is that L max  R minor

n x n y 1 Cpx/2.1 Cpy/2

.x 1/.y 1/  1:

This is interesting in the sense that if either n x OR n y is one but the other islarger than one then the possibility of bistability behavior still persists, while inthe situation of Mackey et al (2011) this is impossible (the same observation hasbeen made by Cherry and Adler (2000) in a somewhat simpler model) However,note from Fig.2.7that this necessary condition is far from what is sufficient since itwould appear from Eq (2.25) that a necessary and sufficient condition is more like

provided that

Œn x R x/ 2C 2n x R x/xC 1

x 1C 1  0and

where y.x/ is either yCor yas given by (2.28)

In Fig.2.8we have plottedd ;x x/ versus  d ;y x/ with x as the parametric variable.

Inside the region bounded by the blue line (below) and red line (above) we areassured of the existence of bistable behavior while outside this region there will beonly a single globally stable steady state Thus, for example, for a constant value

ofd ;ysuch that bistability is possible, then increasingd ;xfrom0 there will be aminimal valued ;xat which bistability is first seen and this will persist asd ;xisincreased until a second valued ;x< d ;xCis reached where the bistable behavioronce again disappears In Fig.2.9we have shown how the change of the parameter

yinfluences the shape and position of the region of parametersd ;yandd ;xwhere

a bistable behavior is possible It is clear that an increase iny corresponds to adecrease in the leakage, and our results show a clear expansion in the size of theregion of bistability as well as a shift in.d ;y; d ;x/ space

Fig 2.8 The parametric plot

of d ;xversus d ;yobtained

from Eq ( 2.29 ) where we

used the following

2.2 The Appearance of Cell Growth Effects and Delays Due to Transcription 23

Fig 2.9 As in Fig.2.8 but

with varying parameter

y 2 f5; 10; 15g, from left to

right

0 1 2 3 4 5 6 7 8 9 0

1 2 3 4 5 6 7 8

for which the flow S t is directed inward on the surface of B All W2 B and

1 If there is a single steady state, then it is globally stable.

2 If there are two locally stable steady states, then all flows S t W0/ are attracted to

one of them.

to Transcription and Translation

The considerations of the previous sections must, however, be tempered by therealization that sometimes cell growth has to be taken into account as well as thefact that significant delays may enter into the dynamical equations (Mier-y-Teran-Romero et al.2010) The effects of growth are obvious in that if a cell increasesits volume then there is an effect on concentrations But where do these delayscome from? Their origin is simple to understand and arises from the fact that thetranscription and translation processes take place at a finite velocity and thereforerequire a non-zero time for completion The existence of these delays has been

known for some time by modelers (Heinrich and Rapoport 1980) and whetherthe incorporation of the delays will potentially change the qualitative nature ofthe model dynamics will depend on the type of regulation Generally when theregulation is that of an inducible operon there will be no change, but if the system

is a repressible one then the inclusion of the transcriptional and translational delaysmay lead to the prediction of limit cycle behavior

Once we take growth and these transcriptional and translational delays intoaccount, our basic dynamical equations are modified to the form

In Eqs (2.30)–(2.32) there are several changes to be noted The first is the

appearance of the terms e M and e I which account for an effective dilution

of the mRNA (M) and intermediate protein (I) because the cell is growing at a rate

1) The second is the alteration of the decay ratesito Ni i

the cell growth leads to an effective increase in the rate of destruction The last is

the altered notation EM t/  E.t  M / and MI t/  M.t  I/ indicating that

both E and M are now to be evaluated at a time in the past due to the non-zero

times required for transcription and translation From a dynamic point of view, thepresence of these delays can have a dramatic effect

Equations (2.30)–(2.32) can be put in a simpler form, just as we did for (2.1)–(2.3), but by now setting

2.2 The Appearance of Cell Growth Effects and Delays Due to Transcription 25

To finish our simplifications, as before rename the dimensionless concentrations

.m; i; e/ D x1; x2; x3/, and subscripts M; I; E/ D 1; 2; 3/ to obtain

Again Eqs (2.34)–(2.36) are not in dimensionless form.

It is important to realize that the appearance of the delaysM andI (or1 and

2) plays absolutely no role in the determination of the steady state(s) of inducibleand repressible systems as discussed above

For an inducible operon in which f0.X/ > 0 a simple extension of the proof inSmith (1995, Proposition 6.1, Chap 6) shows that the global stability properties arenot altered by the presence of the delays 1; 2/

However, for a repressible operon there are, at this point in time, no extensions

of the global stability results of Smith (1995, Theorems 4.1 and 4.2, Chap 3) forinducible systems The best that we can do is to linearize equations (2.34)–(2.36)

in the neighborhood of the unique steady state Xto obtain the eigenvalue equation

g / D P./ C #e D 0 wherein

P./ D N1C / N2C / N3C / and # D d f0.X/ N1N2N3> 0 (2.37)and D 1C 2 Writing out g./ we have

g./ D 3C a12C a2 C a3C #e; (2.38)where

Let / D ˛ / C i! / be the root of Eq (2.38) satisfying ˛ 0/ D 0 and

! 0/ D !0, and set p D a21 2a2, q D a22 2a1a2, r D a23 #2, and let h.z/ D

z3C pz2C qz C r Ruan and Wei (2001, Theorem 2.4) give the conditions for Xto

be locally stable and for the existence of a Hopf bifurcation

Theorem 2.4 (Ruan and Wei 2001 , Theorem 2.4)

1 If r  0 and  D p2 3q < 0, then all roots of Eq (2.38) have negative real

parts for all  0.

2 If r < 0 or r  0, z1> 0 and h.z1/ < 0, then all roots of Eq (2.38) have negative

real parts when 2 Œ0; 0/.

3 If the conditions of 2 are satisfied, D 0and h0.!2/ ¤ 0, then ˙i!0is a pair

of simple purely imaginary roots of Eq (2.38) and all other roots have negative

real parts Moreover,

dRe 0/

d > 0:

Identifying fast and slow variables can give considerable simplification and insightinto the long term behavior of the system A fast variable in a given dynamicalsystem relaxes much more rapidly to an equilibrium than a slow one (Haken1983).Differences in degradation rates in chemical and biochemical systems lead to thedistinction that the slowest variable is the one that has the smallest degradation rate.Typically the degradation rate of mRNA is much greater than the correspondingdegradation rates for both the intermediate protein and the effector.1 2; 3/ so

in this case the mRNA dynamics are fast and we have the approximate relationship

for the relatively slow effector dynamics If instead the effector qualifies as a fast

variable (as for the lac operon) so that 1 3 2 and x3 ' x2 thenthe intermediate protein is the slowest variable described by the one-dimensionalequation

dx2

dt D 2Œd f x2/  x2 : (2.40)Consequently both Eqs (2.39) and (2.40) are of the form

dx

2.3 Fast and Slow Variables 27

where is either 2 for protein (x2) dominated dynamics or 3 for effector (x3dominated dynamics

In the slightly more complicated case of the bistable switch, if it is the case thatthere is a single dominant slow variable in the system (2.17)–(2.20) relative to all

of the other three (and here we assume without loss of generality that it is in the

X gene) then the four variable system describing the full switch reduces to a single

equation

dx

and  is the dominant (smallest) degradation rate (Here, and subsequently, to

simplify the notation we will drop the subscript x whenever there will not be any

confusion when treating the situation with a single dominant slow variable.)Eliminating fast variables, also known as the adiabatic elimination technique(Haken,1983), has been extended to stochastically perturbed systems when theperturbation is a Gaussian distributed white noise, c.f (Stratonovich1963; Titular

1978; Wilemski 1976, Sect 11.1; and Gardiner 1983, Sect 6.4) For the case ofperturbation being a jump Markov process we refer to Yvinec et al (2014)

Dealing with Noise

In all areas of science, when making experimental measurements it is notedthat the quantity being measured does not have a smooth temporal trajectorybut, rather, displays apparently erratic fluctuations about some mean value whenthe experimental precision is sufficiently high These fluctuations are commonlyreferred to as ‘noise’ and usually assumed to have an origin outside the dynamics

of the systems on which measurements are being made—although there have beenmany authors who have investigated the possibility that the ‘noise’ is actually amanifestation of the dynamics of the system under study Indeed, a desire to findways to quantitatively characterize this ‘noise’ is what led, in large part, to thedevelopment of the entire mathematical field loosely known as stochastic processes,and the interaction of stochastic processes with deterministic dynamics is of greatinterest since it is important to understand to what extent fluctuations or noise canactually affect the operation of the system being studied

Precisely the same issues have arisen in molecular biology as experimentaltechniques have allowed investigators to probe temporal behavior at ever finer levels,even to the level of individual molecules Experimentalists and theoreticians alikewho are interested in the regulation of gene networks increasingly focus on trying toassess the role of various types of fluctuations on the operation and fidelity of bothsimple and complex gene regulatory systems Recent reviews (Kaern et al.2005;Raj and van Oudenaarden2008; Shahrezaei and Swain2008b) give an interestingperspective on some of the issues confronting both experimentalists and modelers.Any cell may be seen as a complex chemical reactor in which a very largenumber of inter-linked chemical reactions take place As discussed in Chaps.1and2, an operon may be regarded as a relatively independently functioning chemicalsubsystem in which the reagents are effectors, repressors, operators, etc Once achemical model for an operon is well established, its dynamics can be expressed inequations using different theoretical formalisms For example, under the assumptionthat the operon is a well-stirred chemical subsystem, one may use the law ofmass action to write down ordinary differential equations that describe the timeevolution of the concentrations of all the involved chemical species This formalism

is known as the deterministic approach and has been extensively used and analyzed

30 Part II Dealing with Noise

in Chaps.1and2 Nevertheless, the law of mass action is only strictly valid when thenumber of molecules involved in all chemical reactions is of the order of Avogadro’snumber, as would be the case for a cell culture Unfortunately, this is not thecase in single cell gene regulation For instance, the number of mRNA moleculescorresponding to an specific gene may be as low as a few dozens, or even less.Consequently, the predictions from deterministic models fail to account for singlecell fluctuations in molecule numbers (also known as biochemical noise), whichoriginate from the stochastic nature of individual chemical reactions (Elowitz et al

2002; Kaern et al.2005; Kepler and Elston2001; Lipniacki et al.2006; Oppenheim

et al.1969; Shahrezaei and Swain2008b)

Biochemical noise has become the object of numerous theoretical and mental studies aimed at answering questions like:

experi-• How (and when) does noise affect cell functioning?

• When is noise detrimental? Is it?

• How has the cell biochemical circuitry evolved to minimize noise if/when it isdetrimental?

• How is noise important for the functioning of cellular networks?

Several different methodologies have been developed to answer these questions; likethe chemical master equation (CME) and that of the so-called chemical Langevinequation to which Chap.3 is devoted In Chap.4 another formalism is analyzed,where we also provide examples related to the approaches of Chap.3