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Exercises

8.11 The repressilator.1 2 (Biology, Computation) ©4

The ‘central dogma’ of molecular biology is thatthe flow of information is from DNA to RNA toproteins; DNA is transcribed into RNA, which thenis translated into protein.

Now that the genome is sequenced, it is thoughtthat we have the parts list for the cell. All thatremains is to figure out how they work together!The proteins, RNA, and DNA form a complex net-work of interacting chemical reactions, which gov-erns metabolism, responses to external stimuli, re-production (proliferation), differentiation into dif-ferent cell types, and (when the cell perceives it-self to be breaking down in dangerous ways) pro-grammed cell death, or apoptosis.

Our understanding of the structure of these inter-acting networks is growing rapidly, but our under-standing of the dynamics is still rather primitive.Part of the difficulty is that the cellular networksare not neatly separated into different modules; agiven protein may participate in what would seemto be several separate regulatory pathways. In thisexercise, we will study a model gene regulatory net-work, the repressilator. This experimental systeminvolves three proteins, each of which inhibits theformation of the next. They were added to thebacterium E. coli, with hopefully minimal interac-tions with the rest of the biological machinery ofthe cell. We will implement the stochastic modelthat the authors used to describe their experimen-tal system [37]. In doing so, we will

• implement in a tangible system an example bothof the central dogma and of transcriptional regu-

lation: the control by proteins of DNA expressioninto RNA,

• introduce sophisticated Monte Carlo techniquesfor simulations of stochastic reactions,

• introduce methods for automatically generatingcontinuum descriptions from reaction rates, and

• illustrate the shot noise fluctuations due to smallnumbers of molecules and the telegraph noisefluctuations due to finite rates of binding andunbinding of the regulating proteins.

Figure 8.12 shows the biologist’s view of the re-pressilator network. Three proteins (TetR, λCI,and LacI) each repress the formation of the next.We shall see that, under appropriate circumstances,this can lead to spontaneous oscillations; each pro-tein peaks in turn, suppressing the suppressor of itssuppressor, leading to its own later decrease.

CIλ LacI

TetR

Fig. 8.12 Biology repressilator. The biologist’s viewof the repressilator network. The T-shapes are blunt ar-rows, signifying that the protein at the tail (bottom ofthe T) suppresses the production of the protein at thehead. Thus LacI (pronounced lack-eye) suppresses TetR(tet-are), which suppresses λ CI (lambda-see-one). Thiscondensed description summarizes a complex series ofinteractions (see Fig. 8.13).The biologist’s notation summarizes a much morecomplex picture. The LacI protein, for example,can bind to one or both of the transcriptional regu-lation or operator sites ahead of the gene that codesfor the tetR mRNA.3 When bound, it largely blocks

1From Statistical Mechanics: Entropy, Order Parameters, and Complexity by JamesP. Sethna, copyright Oxford University Press, 2007, page 179. A pdf of the text isavailable at pages.physics.cornell.edu/sethna/StatMech/ (select the picture of thetext). Hyperlinks from this exercise into the text will work if the latter PDF isdownloaded into the same directory/folder as this PDF.2This exercise draws heavily on Elowitz and Leibler [37]; it and the associated soft-ware were developed in collaboration with Christopher Myers.3Messenger RNA (mRNA) codes for proteins. Other forms of RNA can serve asenzymes or parts of the machinery of the cell. Proteins in E. coli by convention havethe same names as their mRNA, but start with capitals where the mRNA start withsmall letters.4RNA polymerase, the molecular motor responsible for transcribing DNA into RNA,needs to attach to the DNA at a promoter site. By binding to the adjacent op-erator sites, our repressor protein inhibits this attachment and hence partly blockstranscription. The residual transcription is called ‘leakiness’.

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the translation of DNA into tetR.4 The level oftetR will gradually decrease as it degrades; henceless TetR protein will be translated from the tetRmRNA. The resulting network of ten reactions isdepicted in Fig. 8.13, showing one-third of the to-tal repressilator network. The biologist’s shorthand(Fig. 8.12) does not specify the details of how oneprotein represses the production of the next. Thelarger diagram, for example, includes two operatorsites for the repressor molecule to bind to, leadingto three states (P0, P1, and P2) of the promoter re-gion depending upon how many LacI proteins arebound.

You may retrieve a simulation package for the re-pressilator from the computational exercises por-tion of the book web site [129].

(a) Run the simulation for at least 6000 secondsand plot the protein, RNA, and promoter states asa function of time. Notice that

• the protein levels do oscillate, as in [37, fig-ure 1(c)],

• there are significant noisy-looking fluctuations,

• there are many more proteins than RNA.

Fig. 8.13 Computational repressilator.

The Petri net version [50] of one-third of the repres-silator network (the LacI repression of TetR). Thebiologist’s shorthand (Fig. 8.12) hides a lot of com-plexity! We have implemented these equations foryou, so studying this figure is optional. The solidlighter vertical rectangles represent binding reactionsA + B → C, with rate kb[A][B]; the open verticalrectangles represent unbinding C → A + B, withrate ku[C]. The horizontal rectangles represent cat-alyzed synthesis reactions C → C+P , with rate γ[C];the darker ones represent transcription (formationof mRNA), and the lighter one represent translation(formation of protein). The black vertical rectanglesrepresent degradation reactions, A → nothing withrate kd[A]. The LacI protein (top) can bind to theDNA in two promoter sites ahead of the gene cod-ing for tetR; when bound, it largely blocks the tran-scription (formation) of tetR mRNA. P0 representsthe promoter without any LacI bound; P1 representsthe promoter with one site blocked, and P2 repre-sents the doubly-bound promoter. LacI can bindto one or both of the promoter sites, changing Pi

to Pi+1, or correspondingly unbind. The unboundP0 state transcribes tetR mRNA quickly, and thebound states transcribe it slowly (leaky repression).The tetR mRNA then catalyzes the formation of theTetR protein.

To see how important the fluctuations are, weshould compare the stochastic simulation to the so-lution of the continuum reaction rate equations (aswe did in Exercise 8.10). In [37], the authors writea set of six differential equations giving a contin-uum version of the stochastic simulation. Theseequations are simplified; they both ‘integrate out’or coarse-grain away the promoter states from thesystem, deriving a Hill equation (Exercise 6.12) forthe mRNA production, and they also rescale theirvariables in various ways. Rather than typing intheir equations and sorting out these rescalings, itis convenient and illuminating to write a routine togenerate the continuum differential equations di-rectly from our reaction rates.

(b) Write a DeterministicRepressilator, derivedfrom Repressilator just as StochasticRepressilatorwas. Write a routine dcdt(c,t) that does the fol-lowing.

• Sets the chemical amounts in the reaction net-work to the values in the array c.

• Sets a vector dcdt (of length the number of chem-icals) to zero.

• For each reaction:

– compute its rate;

– for each chemical whose stoichiometry is

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Exercises 3

changed by the reaction, add the stoichiome-try change times the rate to the correspond-ing entry of dcdt.

Call a routine to integrate the resulting differen-tial equation (as described in the last part of Exer-cise 3.12, for example), and compare your resultsto those of the stochastic simulation.

The stochastic simulation has significant fluctua-tions away from the continuum equation. Part ofthese fluctuations are due to the fact that the num-bers of proteins and mRNAs are small; in partic-ular, the mRNA numbers are significantly smallerthan the protein numbers.

(c) Write a routine that creates a stochastic repres-silator network that multiplies the mRNA concen-trations by RNAFactor without otherwise affectingthe continuum equations. (That is, multiply theinitial concentrations and the transcription ratesby RNAFactor, and divide the translation rate byRNAFactor.) Try boosting the RNAFactor by ten andone hundred. Do the RNA and protein fluctuationsbecome significantly smaller? This noise, due tothe discrete, integer values of chemicals in the cell,is analogous to the shot noise seen in electrical cir-cuits due to the discrete quantum of electric charge.

It scales, as do most fluctuations, as the square rootof the number of molecules.

A continuum description of the binding of the pro-teins to the operator sites on the DNA seems par-ticularly dubious; a variable that must be zero orone is replaced by a continuous evolution betweenthese extremes. (Such noise in other contexts iscalled telegraph noise—in analogy to the telegraph,which is either silent or sending as the operator tapsthe key.) The continuum description is accurate inthe limit where the binding and unbinding ratesare fast compared to all of the other changes in thesystem; the protein and mRNA variations then seethe average, local equilibrium concentration. Onthe other hand, if the rates are slow compared tothe response of the mRNA and protein, the lattercan have a switching appearance.

(d) Incorporate a telegraphFactor into yourstochastic repressilator routine, that multiplies thebinding and unbinding rates. Run for 1000 secondswith RNAFactor = 10 (to suppress the shot noise)and telegraphFactor = 0.001. Do you observefeatures in the mRNA curves that appear to switchas the relevant proteins unbind and bind?