Self-Consistent Proteomic Field Theory of Stochastic Gene Switches

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1 88 Bophyscal Journal Volume 88 February Self-Consstent Proteomc Feld Theory of Stochastc Gene Swtches Aleksandra M. Walczak,* Masak Sasa, y and Peter G. Wolynes* z *Department of Physcs, Center for Theoretcal Bologcal Physcs, Unversty of Calforna at San Dego, La Jolla, Calforna; y Department of Complex Systems Scence, Graduate School of Informaton Scence, Nagoya Unversty, Nagoya, Japan; and z Department of Chemstry and Bochemstry, Unversty of Calforna at San Dego, La Jolla, Calforna ABSTRACT We present a self-consstent feld approxmaton approach to the problem of the genetc swtch composed of two mutually repressng/actvatng genes. The proten and DNA state dynamcs are treated stochastcally and on an equal footng. In ths approach the mean nfluence of the proteomc cloud created by one gene on the acton of another s self-consstently computed. Wthn ths approxmaton a broad range of stochastc genetc swtches may be solved exactly n terms of fndng the probablty dstrbuton and ts moments. A much larger class of problems, such as genetc networks and cascades, also reman exactly solvable wth ths approxmaton. We dscuss, n depth, certan specfc types of basc swtches used by bologcal systems and compare ther behavor to the expectaton for a determnstc swtch. INTRODUCTION Genetc swtch systems are an elementary means of regulatory control present n every lvng organsm. Ther complexty and detals dffer, but the general mechansm, that of the expresson of a gven gene beng regulated by protens, s beleved to be unversal (Ptashne and Gann, 00). They are buldng blocks of larger regulatory elements: genetc networks and sgnalng cascades. The pathways by whch these systems operate are passed on from generaton to generaton. Understandng ther stablty and characterstcs s therefore fundamental. A lot of prevous work has consdered a determnstc descrpton of genetc swtches (Ackers et al., 98; Hasty et al., 00). The need for a stochastc treatment of genetc swtches, due to the sngle copy of the DNA molecule and multple proten molecules n the cell, has been largely recognzed (Sneppen and Aurell, 00; Kepler and Elston, 00). The most general way of accountng for nondetermnstc processes s to wrte down the master equaton for a gven system. To defne the state of the swtch, one must specfy the DNA bndng states of partcular genes and the number of protens of each type. The probablty dstrbuton for even a sngle swtch consstng of two genes, the product protens of whch act as regulator protens for the system, may not be determned exactly and approxmatons must be consdered (Balek, 00; Hasty et al., 000; Sneppen and Aurell, 00). Several approaches to account for the probablstc nature of chemcal reactons have been undertaken, rangng from the Langevn descrpton of sngle genes (Balek, 00), and two nteractng gene swtches (Hasty et al., 000), to the master equaton reduced to Fokker-Planck equaton consderatons (Kepler and Elston, 00; Hasty et al., 00a). A dynamcal acton formulaton has also been used (Sneppen and Aurell, 00) to determne the lfetmes of states of the swtch. A popular alternatve to purely analytcal methods, whch often need to make approxmatons or are lmted to very smple model systems, has been to conduct stochastc smulatons of genetc swtches. Two types of smulatons are mostly used. In the frst, the randomness of the system s ntroduced by means of a Monte Carlo algorthm wth a fxed tme step (Paulsson et al., 000). The second s based on the Gllespe algorthm (Gllespe, 977) to predct the probablty of a gven reacton occurrng (Arkn et al., 998). For sngle-gene systems, stochastc smulatons have shown that the stochastcty n the system s responsble for the bmodal probablty dstrbutons (Cook et al., 998) that have been expermentally observed. These methods prove very useful, because they allow us to test the theoretcal predctons on model systems that mght be hard to buld expermentally. However, ths approach often does not enable us to gan ntuton or nsght nto the mechansms behnd the functonng of the system. The am of the present work s to gan a better and deeper understandng of the devce physcs of genetc swtches. We therefore, contrary to many mportant prevous dscussons (McAdams and Arkn, 997; Aurell et al., 00; Vlar et al., 003), do not present a specfc concrete bologcal system, but dscuss generc behavor and try to understand ts sources. Our approxmaton also allows for an exact soluton of a broad class of genetc swtch systems wthout any further assumptons and wth lttle computatonal effort. Hasty et al. (00b) present an overvew of the exstent theoretcal approaches. A popular approxmaton assumes that the DNA bndng state reaches equlbrum much faster than the proten number state. Therefore the adabatc approxmaton s often consdered (Ackers et al., 98; Sneppen and Aurell, 00; Darlng et al., 000), allowng for a thermodynamc treatment (Ackers et al., 98) of the DNA bndng state. The proten number fluctuatons are then treated stochastcally. Even before the statstcal thermodynamcs approach Submtted July 30, 004, and accepted for publcaton October, 004. Address reprnt requests to Peter Wolynes, Unversty of Calforna at San Dego, Chemstry and Bochemstry, 9500 Glman Drve, MC 037, La Jolla, CA Tel.: ; E-mal: pwolynes@ucsd.edu. Ó 005 by the Bophyscal Socety /05/0/88/3 $.00 do: 0.59/bophysj

2 SCPFT of Stochastc Gene Swtches 89 of Ackers et al. (98) usng partton functons, much prevous work assumed the DNA bndng and unbndng can smply be accounted for by an equlbrum constant, snce the relaxaton tmescales for equlbraton of the DNA state are much larger than those of the proten numbers, whch requre proten synthess and degradaton to change. The partton functon approach has also been successful for lookng at logc gates bult from swtches (Buchler et al., 003). The adabatc approxmaton s beleved to hold true n many cases, judgng by the expermental parameters of bologcal swtches (Darlng et al., 000). But as the experments of, for example, Becske et al. (00) show, not all swtches need to functon wthn the adabatc lmt and the nonadabatc lmt may result n new phenomena. We therefore consder a wde range of parameter ratos n our dscusson. In ths artcle we explore more fully an approxmaton, prevously used by Sasa and Wolynes (003) for the varatonal treatment of the problem, the self-consstent proteomc feld (SCPF) approxmaton. Wthn ths approxmaton one assumes that the probablty of fndng the swtch n a gven state s a product of the probabltes of states of ndvdual genes. One can then solve the steady-state master equaton for the probablty dstrbuton of many regulatory systems exactly. We dscuss the approxmaton and present a detaled study of dfferent classes of genetc swtches, some of whch have never been consdered theoretcally. We consder separately several partcular features of such systems that are found n known swtches, to be able to characterze ther contrbutons to the behavor of the whole system. To be specfc, startng from a symmetrc toggle swtch, we go on to compare the effects of multmer bndng, and of the producton of protens n bursts, on the stablty of the swtch. The stochastc effects prove to be modest for symmetrc swtches wthout bursts, especally f the genes have a basal producton rate. We fnd the determnstc and stochastc SCPF solutons to have smlar probabltes of partcular genes to be on, and smlar mean numbers of protens of a gven speces n the cell. However, n the nonadabatc lmt, when the unbndng rate from the DNA s smaller than the death rate of protens, the probablty dstrbutons have two well-defned peaks, unlke n the determnstc approxmaton or adabatc lmt of the stochastc SCPF soluton. We also show that the effect of stochastcty on the observables becomes more apparent when protens are produced n bursts. In these types of swtches, the defnton of the adabatc lmt, whch was clear for the swtches n whch protens are produced separately, s no longer smple. Our dscusson shows that the propertes of genes often analyzed n the determnstc lmt, may be strongly nfluenced by stochastcty n ths case. Randomness n a bologcal reacton system leads to quanttatve and, n many examples, even qualtatve changes, from predctons of determnstc models. We also dscuss the dfferences n the behavor of asymmetrc and symmetrc swtches. We pont to the mechansms resultng n dfferent types of bfurcatons and show how they are nfluenced by nose. Wthn the SCPF approxmaton, swtches that are regulated by bndng and unbndng of monomers do not have regons of bstablty. Ths holds true for both symmetrc and asymmetrc swtches. When protens are produced ndvdually rather than n bursts, fast unbndng from the DNA can effectvely mnmze the destructve effect of proten number fluctuatons on the stablty of the DNA bndng state. Furthermore, a detaled analyss of the probablty dstrbutons shows that they have long tals, and are far from Possonan n both the adabatc and nonadabatc lmts. We dscuss the propertes of the system n terms of clouds of protens bufferng the DNA. We show how fast or slow DNA bndng characterstcs and proten number fluctuatons nfluence the stablty of the bufferng clouds, leadng to specfc emergent behavor of observables. Throughout the artcle, a comparson s made between results of the exact stochastc soluton, to solutons of the determnstc knetc equatons for the system wthn the self-consstent proteomc feld approxmaton. We establsh a base of potental buldng blocks of more complcated swtches and systems, such as networks and sgnalng cascades, for whch an exact soluton wthn the present approxmaton can also be obtaned. A detaled dscusson of these larger systems wll be the topc of another artcle. We also present lmtatons of the present style of analyss where exact solutons are not possble. There are two ams of ths artcle. The frst s to dscuss the self-consstent feld approxmaton and show that t has an exact soluton that could be extended to a large class of systems. Ths approxmaton lets one deal n a straghtforward and computatonally nexpensve manner wth the effect of random processes on genetc networks. The second s to dscuss the many components of bologcal swtches present n nature and n engneered systems, n the necessary stochastc framework. THE SELF-CONSISTENT PROTEOMIC FIELD APPROXIMATION The basc mechansm of gene transcrpton regulaton n prokaryotes may be reduced to the bndng and unbndng of regulatory protens, repressors, and actvators, to the operator ste of the DNA. If we use ths smplfed treatment, whch neglects extra levels of regulaton such as the bndng of RNA polymerase, effectvely each gene can be descrbed as beng ether n an actve (on) state, when the repressor s unbound (actvator bound); or n an nactve (off) state, wth the repressor bound (actvator unbound). The stochastc system of a sngle gene and ts product protens s descrbed by the jont probablty dstrbuton P~ðn; tþ ¼ðP ðn; tþ; P ðn; tþþ of the number of product protens n the cell n, and the DNA bndng-ste state, as on (proten not bound) ¼ ; and off (proten bound) ¼. To conserve probablty, + n ðp ðn; tþp ðn; tþþ ¼ : Bophyscal Journal 88()

3 830 Walczak et al. If one consders two nteractng genes, the descrpton n terms of a jont probablty vector needs to be extended to four states: both genes may be on, oroff; or one of the genes may be on, and the other off. If the two genes do not nteract, as would be the case for two self-regulatory protens, the probablty of fndng the two-gene system n a gven state, defned by both the number of product protens and the DNA bndng-ste state, would be the product of the states of partcular genes P jj# (n,n ;t) ¼ P j (n ;t)p j# (n ;t). Ths s generally not true for two nteractng protens, as s the case n a genetc swtch. However, as a frst approxmaton to the problem, one can gnore correlatons between the spaces of the two genes and assume the space of the swtch s a sum of spaces of the genes that compose t. Snce we are lookng for solutons n whch the symmetry of the system s broken and dfferent behavors of the on- and off-state of a gene are possble, we must allow for dfferent probablty dstrbuton functons for the on- and off-states. Ths s analogous to the unrestrcted Hartree approxmaton n quantum mechancs, where allowng dfferent spatal functons for spn-up and spn-down states results n breakng of the symmetry of the bound molecular orbtal soluton to the dssocated soluton of two separate hydrogen atoms wth opposte spn-states for large nternuclear dstances. We therefore allow for multple solutons for a gven set of parameters. The total probablty of havng a gven gene state j and n protens of that type s smply gven by P j (n,n # ) ¼ P j,j#¼0 (n,n # ) P j,j#¼ (n,n # ). The self-consstent approxmaton s a crude one, snce n the case of the genetc swtch, the state of a gven gene s often determned by the number of proten products of the other gene. However, wthn ths approxmaton, one can solve the master equaton for the probablty dstrbuton exactly, wthout any further approxmatons. Ths yelds a powerful computatonal tool, whch smultaneously gves useful nsght. THE TOGGLE SWITCH For clarty of exposton, we show how the problem may be solved exactly wthn the self-consstent proteomc feld approxmaton on a well-defned system of the toggle swtch. We then expand the method to apply to other systems. The elementary system we use as an example s composed of two genes, labeled and, as presented n Fg.. Gene produces protens of type, whch act as regulatory protens,.e., repressors, on gene. The product of gene, protens of type, n turn repress gene. In ths smplfed model, we assume that proten producton occurs nstantaneously upon unbndng of the repressor. For now, we assume that repressor protens bnd as dmers, snce that s a common scenaro n bologcal systems, but we do not treat dmerzaton knetcs explctly. For smplcty, the couplng form between the genes responsble for bndng wll be taken to be of the form h n p 3ÿ ; where p s the order of the multmerzaton of the repressor. Ths form s a small approxmaton to the more exact h n 3ÿ (n 3ÿ ÿ )...(n 3ÿ ÿ p ). We have checked that FIGURE A schematc representaton of the toggle swtch. Gene produces protens of type, whch repress gene ; and gene produces protens of type, whch repress gene. usng the smpler monomal does not nfluence the results n any regme dscussed. We also do not account for the exstence of mrna molecules and the consequent tme delays owng to ther synthess as ntermedates. The extensons of the model are dscussed later. Wthn the self-consstent proteomc feld approxmaton, the set of master equatons for the correspondng system s of the ðn ðn Þ ¼ g ðþ½p ðn ÿ ÞÿP ðn ÞŠ k ½ðn ÞP ðn Þ ÿ n P ðn ÞŠ ÿ h n 3ÿ P ðn Þ f P ðn Þ; ¼ g ðþ½p ðn ÿ ÞÿP ðn ÞŠ k ½ðn ÞP ðn Þ ÿ n P ðn ÞŠ h n 3ÿ P ðn Þÿf P ðn Þ; () for n $ where the ¼, refers to the gene label. P (n ) descrbes the probablty of gene beng n the on-state and there beng n proten molecules of type n the cell. The frst term on the rght-hand sde of Eq. descrbes the producton of protens of type wth a producton rate g j (), where j ¼,, dependng on whether the gene s n the on- or off-state. The second term accounts for the destructon of protens wth rate k. The bndng of repressor protens produced by the other gene s proportonal to the number of dmer molecules present n the system n 3ÿ wth rate h. We assume unbndng occurs wth a constant rate f. Bndng and unbndng contrbutes to the knetcs of the DNA bndng states, as descrbed by the last two terms. Ths set s supplemented by the P j (n ¼ 0) equatons to account for boundary ðn ¼ 0Þ ¼ÿg ðþp ðn ¼ 0Þ k P ðn ¼ Þ ÿ h n 3ÿ P ðn ¼ 0Þ f P ðn ¼ 0Þ; Bophyscal Journal 88()

4 SCPFT of Stochastc Gene Swtches ðn ¼ 0Þ ¼ÿg ðþp ðn ¼ 0Þ k P ðn ¼ Þ h n P 3ÿ ðn ¼ 0Þÿf P ðn ¼ 0Þ: () For convenence, let us defne + n P j ðn Þ¼C j ; the probablty of fndng the DNA bndng ste n a gven state. One can now sum the P j () equatons over the number states of the second proten wth P () P (), and lkewse the P j () equatons. Due to the SCPF approxmaton, the only term affected s the repressor bndng term h ðn Þ; and snce + n P ðþ P ðþ ¼; the summaton results n + n h ðn ÞðP ðþ P ðþþ ¼ h ðc ðþæn æc ðþæn æþ¼h FðÞ, where Æn j æ s the second moment of the number dstrbutons of type protens produced when gene s n the j th state. The equatons of moton of the moments of the probablty dstrbuton are of the j ðþæn k æ j ¼ g j ðþ½æðn j Þ k æ ÿ Æn k j æšc jðþ k ½Æn j ðn j ÿ Þ k æ ÿ Æn k j æšc j ðþ ðÿþ j h Fð3 ÿ ÞÆn k æc ðþ ðÿþ j f Æn k æ C ðþ: (3) The steady-state equatons for the moments of the dstrbutons that follow are closed-form; the n th order moment equaton of moton depends only on the lower moments of the th gene and n 3ÿ : To analyze the behavor of swtches we ntroduce the followng scaled parameters: the adabatcty parameter v ¼ f /k, whch represents the characterstc rate of change of the DNA state compared to the characterstc rate of change n proten number, X eq ¼ f =h measures the tendency for protens to be unbound from the DNA; X ad ¼ðg ðþ g ðþþ=ðk Þ the effectve producton rate; and dx sw ¼ ðg ðþÿg ðþþ=ðk Þ dstngushes between the two DNA states n terms of proten dynamcs. We present a detaled dervaton of the moment equatons n Appendx A. The resultng equatons for the 0 th moments couple to the hgher moments by the nteracton functon F(). These lower moments can be solved self-consstently. The resultng soluton predetermnes all the other moments, whch completely descrbe the probablty dstrbuton. Each gene therefore couples to the other gene by the nfluence of the self-consstently generated proteomc feld. One could defne the generatng functon and calculate the probabltes of havng a gven DNA bndng state j for the th gene when there are n protens of type n the cell. In practce, t s easer to go back to the steady-state master equaton and solve drectly for the probablty dstrbutons than sum an nfnte number of moments. Rewrtng the steady-state master equaton (Eq. ) one gets P ðn Þ¼ X ad dx sw Fð3 ÿ Þ v X eq n ðn ÞP ðn Þv P ðn ÞŠ: ½ðX ad dx sw ÞP ðn ÿ Þ P ðn Þ¼ X ad ÿ dx sw v n ½ðXad ÿ dx sw ÞP ðn ÿ Þ Fð3 ÿ Þ ðn ÞP ðn Þv X eq P ðn ÞŠ: ð4þ These sets of equatons gve recurson relatons for P j (n ) that one can use to express P j (n ) as a functon of P (0) and P (0). The normalzaton condton+ n ðp ðn ÞP ðn ÞÞ ¼ gves P j (0) n term of constants and the result s the probablty functon P j (n ) as a seres. The SCPF approxmaton reduces the two-gene problem to a one-gene problem parameterzed by the moments of the second gene, whch can be worked out ndependently, as we have already shown, and these are represented by F(3ÿ) whch s a constant n terms of ths calculaton. To see the effect of the stochastc nature of the system, we compare the exact solutons of the self-consstent feld approxmaton equatons to the results that would follow from determnstc knetc rate equatons for the number of protens of each type and the fracton of on/off DNA bndng states for each gene, C ðþ¼ X eq nðþ¼x ad X eq n ð3 ÿ Þ dx sw ðc ðþÿc ðþþ; (5) where n() s the number of protens of type present n the cell. The exact SCPF equatons reduce to the determnstc knetc equatons n the lmt of large v and X ad for the case dscussed above. The F(3 ) term n the stochastc SCPF equatons s replaced by the n (3 ) term n the determnstc knetc rate equatons. For the toggle swtch, where repressors bnd as dmers, t s easly shown that the nteracton functonal may be rewrtten n the form FðÞ¼ðX ad Þ X ad ðdx sw ÿ C ðþþðx ad Þ dx sw ðc ðþ Þÿ4v ðdx sw Þ C ðþc ðþ v C ðþ ¼ ÆnðÞæ v ÆnðÞæ; (6) v C ðþ whch n the large v-lmt reduces to F() ¼ Æn()æ Æn()æ. So for large mean numbers of protens present n the cell, whch corresponds to large effectve producton rates X ad, Æn()æ of the order of hundreds s a small correcton to Æn()æ. We therefore reproduce the determnstc knetcs result. As shown by Sasa and Wolynes (003), the dfference n the probablty that gene s actve and that gene s actve, DC ¼jC () C ()j, plays the role of an order parameter. We can now consder a famly of swtches and dscuss ther stablty, senstvty of regons of bstablty to control parameters, and types of bfurcatons. THE SYMMETRIC TOGGLE SWITCH For pedagogc purposes we wll start by analyzng the sngle symmetrc toggle swtch, such as dscussed above, n whch Bophyscal Journal 88()

5 83 Walczak et al. repressors bnd as dmers, wth v ¼ v ¼ v, X ad ¼ Xad ¼ X ad ; dx sw ¼ dx sw ¼ dx sw ; and X eq ¼ Xeq ¼ Xeq ; as t s the most ntutve and shows the most generc behavor. It s an academc example, as even ndvdual genes n swtches engneered n the laboratory mostly have dfferent chemcal parameters. Yet a lot can be learned from ths smple system. The general mechansm of the phase transton Fg. shows the phase dagrams for the system, jdcj, as a functon of reservor proten number and the adabatcty parameter for the exact SCPF equatons for growng values of the parameter descrbng the tendency that protens are unbound from the DNA, X eq. The determnstc knetcs and exact SCPF approxmatons gve qualtatvely smlar results. The analogous determnstc knetc phase dagrams agree wth the SCPF solutons n the large v- and X ad -lmt, hence they become more smlar wth growng X eq, as the bfurcaton occurs at larger effectve producton rates for larger X eq. For large fluctuatons and a small unbndng rate, nether gene nor gene s favored and the probablty of a gven gene to be on s determned solely by the effectve producton rate of the other gene and decreases n a quadratc manner as the number of repressor protens grow (Fg. 3). Snce the swtch s symmetrc, the system has one stable state, DC ¼ 0, where the probabltes of the genes to be on are equal. As the relatve proten number fluctuatons get smaller and the DNA unbndng rate grows, a proteomc cloud buffers the repressed gene, keepng t repressed. The symmetry of the system s broken and the soluton bfurcates nto two separate basns of attracton. For the stochastc SCPF equatons the bfurcaton takes place for larger effectve producton rates (larger X ad ), than for the determnstc equatons, even n the large v-lmt, whch depcts ther senstvty to fluctuatons. The crtcal number of reservor protens necessary for the bfurcaton of the soluton to take place s the same n both approxmatons and s determned by Ænæ c ¼ (X eq ) ½ (Fg. 3). In the dscussed example, Ænæ c ¼ 3 ¼ 000 ½, for X eq ¼ 000. For the determnstc knetc swtch the bfurcaton takes place when C () ¼ ( Æn(3 )æ /X eq ) ÿ ¼ 0.5, due to the smple form of the nteracton functon equal to Æn(3 )æ ¼ (X ad C (3 )).SoC () ¼ 0.5 s equvalent to the Æn(3 )æ /X eq ¼. In a nosy system larger effectve producton rates are needed to acheve the crtcal value of protens. The nteracton functon n ths case may be wrtten as FðÞ ¼ÆnðÞæ ðv= ðvc ðþþþænðþæ; and ðv= ðvc ðþþþ$; always. So at Ænæ c, F(3 )/X eq. and the probablty of the genes to be on s,0.5, therefore C bff;scpf ðþ, C bff;kn ðþ: The mechansm of the bfurcaton requres the two genes to be more lkely to be unbound than bound for the phase transton to take place. The curvature of the null clnes presented n Fg. can be smply worked out to be of the form v ¼ðz =ðj X ad j X ad z ÞÞ ÿ j ; wth z,j constants determned by the specfc value of C (), C (). Adabatcty parameter dependence As the adabatcty parameter decreases, the area of phase space correspondng to multple solutons decreases (Fg. ). For very small values of the adabatcty parameter, there exsts only one soluton that corresponds to a state n whch the two genes are off. The value of v below whch only one soluton exsts decreases wth the tendency for protens to be bound, but exsts for all values of X eq. Therefore f the two genes have very hgh repressor bndng affntes, the crtcal number of protens necessary for the phase transton to take place cannot be formed, even for very hgh producton rates. Ths regon of parameter space where one soluton s possble corresponds to a stuaton n whch a bufferng proteomc cloud may not form, due to a very fast destructon rate of protens or a very small unbndng rate from the DNA. The crtcal number of protens necessary for the bfurcaton to occur grows wth the tendency for protens to be unbound from the DNA (X eq ), as the cloud bufferng the genes needs to be bgger and exhbt smaller relatve proten number fluctuatons, whch effectvely decrease wth the growth of the adabatcty parameter. Ths s further dscussed n terms of the probablty dstrbutons. Therefore a monostable soluton exsts at all values of the effectve growth rate, X ad, for larger values of v at large X eq than at smaller X eq values. The bfurcaton pont s a result of competton between the number of reservor repressor protens and the tendency for protens to be unbound from the DNA. Ths s clear from the dependence of the number of protens present n the cell at FIGURE Phase dagram obtaned as an exact soluton wthn the SCPF approxmaton for the sngle symmetrc swtch when repressors bnd as dmers wth X eq ¼ (A), 00 (B), and 000 (C). Contour lnes mark values of DC. Bophyscal Journal 88()

6 SCPFT of Stochastc Gene Swtches 833 FIGURE 3 Probablty that genes are n the actve state (A), the mean number of protens of each type present n the cell Æn()æ(B), and the mean number of protens of each type present n the cell f gene s n the on-state Æn ()æ (C) as a functon of X ad ¼ dx sw for a symmetrc swtch. Exact solutons of the SCPF approxmaton equatons compared wth determnstc knetc rate equatons solutons, for a sngle symmetrc swtch, X eq ¼ 000, v ¼ 0.5. the bfurcaton pont on the relatve values of X ad and X eq, but not the adabatcty parameter v. Mean proten numbers The total number of protens present n the cell, produced both n the on- and off-states, asymptotcally away from the bfurcaton ponts s the same for the determnstc and stochastc approxmatons, and t s gven by Æn()æ ¼ X ad, when C () the probablty of the gene to be on s close to unty. The number of protens of a gven type present n the cell, when the gene that produces them s n the on-state, s always consderably smaller n the nosy system than n the determnstc case (Fg. 3 C). Snce the producton rate n the off-state was assumed zero, n the determnstc case no protens of a gven type are present n the cell f the gene s n the off-state, unlke n the nosy system. Therefore the number of protens n the determnstc system s nonzero only f the gene s on. But nteracton of the DNA bndng state wth the protens bufferng t results n a resdual number of protens present n the off-state for all values of v. The regon of bstablty of the swtch n parameter space grows as the bndng rate ncreases wth respect to the unbndng rate, stablzng the DNA bndng states. As the susceptblty of the system to fluctuatons ncreases, the determnstc equatons prove to be a poor approxmaton to descrbe the state of the system. Gene-bufferng proteomc cloud nteractons The stochastc nature of the system also manfests tself at the DNA level (Fg. ). As the tendency for protens to be unbound from the DNA grows, the area of parameter space wheren multple solutons are possble decreases snce a larger number of protens s needed to reach a state n whch two genes are more lkely to be repressed (proten bound state) than at small X eq. For small unbndng rates or large bndng rates, regardless of the rato of the rate of unbndng of repressors from the DNA to proten degradaton, bstablty requres smaller numbers of protens, whch correspond to larger relatve fluctuatons, than for large X eq. Therefore a larger unbndng rate relatve to the bndng rate makes the system more susceptble to proten number nose. Competton between X eq and Æn()æ results n X eq, for a gven null clne, beng a parabolc functon of X ad, for the dmer bndng case, wth coeffcents determned by v and C (). Ths s easly generalzed to hgher order functons for hgher order (p) olgomers, and results n p-order dependence. The swtchng regon, by whch we mean that the regon of parameter space between the bfurcaton pont and DC. 0.9 decreases as the bndng and unbndng rates become comparable (X eq decreases). As dscussed above, the probablty of the genes to be on at the bfurcaton pont tends to 0.5 as the adabatcty parameter grows (Fg. 3), therefore the probablty to be on has to ncrease by a smaller DC to reach C () ¼. Therefore the swtchng regon decreases also as the unbndng rate from the DNA grows, snce smaller effectve producton rates are needed to reach DC ¼, than for small v. Small values of v correspond to large fluctuatons n the DNA bndng state as well as the proten number state, and result n destablzng the gene-bufferng proten cloud nteractons. Hence very large effectve producton rates are needed for DC Therefore the DNA unbndng rate must become consderably faster compared to the proten degradaton rate for the swtch to have two stable solutons n a large regon of parameter space. The probablty dstrbutons A better understandng of the bfurcaton can be ganed from examnng the probablty dstrbutons. Fg. 4, A and B, and Fg. 4, C and D, show the evoluton of the probablty dstrbutons of gene and gene, respectvely, to be on and off as functons of X ad. The peak of the dstrbuton decreases and the wdth spreads out as the control parameter grows, untl t reaches the bfurcaton pont at X ad ¼ 44. Then the value of the probablty functon correspondng to the most probable number of protens grows agan. The spread of the functons grows as the effectve producton rate n the onstate ncreases; t narrows, however, wth the ncrease of the adabatcty parameter, as would be expected, snce the DNA state fluctuatons become smaller wth v. The average number of protens n the cell n the on-state (DC. 0.9) does not show a dependence on v. Yet as the unbndng rate from the DNA becomes very fast compared to the proten number Bophyscal Journal 88()

7 834 Walczak et al. FIGURE 4 Evoluton of probablty dstrbutons for the probablty of the gene that wll be actve (on) after the bfurcaton to be on (A) and off (B) and the gene that wll be nactve (off) tobeon (C) and off (D) as a functon of the order parameter X ad for a symmetrc swtch. The bfurcaton occurs at X ad ¼ 44, X eq ¼ 000, v ¼ 0.5. fluctuatons, the system swtches often between the two states, hence a large number of protens s present even n the off-state. If the DNA unbndng rate s small, the proten number characterstcs follow the DNA state havng tme to reach a steady state wthn each well, before the DNA bndng ste swtches nto the other state, so the number of protens n the off-state falls to zero (Fg. 5, A and B). Ths results n a two-peak, bmodal probablty dstrbuton (Fg. 4). If v s large, random fluctuatons n the DNA state do not change the effectve state of the system, snce a resdual hgh mean proten number s present even n the off-state. In such a case, lower effectve producton rates than for small v result n hgher proten yelds, and hence smaller swtchng regons. For small v one mght expect Posson dstrbutons of protens n each of the DNA states, snce the unbndng rate from the DNA s smaller than the proten degradaton rate, so the protens may reach a steady state wthout the DNA state changng. Hence, effectvely protens would feel the effect of only one well and be subject to a brth/death process. Ths s not true, however. The dfference between the exact soluton and a soluton obtaned wthn a Possonan approxmaton to the state of the system s surprsngly large, owng to the skewed tals of these dstrbutons. Fg. 5, C and D, compares these probablty dstrbutons wth dstrbutons for the same system f one assumes a Possonan probablty functon. The dstrbutons obtaned as an exact soluton wthn the SCPF approxmaton are clearly not symmetrc, but exhbt long tals toward zero. Therefore, although the most probable values of the two types of dstrbutons are smlar, nose has a destructve mpact on the system, resultng n a larger probablty of havng a smaller number of protens n the cell than expected based on a Possonan dstrbuton, whose hgher moments are equal to the mean. Therefore, a larger producton rate s needed for one of the states to be favored as a result of nose, than that predcted from a symmetrc probablty dstrbuton. The most probable number of protens n the on-state, f the unbndng from the DNA s slow, s zero, unlke the number predcted by Possonan dstrbutons. The nfluence of nose on proten number fluctuatons brngs the proten-number means down, as can also be seen from Fg. 3 C. Overall, the spread of the probablty dstrbutons s large, and ther characterstcs for small values of the control parameters are dfferent from those predcted by Possonan dstrbutons, let alone by determnstc knetc equatons; therefore the effects of stochastcty may not be neglected. FIGURE 5 Probablty dstrbutons for the gene to be n the on-state (A) and off-state (B) for a gene n the actve state for dfferent values of the adabatcty parameter v ¼ 0.5, 0, 00. X eq ¼ 00, X ad ¼ dx sw ¼ 00. Comparson of probablty dstrbutons obtaned by exactly solvng the steady-state equatons n the SCPF approxmatons wth analogous Possonan dstrbutons (C and D). Symmetrc swtch, X ad ¼ 44, X eq ¼ 000, v ¼ 0.5. The nonzero basal effectve producton rate case The above analyss concerns a swtch wth a zero basal producton rate, so protens were not produced n the off-state. In a number of bologcal systems (Ptashne and Gann, 00) a nonzero basal producton rate exsts and we now turn to consder the effect of ths on a symmetrc swtch. Fg. 6 B shows the dependence of the bfurcaton curves for dfferent values of the effectve basal producton rate g /(k). Values,, when the death rate s larger than the producton rate, show that, for the symmetrc swtch, assumng the effectve producton rate to be zero n the off-state s a reasonable approxmaton. If the on-state has a postve nput to the number of reservor protens present due to g /k., the probablty of the actve gene to be on, even for very large Bophyscal Journal 88()

8 SCPFT of Stochastc Gene Swtches 835 FIGURE 6 Nullclnes for a symmetrc swtch, where protens bnd as dmers, when the effectve base producton rate s g /(k) 6¼ 0. (A) Dependence on the adabatcty parameter v ¼ 0.005, 0.05, 0.5, 5, and 50, compared to the determnstc equatons soluton, g /(k) ¼ 5. (B) Dependence on g /(k) ¼ 0.0, 0., 0.5,.0, 5, 0, and 0, v ¼ 0.5. X eq ¼ 000. on-state effectve producton levels X ad,s,. Hence the offstate contrbutes consderably to the steady-state number of protens. The soluton that corresponds to the more actve of the two states may effectvely be an off-state, snce t has C (), 0.5, although the effectve producton rate n the on-state n the bfurcated regon of parameter space s much larger than n the off-state (for example, the g /(k) ¼ 0 lne n Fg. 6 B). As the effectve basal producton rate ncreases, a larger producton rate n the on-state than for small g /(k). s requred to reach the crtcal number of protens for the bfurcaton to take place, whch s gven by Æn()æ¼ X ad C () ÿ g /k(c () ). For ths reason, even for the determnstc approxmaton at the bfurcaton pont, the two genes must be more probable to be off, as can also be seen for the exact SCPF solutons from the probablty dstrbutons (Fg. 7, B, C, E, and F). Fg. 6 A shows the dependence of the bfurcaton curves on the adabatcty parameter, whch tend to the determnstc case for large v. A closer analyss of the g /k. case, snce the g /k, s analogous to the zero basal producton rate case, whch has already been dscussed, shows that mean propertes of the system are n even better agreement wth the determnstc soluton than the g ¼ 0 case (Fg. 7, A and D). The genes have a nonzero probablty of beng n the off-state, wth the probablty dstrbuton of the off-gene havng a long tal toward hgher proten numbers (Fg. 7, E and F). In the off-state the effectve producton rate g /(k) s small and the nose nput s small, relatve to the large proten numbers present n the system. The small effect of stochastcty results n the observed smlar mean characterstcs. Yet the form of the probablty dstrbutons for the genes to be on before the transton s especally broad, wth a far smaller probablty than those of the off-state (Fg. 7, B, C, E, and F). These clearly show that the two genes are more probable to be n the off-state before the bfurcaton pont. Therefore, although the average observables are smlar for the determnstc and SCPF stochastc solutons, the predcted dstrbutons are unusual. Summary The symmetrc swtch s based on a competton between the accessblty of the repressor ste and the number of repressor protens present n the cell. The bfurcaton s solely a result FIGURE 7 Probablty of genes to be on (A) and mean number of protens of a gven type present n the cell (D) for a symmetrc swtch wth an effectve base producton rate. Evoluton of probablty dstrbutons for the probablty of the gene that wll be actve after the bfurcaton to be on (B) and off (C) and the gene that wll be nactve to be on (E) and off (F) as a functon of the order parameter X ad for the same system. The bfurcaton occurs at X ad ¼ 6, g /(k) ¼ 5, v ¼ 0.5, X eq ¼ 000. Bophyscal Journal 88()

9 836 Walczak et al. of the nonlnearty of the system and ntroducng nose smply affects the regon n parameter space where gven states occur. The proten number fluctuatons have a destructve role n determnng the stablty of the bfurcated soluton; however, fast DNA unbndng rates can compensate for the destablzng effect of proten number fluctuatons. In ths regon the stochastc soluton predcts smlar means to the determnstc case, but the form of the probablty dstrbutons whch depends on a large number of hgher moments s nontrval. It s a result of the nterplay of the DNA bndng and proten degradaton knetcs. THE ASYMMETRIC TOGGLE SWITCH Most swtches found n nature are not symmetrc. For asymmetrc swtches, when protens bnd as dmers, the two genes nteract, resultng n probabltes to be on, dfferent from those mposed purely by the equlbrum between bndng and unbndng. The steady-state soluton s a compromse between the tendency that repressors are unbound from the ntally off-gene (X eq for the forward transton, Xeq for the backward n the followng dscusson) and the effectve producton rate of the ntally on-gene (X ad; forward; X ad ; backward transton), at least for the determnstc case. Ths results n the characterstc S-curve bfurcaton dagram, as presented n, for example, Fg., wth possble forward and backward transtons, hence hysteress. We refer to the transton that occurs wth ncreasng X ad as the forward transton, and that wth decreasng X ad as the backward transton. Snce X ad s a well-defned functon of the probabltes that the genes are on, the smplcty of the determnstc equatons allows for a completely analytc dscusson of the asymmetrc swtch. The more complcated form of the exact SCPF equatons makes ths approach mpossble. However, the determnstc rate soluton offers valuable nsght nto the basc mechansm behnd the transton. The general mechansm By combnng the steady-state equatons of moton for the probabltes of the two genes to be on and notng that, wth a zero basal producton rate ÆnðÞæ ¼ X ad C ðþ; one can derve the form of the determnstc bfurcaton curves as X ad ðc ðþþ ¼ Xeq ðxadc ðþþ X eq C ðþ ÿ (7) as a functon of C (), and 0 X ad ðc ðþþ ¼ Xeq X ad B C ðþ ÿ C ; C ðþ ÿ A X eq as a functon of C (). The transton ponts are determned as the extrema of Eqs. 7 and 8, whch are functons solely of the scaled parameter X ad =X eq and are plotted on the bfurcaton graphs. It s worth notcng that the bfurcaton ponts C () do not depend on the value of X eq ; the parameter descrbng the gene bndng knetcs of the gene that s on ntally. Ths s not true for the exact SCPF soluton, whch cannot be solved analytcally, but the bfurcaton curve has the more complex form of 0! ðc ðþþ ¼ C ðþ X eq v C ðþ v ÿ v C ðþ A v C ðþ ; (9) X ad where C () s a functon of v, X eq ; C (), and X ad: The bfurcaton pont s therefore determned by the proten (X ad ) and DNA (X eq ) characterstcs and mutual nteractons (v )of ther two genes. The determnstc approxmaton therefore greatly smplfes the mathematcal mechansm of the transton. Ths may lead to large errors when studyng more complcated bologcally relevant systems, where one consders asymmetrc swtches wth nonzero basal producton rates and protens are produced n bursts. The case of the nonzero basal producton rate wthn the determnstc approxmaton also cannot be solved analytcally. The general pcture behnd the transton s seen from the determnstc approach. The larger the tendency for protens to be unbound from the DNA, the larger the effectve producton rate X ad must be for the transton, from one gene to be actve, to the other to be actve, to take place nasmuch as repressor protens are less lkely to bnd to the on-gene () at large X eq than at small X eq : However, f one consders a nosy system, t s effectvely harder for protens to stay bound to the ntally off-gene due to the destablzng effect of DNA bndng nose (Fg. 8). For the stochastc system, apart from very low values of the adabatcty parameter (v, 0.) (Fg. ), there s a threshold number of reservor protens that wll cause a rapd transton. If we start wth a small effectve producton rate for one type of proten and ncrease ths rate, keepng the producton rate of the other gene fxed at an ntally hgher value, the protens produced by the gene wth the ntally smaller producton rate repress t gradually and neffectvely, untl they reduce the probablty of the gene to be on to one-half, for the exact SCPF soluton. The number of protens present n the onstate decreases much more rapdly wth the change of X ad whether t be an ncrease for the forward transton or a decrease for the backward transton n the examples presented than the number of protens n the off-state grows (Fg. 0). Hence, the probablty of the ntally actve gene to be on shows a larger senstvty to the change of X ad than does the off-state probablty. Ths leads to a rapd Bophyscal Journal 88()

10 SCPFT of Stochastc Gene Swtches 837 FIGURE 8 Dependence of the probablty of genes to be on n an asymmetrc swtch as a functon of ncreasng parameters of one gene X ad ¼ dxsw n the forward (top) and backward (bottom) transton for dfferent values of X eq : 5, 50, and 500. All other parameters fxed at X eq ¼ 000; v ¼ v ¼ 0:5; and X ad ¼ dxsw ¼ 80: Comparson of solutons of determnstc and exact SCPF equatons. transton of the prevously actve gene to an nactve state (Fg. 9). Such behavor s descrbed by Ptashne (99) and Ptashne and Gann (00) n the l-phage swtch; they pont out ts role as a buffer aganst ordnary fluctuatons n repressor concentraton. The observed system swtches when the represson probablty drops to 50%, as n the solutons of ths model. Our analyss seconds the hypothess of Ptashne and Gann, nasmuch as the determnstc system lacks ths behavor, the transton s rapd, and for certan values of parameters, takes place when the probablty of the ntally on-gene drops to 80% (Fg. 8). The bufferng capabltes of the stochastc system are clearly seen n the long tals toward n ¼ 0 of the probablty dstrbutons of the gene that s swtchng from the on- to the off-state (Fg. 9, A and B). FIGURE 9 Evoluton of the probablty dstrbutons for the two genes to be actve for the forward transton (A and B) and the backward (C and D)as a functon of X ad ¼ dxsw for X eq ¼ 50 wth Xeq ¼ 000; v ¼ v ¼ 0.5; X ad ¼ dxsw ¼ 80 for an asymmetrc swtch. The effect of nose on the bfurcaton mechansm The mean number of protens at the transton pont dffers for the determnstc and exact SCPF soluton (Fg. 0). More repressors are needed to nduce the transton n the determnstc approxmaton than n the stochastc system, snce, due to the form of the nteracton functon for the exact case, F() ¼ Æn()æ (v )/(v C ()) Æn()æ. Æn()æ.A smaller number of protens s therefore needed for the nactve gene to become compettve wth the actve gene. The mechansm of the transton s dfferent from the symmetrc gene case, where a crtcal number of protens needs to be reached. The asymmetrc swtch s based on the competton between the probablty that protens of one knd wll repress the opposng genes and the analogous probablty for the other knd of protens. The represson capablty s governed by X3ÿ ad=xeq ; whch mght be looked upon as the product of the probablty of havng a certan number of repressor protens (3 ) n the cell and the tendency for them to be bound to the opposng gene (). In fact, the transton pont n the determnstc case s purely a functon of such ratos, X3ÿ ad=xeq ¼ f ðx ad =X eq 3ÿÞ: In both the stochastc and determnstc cases, the transton ponts are set by the nteracton functon whch regulates the on- and off-state probabltes of a gven gene Fð3 ÿ Þ=X eq ¼ C ðþ=c ðþ: Incluson of nose n the system effectvely ncreases the nonlnearty of the system, whch results n the already dscussed bufferng capabltes of the system. Stochastcty alters the very smple compettve mechansm seen n the determnstc knetcs to allow for more levels of control of the stablty of the state of the system aganst random fluctuatons. Further comparson of solutons of the determnstc and stochastc equatons leads to the same conclusons as for a symmetrc swtch. As the tendency for protens to be unbound from the DNA grows, the dfference n the crtcal number of reservor protens necessary for the transton to take place ncreases for both approxmatons. The crtcal Bophyscal Journal 88()

11 838 Walczak et al. FIGURE 0 Mean number of protens of each type present n the cell, accordng to exact solutons of the SCPF approxmaton and determnstc knetc rate equatons for an asymmetrc swtch, wth X eq ¼ 000; v ¼ v ¼ 0.5, X ad ¼ dxsw ¼ 80, and X eq ¼ 50 and 500 durng the forward (A) and backward (B) transtons n an asymmetrc swtch. number of protens produced by a gven gene necessary for the transton to take place for both genes s, n most cases (see v dependence dscusson), smaller for the exact solutons of the SCPF equatons and the dfference between the stochastc and determnstc result grows wth both X eq and decreases wth v (Fg. 0). It has a value of 5 for X eq ¼ 500; v ¼ v ¼ 0.5 and for X eq ¼ 500; v ¼ v ¼ 0. Consder the forward transton. The ntally nactve gene s buffered by a cloud of repressor protens. As one ncreases the effectve producton rate of the protens produced by the nactve gene (X ad ), the number of protens that are able to repress gene grow slowly and lnearly ÆnðÞæ ¼ X adc ðþ; where C () ;const, and form a bufferng proteomc cloud around t. In the results presented n the fgures of ths artcle, the tendency that protens are unbound from gene, (X eq ), s smaller than X eq ; so gene s able to produce enough repressors to form a stable bufferng cloud around gene and turn t nto the nactve state at qute modest values of X ad: If X eq, Xeq ; gene produces protens less effectvely, as the probablty of t beng repressed s larger than n the prevous case, and larger values of X ad are needed to produce enough repressors to acheve a hgh effectve probablty of bndng, X ad =X eq : An example of how Xad;crt grows as X eq /Xeq ; s seen by comparng the X ad ;33 for Xeq ¼ 000; Xeq ¼ 50 (Fg. 8) and X ad ;300 for Xeq ¼ 00; Xeq ¼ 50 (Fg. ). Adabatcty parameter dependence The nteracton of the bufferng proteomc cloud wth the DNA can be altered when the rato of the DNA unbndng rate compared to the proten degradaton rate s changed. For small v values the unbndng rate of repressors from the DNA s slower than the destructon of the produced protens. Apart from very small v-values, as long as there s a crtcal number of repressor protens n the bufferng cloud, the offgene s repressed and t responds by turnng on, but only once the ntally on-gene s nearly totally repressed. Large adabatcty parameters result n the effcent formaton of the bufferng proteomc cloud. For the ntally off-gene, a small DNA unbndng rate of the off-gene decreases the effectveness of the bufferng proteomc cloud around t, as the proten number state can reach a steady state before the DNA state does. The hndered DNA reacton to the proten-number state effectvely ncreases the tendency of repressor protens to be unbound from the DNA for a gven X ad : Ths, n turn, decreases the probablty of the ntally on-gene to be on, leadng to rapd swtchng behavor as can be seen for gene n the forward, or gene n the backward, transton for v. 0. n Fg. A. The ntally on-gene reacts to the nteracton functon of the ntally off-gene, for whch F() / Æn()æ / C () Æn()æ n the small v-lmt. Therefore, the nteracton functon s effectvely ncreased for C () 0, leadng to the FIGURE Bfurcaton dagrams for an asymmetrc swtch, presentng X ad ¼ dxsw as a functon of C () (A C), and C () (D F) for dfferent values of the adabatcty parameter: v ¼ v (A,D), v, wth v ¼ 0.00 ¼ const (B,D), v, wth v ¼ 0.00 ¼ const (C,F). X eq ¼ 00; Xeq ¼ 50; and Xad ¼ dx sw ¼ 80: Bophyscal Journal 88()

12 SCPFT of Stochastc Gene Swtches 839 enhanced bufferng. The reacton of the ntally off-gene s unaltered, as for C () F() ¼ Æn()æ Æn()æ ;const, f C () remans close to. However, f v s very small (black dash-dot curve n Fg., A and D), the bufferng proteomc cloud s not gven a chance to form due to a very hgh degradaton rate of protens and gene s smply repressed n a gradual transton. If v s extremely small and v large, the bufferng proteomc cloud around gene cannot form and the probablty of t to be off n the forward transton decreases gradually. A bufferng proteomc cloud exsts around gene, hence the backward transton s remnscent of the determnstc result (Fg., B and E). The most nterestng case s shown n Fg., C and F, where a large v acts as a buffer aganst fluctuatons n the number of protens, whch repress gene. For large producton rates of repressors the probablty of gene to be on for the forward transton decreases faster than n the determnstc soluton; however, the bufferng cloud repressng gene allows gene to reman n the on-state. A bufferng proteomc cloud does not form around gene, and t remans on untl the number of protens produced by gene grows consderably, as the effectve producton rate, X ad ; s ncreased. The effectve producton rate of gene must be very large to sustan a suffcent steady-state number of protens to repress gene to the pont that C (), 0.5, whch leads to swtchng. For the backward transton, the lack of a bufferng proteomc cloud around gene results n destablzng gene for larger X ad effectve producton rates than for large v values. These examples show how certan combnatons of values of adabatcty parameters can lead to a system wth a larger swtchng regon than the determnstc model predcts. Ths property may be useful when engneerng artfcal swtches. If one has a constrant on the producton rates of the genes, one can use repressors wth dfferent bndng affntes to acheve swtchng n the desred regon of parameter space. In ths smple system slow unbndng from the DNA can compensate for the destablzng of the DNA state by proten number fluctuatons. As the probablty of the ntally actve gene to be on gradually decreases, the ntally repressed gene becomes actve only once the probablty of the other gene to be on has fallen below a certan value, a. The susceptblty of the system to proten number fluctuatons may be estmated by the value of a. For small v, whch s stll able to sustan a bufferng proteomc cloud, ths value tends to be 0.5. The ncapablty of the system to form a bufferng proteomc cloud s much stronger f both adabatcty parameters are small, snce the reacton of both genes to the change n the number of protens s hndered (Fg., A and D). DNA state fluctuatons contrbute to effectvely faster proten number fluctuatons, therefore the exact soluton exhbts the very small v-characterstcs, where a bufferng proteomc cloud cannot form, for a slghtly wder range of the adabatcty parameter than one would expect wth a Possonan dstrbuton (results not shown). Combnng these observatons, a swtch works most effectvely f the change of the DNA state compared to the proten number fluctuatons of one gene s suffcently smaller than that of the other gene, to allow for effectve bufferng. The nonzero basal producton rate The asymmetrc swtch, n whch both genes have a nonzero basal effectve producton rate, proves to be susceptble to nose. In Fg., we show the dependence of C (), wth g ()/(k) ¼ g ()/(k) ¼ 5 and C (), wth g ()/(k) ¼ g ()/(k) ¼ 0.5 n the small v lmt. The stochastc solutons converge to the determnstc solutons for large v. If gene s ntally n the on-state, the majorty of protens are produced wth the hgh fxed rate n the on-state, as g () g (). The represson of gene s, n turn, governed by the nteracton functon of gene. If X ad s small the number of protens produced n the on- and off-states by gene are comparable. Snce the number of protens produced by gene grows faster the larger g s, gene gets repressed more effectvely at smaller X ad values. Ths results n a smaller number of repressors produced by gene, and the transton from gene beng on to ts beng off, takes place for smaller X ad ; effectve growth rate values, than for small g. The determnstc soluton s much more nfluenced by the producton of protens n the off-state than the stochastc FIGURE Bfurcaton dagrams as a functon of X ad ¼ dxsw g =ðkþ for C (), wth g ()/(k) ¼ g ()/(k) ¼ 5(A) and C () g ()/(k) ¼ g ()/ (k) ¼ 0.5 (B) for X eq ¼ 5; 50, and 500. Comparson of exact solutons of the SCPF and determnstc knetc equatons for an asymmetrc swtch. v ¼ v ¼ 0.5, X eq ¼ 000; and Xad ¼ dxsw ¼ 80: Bophyscal Journal 88()

13 840 Walczak et al. soluton. In the exact SCPF soluton, slow DNA unbndng rates compared to proten degradaton rates are another means of control of the stablty of the DNA state aganst random proten number fluctuatons. The state of the system s far less nfluenced by the exact proten numbers than n the determnstc soluton. So untl the probablty of a gene to be on s larger than that of beng off, the fracton of protens produced wth a smaller effectve producton rate n the offstate s treated as a random fluctuaton by the system. Once agan, the SCPF system demonstrates ts susceptblty to proten number fluctuatons. The nfluence of the off-state proten producton on the total repressor yeld may also be seen n the fast decrease of C () and ncrease of C () n the forward transton. If g s consderably large, ts effect can also be seen n the stochastc soluton; hence even when gene s n the onstate, t never reaches C () ¼, although gene s totally repressed (Fg. B; results not shown for gene ). The magntude of the probablty of gene to be on for very large effectve producton parameters strongly depends on the tendences of the protens to be unbound from gene. As X eq ncreases, the asymptotc X ad lmt of C () becomes smaller, as t s effectvely harder for repressors to stay bound to the DNA. The gene s more lkely to be n the off-state, whch, however, manages to sustan the necessary number of protens produced by gene to repress gene. As g ncreases, the regon of bstablty grows nto areas of parameter space, n whch the tendency of protens to be unbound, X eq ; s larger than for small g. For small values of X eq ; the number of repressors produced by gene n the off-state s suffcent to repress gene, and one observes a smooth and slow transton n terms of X ad: If g s consderably large, the transton takes place for larger values of X ad n the stochastc soluton than n the determnstc soluton, hence showng the large bufferng regon that the nterplay of DNA and proten number fluctuatons provdes. Ths also results n an effectve smlarty of the determnstc and stochastc solutons. In regons of parameter space, n whch the transton takes place, the determnstc and stochastc solutons dffer, apart from the large v-lmt. Most expermentally observed protens have very small basal producton rates, whch seconds our analyss that t s functonally unfavorable for large basal producton to occur. The dependence on other parameters s analogous to the case wthout a basal producton rate. The regon of bstablty The backward transton, as already dscussed, s analogous to the forward transton. In most cases, the regons of bstablty (Fg. ) n parameter space are reduced n sze by nose. When engneerng artfcal swtches, one may be nterested n makng sure the forward and backward transton takes place for consderably dfferent producton rates. We therefore consder how the regon of bstablty, defned as the dfference n the crtcal effectve producton rate for the forward and backward transton, depends on the parameters of the model. For the determnstc case the regon of bstablty depends on the tendences that protens are unbound from the DNA n a quadratc manner, as can easly be seen from the bfurcaton equatons (Eqs. 7 and 8) and whch s demonstrated n Fg. 3. The SCPF soluton shows the same behavor. For large values of the adabatcty parameter the sze of the regon of bstablty s ndependent of v, as s the form of the bfurcaton curve (Fg. 3). The approach to ths plateau s very rapd and s gven by the rato of polynomals. However, the sze of the regon of bstablty for the v ¼ v never reaches that of the determnstc soluton, as even n the large v-lmt the greater nonlnearty of the nteracton functon F() results n a more complex SCPF curve that does not reduce to determnstc soluton, but X ad ðc ðþþ/ððððc ÿ ðþÿþxeq Þ ÿ Þ=ð4C Þ ðþþ 6¼ X eq ððxadc ðþþ =X eq ÞðCÿ =: (0) ðþÿþ Ths effect s true for both curves, as the presented graphs show C () hysteress and the chosen equatons C (). The same behavor s observed for the case wth a zero and a nonzero basal producton rate. The ncrease wth X eq s slghtly slower n the g 6¼ 0 case as the bfurcaton curve s smaller by jg /k(c f () C n ()) ln(c f ()/C n ())/j. Summary After the transton, the number of protens produced by the now on-gene follows a lnear dependence on X ad, smlarly to FIGURE 3 Regon of C () hysteress for an asymmetrc swtch for the SCPF and determnstc approxmatons as a functon of v ¼ v, wth ¼ 50 ðaþ and Xeq wth v ¼ v ¼ 00 (B). ¼ 00; Xad ¼ dxsw ¼ 80; g /(k) ¼ 0.5. X eq X eq Bophyscal Journal 88()

14 SCPFT of Stochastc Gene Swtches 84 the symmetrc swtch. The number of protens n the cell s ndependent of the DNA dynamcal characterstcs, as those reman constant n that regon of parameter space. The number of protens of the on-gene rapdly falls before the transton takes place. Based on the bfurcaton dagram of Fg. the phase transton s dscontnuous. The regon of parameter space where swtchng may occur may be roughly estmated by the parameters of the genes whch must be compettve, (X ad=xad Þ X eq =Xeq : Ths has a major mplcaton for bologcal systems, such as the l-phage, where many mechansms are used to acheve balance between two genes. The frst-order phase transton, as opposed to the second order present n the symmetrc system, s a result of the breakng of symmetry and s clearly seen n the evoluton of probablty dstrbutons n phase space (Fg. 9). The gene that s on after the transton rapdly ncreases ts probablty of beng on, whereas the off-gene decreases wth a rapd drop n the number of protens t produces. THECASEWHENPROTEINSBIND AS MONOMERS Equatons and can easly be augmented to descrbe the bndng of monomers or hgher order olgomers by changng the form of the bndng term to h n p 3ÿ ; where p ¼ for monomers. The equatons reman solvable for any value of p. Monomers do not make good repressors/actvators The behavor of the system s qute dfferent f we consder the case when protens bnd as monomers. For a symmetrc swtch there s no regon of the parameter space n whch one observes swtchng. The SCPF equatons may be reduced to a sngle quadratc equaton, dx sw C ðþ ðx eq X ad ÿ X sw ÞC ðþÿx eq ¼ 0; () whch has, at most, only one postve soluton. Therefore the probablty of one gene to be n the actve state s always equal to that of the other to be n the actve state, and no swtchng s observed. Equaton, above, s ndependent of v, the adabatcty parameter; therefore, t s solely a consequence of the lack of nonlnearty n the bndng of protens and cannot be nfluenced by very slow DNA unbndng rates. By wrtng down determnstc equatons we can also show that when protens bnd as monomers, swtchng does not occur. A smlar equaton to Eq., also ndependent of v, holds for asymmetrc swtches. It also has one postve soluton, and, therefore, the parameters of the model predetermne the soluton and each gene has a probablty to be on, determned by ts knetc rates. Snce the rates are dfferent for the two genes, the gene wth the larger producton rate wll be n the actve state, repressng the weaker gene (Fg. 4 A). In naturally occurrng bologcal swtches and those developed expermentally, protens bnd as dmers, or hgher order multmers (Ptashne, 99). We see cooperatvty contrbutes to mprovng the effcency of a swtch. A swtch controlled by monomers s shown to react neffectvely to changes n the repressor concentraton, just as n the case of the asymmetrc swtch n our model dscussed above. Monomers do not have the ablty to stablze a broken symmetry state; therefore, the soluton s fragle to knetc rates and neffcent. Effectvely monomers do not make good repressors/actvators. Ptashne and Gann (00) explan the cooperatvty process between two monomers by clamng that one monomer bound to the DNA ncreases the local concentraton of protens around the bndng ste through weak proten-proten nteracton, thus causng the second to bnd cooperatvely. Our model lacks spatal dependence, whch therefore shows that ths effect need not be thought of as due to changes n local concentraton, but actually s requred by the nsuffcent nonlnearty for monomers, whch cannot produce bstablty. Bmodal probablty dstrbuton Although the probabltes of the two genes to be on are equal for the whole regon of parameter space, and the mean number of both types of protens n the cell s the same as n the determnstc case, the probablty dstrbutons are bmodal when the DNA unbndng rates are slower than the proten number fluctuatons (Fg. 4, B and C). The mechansm of ths FIGURE 4 (A) Probablty of genes n an asymmetrc swtch to be actve when protens bnd as monomers, for dfferent values of X eq : X ad ¼ dxsw ¼ 80: Probablty dstrbutons for the gene to be n the onstate (B) and off-state (C) for a gene n the actve state for dfferent values of the adabatcty parameter v ¼ 0.5, 5, and 00, when protens bnd as monomers to a symmetrc swtch. X ad ¼ dx sw ¼ 50, X eq ¼ 000. Bophyscal Journal 88()

15 84 Walczak et al. small v-behavor has already been dscussed n the example of the symmetrc swtch, when protens bnd as dmers. Ths s analogous to the case when DNA fluctuatons nduce a probablty dstrbuton wth two peaks for the sngle gene wth an external nducer (Cook et al., 998). In fact, the SCPF approxmaton has reduced ths two-gene system to an effectve one-gene system wth an external nducer. A bmodal dstrbuton n the small v-case s also observed for the asymmetrc swtch, when protens bnd as monomers. THECASEWHENPROTEINSBINDASHIGHER ORDER OLIGOMERS Swtches n whch effector protens bnd as hgher order olgomers are omnpresent n nature and have been realzed expermentally n artfcal swtches (McLure and Lee, 998). We consdered the bndng of trmers ðh ðn 3ÿ Þ¼h n 3 3ÿ Þ and tetramers ðh ðn 3ÿ Þ¼h n 4 3ÿÞ n symmetrc swtches. The equatons of moton have the same form as before, but the nteracton functon F() accounts for the hgher moments. For protens bndng as k th order olgomers, t has the form FðÞ ¼C ðþ Æn k ðþæc ðþ Æn k ðþæ: As shown when dscussng the dmer bndng swtch, the k th order moments have a smple form n the creaton operator representaton. The general mechansm From Fg. 5 one notes that, for the system to act as a bstable swtch, a consderably smaller number of reservor protens s needed than n the case of the dmer bndng swtch. As the multmercty number grows, the area of bstablty of the swtch n parameter space grows. Snce we assumed only one type of proten repressed a gven gene, bndng of hgher order multmers s an effectve model of cooperatvty. Therefore, we expect the system to have a larger regon of bstablty, the hgher the order of the bndng multmer. The evoluton of the system n parameter space when trmers bnd s qualtatvely smlar to the dmer bndng scenaro (Fg. 6, B andc). Fast DNA unbndng rates stablze the system and the bfurcaton takes place for smaller effectve producton rates, for large v than for small v (Fg. 6, A and D). The crtcal number of protens necessary for the bfurcaton to take place s ndependent of the adabatcty parameter and decreases wth multmercty: Ænæ c ¼ 3 for dmers bndng, Ænæ c ¼ 8 for trmers bndng, and Ænæ c ¼ 4 for tetramers bndng. Ths along wth the narrow probablty dstrbutons (Fg. 6, E and F), small v-dependence when tetramers bnd (Fg. 5) shows that one bndng event determnes the result, hence DNA bndng rates do not play a role. Once there are Ænæ c protens of a gven type n the cell, a tetramer repressor wll bnd and stay bound. In the determnstc case the probablty of the genes needs to fall to (p )/p, where p s the order of multmerzaton of the repressor, for the bfurcaton to take place. That, along wth the need for the number of repressors to be comparable wth the tendency for protens to be unbound from the DNA, sets the crtcal number of protens necessary for the bfurcaton. Hence, the bfurcaton occurs when both genes are more probable to be on than off, for both tetramers and trmers. Therefore, for the tetramer system, a large bufferng proteomc cloud s not needed to stablze the DNA bndng state of the swtch, and the characterstcs of the system are practcally ndependent of the adabatcty parameter. Tetramer bndng results n nearly determnstc characterstcs In naturally occurrng systems the producton of the crtcal number of protens s slowed down by relatvely hgh multmerzaton rates and spatal dependence arsng from the need of a large number of partcles to dffuse together. These elements, whch we neglect n our smple model, consttute what mght be called the cost of multmerzaton. Ths analyss also explans why most repressors and actvators bnd as dmers and tetramers, not trmers or pentamers. The effect of trmers bndng s not dfferent from that of dmers: a bufferng proteomc cloud needs to be formed; the state of the system s qute nfluenced by nose; and the swtchng regon (regon n X ad parameter space from the bfurcaton pont to DC. 0.9) s qute large. Yet n a real system there s an effectve cost of trmerzaton: the energy of trmer formaton and a need for the dffuson of partcles. For tetramers the effect of stochastcty becomes neglgble. Effectvely one tetramer s suffcent for the bfurcaton to take place. The bndng of tetramer repressors may be thought of as a mechansm for ncreasng the determnstc nature of the swtch. FIGURE 5 Phase dagram for the SCPF approxmaton for a sngle symmetrc swtch to whch protens bnd as trmers (A) and tetramers (B), wth X eq ¼ 000. Contour lnes mark values of DC. Bophyscal Journal 88()

16 SCPFT of Stochastc Gene Swtches 843 FIGURE 6 Mean number of protens n the cell, for each type when protens bnd as trmers (A) and tetramers (D), v ¼ 0.5, 0, symmetrc swtch. The evoluton of the probablty dstrbuton for the probablty of the gene that wll be actve and nactve after the bfurcaton to be on as a functon of X ad for a swtch when protens bnd as trmers (B and C) and tetramers (E and F). X eq ¼ 000, v ¼ 0.5. Bndng of hgher order olgomers as a compettve mechansm Ths analyss, although t neglects some mportant features, allows for a more quanttatve formulaton of cooperatvty. Snce most bologcal swtches are asymmetrc, cooperatvty s also used as a means of makng genes wth smaller chemcal rates more compettve. Tetramer bndng seems to have a dfferent role than that of lower order multmers. It may be used by genes that need to react to very small concentratons of protens; for example, they turn on degradaton mechansms when even a small number of toxc molecules s present. Or they may act as an extra mechansm stablzng the exstent state of a gene, as seems to be the case for the ci gene of the l-phage. It seems that tetramers are used ether n a stablzng role or as a drastc, all-or-none response to the proten dstrbutons n the system. Ths formulaton of the problem s naturally oversmplfed, but t allows for general observatons. parameter X ad compared to when protens are produced separately. Therefore, even n the large v-lmt, nose resultng from large proten number fluctuatons plays a role n defnng the regon of stablty of the swtch, as the crteron of large X ad s not reached. The number of protens n the cell when the bfurcaton occurs s determned by the tendency that protens are unbound from the DNA and does not change when protens are produced n bursts. For the rates dscussed n Fg. 7, the crtcal mean number of protens present n the cell at whch the bfurcaton occurs s n c ¼ 0 ¼ X eq ¼ 00 ½. If protens are produced n bursts of N ¼ 0, as n the left-hand fgures, ths value of n c s acheved when X ad. (that s, protens must get produced at a hgher rate than they are destroyed, to be able to sustan the steady-state number of 0 protens n the cell). In the fgures THE CASE WHEN PROTEINS ARE PRODUCED IN BURSTS Many protens n bologcal systems for example, the Cro proten n l-phage are produced n bursts of N of the order of tens. We consder a symmetrc swtch, where protens bnd as dmers and are produced n bursts of N. The dervaton of the moment equatons for ths case s presented n Appendx B. The general mechansm We dscuss the effect of burstng phenomena based on the example of a symmetrc toggle swtch n whch protens bnd as dmers, as that can offer the most nsght, when compared to prevous results. In ths case, swtchng takes place for much smaller values of the effectve producton rate FIGURE 7 Probablty that gene s on when protens are produced n bursts of N ¼ 0 (A) and N ¼ 00 (B). Mean number of protens of each type present n the cell when protens are produced n bursts of N ¼ 0 (C) and N ¼ 00 (D). Symmetrc swtch protens bnd as dmers, X eq ¼ 00, v ¼ 00. Comparson of determnstc and stochastc solutons. Bophyscal Journal 88()

17 844 Walczak et al. on the rght-hand sde of Fg. 7, protens are produced n bursts of N ¼ 00. In ths case even when the degradaton rate s larger than the producton rate, the crtcal steady-state number of protens necessary for the bfurcaton to take place can be reached and a bstable swtch s possble. A bstable swtch can exst f the degradaton rate exceeds the producton rate even for burst szes present n bology. For X eq ¼ 00, the order of the tendences for protens to be unbound from the DNA n the l-phage, the value of N for whch X ad c, s smaller than N ¼ 0, the burst sze for Cro protens n the l-phage. X ad at the crtcal pont decreases as a functon of N (Fg. 8 A) and depends on the tendency that protens are unbound from the DNA X eq (Fg. 8 B) and the adabatcty parameter, v (Fg. 9). If protens are produced ndvdually, the span of the nonadabatc regme s clear from Fg. 9. It corresponds to v,. The bfurcaton curves show small dscrepances for larger values of the adabatcty parameter. However, for larger burst szes, there s a contnuous change n the form of the bfurcaton curves wth v. All of the solutons dffer substantally from the determnstc treatment, as shown n Fg. 7 A. FIGURE 8 Bfurcaton curves as a functon of X ad ¼ dx sw, v ¼ 00 for dfferent burst sze values N ¼,, 5, 0, 50, and 00, wth X eq ¼ 00 (A) and for protens produced n bursts of N ¼ 00 (B) for dfferent values of X eq ¼, 0, 00, and 000. The nfluence of the adabatcty parameter on the bfurcaton mechansm Contrary to the N ¼ case, the effectve producton rate at the bfurcaton pont Xc ad grows wth the ncrease of the adabatcty parameter, for consderably large burst szes, as n the N ¼ 00 example n Fg. 9. In ths case each gene produces a large number of repressors at a tme. The bfurcaton takes place n aregonwthx ad,, whch corresponds to very small effectve producton rates, whch denote very large death rates. Therefore, n the regon of parameter space before the bfurcaton takes place, both genes reman repressed (C (), 0.5) n the steady state, as opposed to the prevously dscussed stuatons, n whch both genes had equal probabltes to be actve (C (). 0.5). For large N bursts, the bfurcaton takes place when one of the genes becomes unrepressed n the steady state. That s, when the repressor cloud bufferng the DNA becomes destablzed, not when the cloud forms as n the smaller N examples. For large N bursts, f the rate of unbndng from the DNA s fast compared to the proten degradaton rate, larger effectve producton rates are needed for the bufferng proteomc cloud to stablze the DNA state than for small v (Fg. 9 C). The larger X ad s, the more repressor molecules are present n the system, whch corresponds to the larger protennumber fluctuatons, that are necessary for one of the genes to become unrepressed. For slower DNA unbndng rates, the bufferng proteomc cloud s smaller, snce the proten number reaches a steady state before the DNA state does. Therefore the bufferng proteomc cloud s destablzed at smaller values of X ad. Hence, n the case of small v the un-repressng bfurcaton takes place for smaller effectve producton rates than for large v. However, f the unbndng rate from the DNA s very small (.e., v, 0.0), Xc ad as a functon of the adabatcty parameter grows agan as ths corresponds to effectvely large death rates that need very hgh producton rates to sustan a proteomc cloud bufferng the DNA. If the effectve producton rate s too small n ths case, the steady-state number of protens s too small to form the bufferng proteomc cloud, although the burst sze s enormous. In the very small v-lmt the bfurcaton cloud needs to be formed for the bfurcaton to be possble, as n the mechansm present n the small N case. The value of X ad at the bfurcaton pont n both the large and small v-lmts s strongly governed by proten and DNA bndng-state fluctuatons n the system. For ths reason, the determnstc soluton fals. It assumes the ncorrect mechansm, n whch the bfurcaton s a result of repressng one of the genes. Such a scenaro s possble f the death rate of protens s slow enough to allow for the exstence of Æn() c æ repressor molecules n the system at very small producton rates (C () bff,kn ¼ 0.5) (Fg. 7, A and B). One can see that the order of takng the adabatc lmts n the steady state for protens produced n large bursts s subtle and depends strongly on the parameters of the system, as the bfurcaton s governed manly by relatve proten and DNA fluctuatons, both of whch are very large. Furthermore, the determnstc soluton s closer to the small v-lmt, whch Bophyscal Journal 88()

18 SCPFT of Stochastc Gene Swtches 845 FIGURE 9 Bfurcaton curves for protens produced separately N ¼ (A), n bursts of N ¼ 0 (B) and N ¼ 00 (C) as a functon of X ad ¼ dx sw for dfferent values of the adabatcty parameter. corresponds to slow DNA unbndng rates compared to proten number fluctuatons. Determnstc results may therefore be msleadng n the burstng stuaton, even for large v. The steady state comes about as a result of dfferent mechansms, dependng on the burst number N and the order of reachng the steady state by the proten, and DNA bndng ste dynamcs changes dependng on v. For small burst szes, slower DNA unbndng rates requre larger effectve producton rates to reach the steady-state number of protens necessary to form the bufferng proteomc cloud than for large N. For larger burst szes, faster DNA unbndng rates destablze the bufferng cloud of protens for smaller effectve producton rates than n the small N case (Fg 8 A). Consequences of bfurcaton at smaller X ad values The dvergence from the determnstc soluton at the bfurcaton pont ncreases wth the burst sze, as s expected due to the enormous nose effect due to large N, on a system wth a constant, and ndependent of the burst sze number of protens at the bfurcaton pont. As already noted, the number of protens n a cell s n the range of tens to hundreds, even f they are produced n bursts. Ths number s reached for smaller effectve producton rates for larger burst szes than for small N-values. Therefore systems where protens are produced n bursts dsplay smaller values of X ad and are more susceptble to nose f the number of protens n the cell s to be of the order whch s observed expermentally. Furthermore, the nosy-burst systems, even for very large values of X ad,do not converge as closely to the determnstc soluton as they do for the sngle proten producton example. Ths can be seen from the form of the steady-state moment equatons. The nteracton functon F() for the N ¼ case n the lmt of large v and X ad converges to F() / Æn()æ Æn()æ, whereas the determnstc soluton corresponds to F() ¼ Æn()æ. Therefore for large mean values of protens the two are equal. However, n the case when N., F() / Æn()æ ( (N )/) Æn()æ, whch requres N Æn()æ for the effect of burstng to be neglgble at very large N. The values of the effectve producton rate that correspond to values of the protens seen expermentally seem to be small. Therefore we can say that, effectvely, the role of burstng s to enable the exstence of a bstable soluton at lower effectve producton rates, whch determnes a regon of parameter space that has been prevously unstuded. In ths regon, one cannot make the adabatc assumpton that the change n the DNA state can be ntegrated-out due to a separaton of tmescales. That assumpton leads to erroneous results, predctng a regon of bstablty where explct treatment of both tmescales suggests monostablty. Furthermore, for very large N, the regon of bstablty decreases wth the adabatcty parameter, makng the dsagreement of the stochastc solutons wth those of the determnstc rate equatons larger. The adabatc approxmaton and the full solutons converge only n the regme of large v and X ad, the second of whch s never fulflled at the bfurcaton pont or for bologcal concentraton for systems n whch protens are produced n large bursts. Dependence on the DNA bndng coeffcent Just as ncreasng the burst sze, decreasng the tendency for protens to not be bound to the DNA results n a dfferent swtchng mechansm. The probablty of the genes to be on falls to far smaller values than the 0.5 of the N ¼ case. If the burst sze s large, both genes have a very low probablty of beng on before the crtcal number of protens necessary for bfurcaton s acheved. The same effect s observed f protens are more lkely to bnd to the DNA (small X eq ) (Fg. 8 B). When the genes are more probable to bnd a repressor and successful unbndng events are rare, earler bfurcatons n terms of X ad result. As X eq ncreases, the probablty of the genes to be on at the bfurcaton pont ncreases, snce repressors have a hgher tendency of unbndng. For very hgh values of the adabatcty parameter, correspondng to hgh unbndng rates from the DNA bndng ste, the stable soluton that corresponds to the off-state and the unstable state merge and the system s monostable agan, wth only the on-state present. Ths lmt s also reached by keepng X ad fxed but takng the burst sze N / N. Probablty dstrbutons In the case of the rates used n Fg. 0, n c ¼ 3 s the same as for N ¼, but we note a 0-fold decrease n Xc ad compared to when protens are produced separately. When protens are produced n bursts, the probablty dstrbutons have tals toward larger n, as opposed to the dstrbutons for ndvdual proten producton. The mean number of protens n the system for gven states of the swtch s smlar to that of the N ¼ case; however, the dstrbutons wth bursts are much Bophyscal Journal 88()

19 846 Walczak et al. FIGURE 0 The evoluton of the probablty dstrbuton of the gene that s actve after the bfurcaton, to be on (A) and off (B) and the gene that s nactve to be on (C) and off (D) as a functon of X ad for a swtch when protens are produced n bursts of N ¼ 0, X eq ¼ 000, v ¼ 00. Bfurcaton pont at X ad ¼ dx sw ¼ 35. broader, as could be expected. In ths case even very fast unbndng rates from the DNA cannot correct for the enormous proten number fluctuatons and one must explctly keep track of the change of the DNA bndng state. A system n whch protens are produced n bursts s very nosy, especally compared to the nearly determnstc case of protens bndng as tetramers. Nonzero basal effectve producton rate If there s a nonzero basal producton rate, the dfference between the determnstc and stochastc solutons s also qualtatve even for relatvely small burst szes. In ths case, protens are also produced n the off-state so that the number of repressors produced by the off-gene after the bfurcaton s nonzero, but equal to the burst sze N, snce ÆnðÞæ ¼ NðX ad dx sw ðc ðþÿþþ/ CðÞ/0 Ng =k: Ths number s equal for both the stochastc and determnstc solutons and s equal to 0 n the examples presented n Fg., C and D. So producton n bursts mantans a hgh level of repressor protens, even for very small g /k values, f the burst sze s large. When usng expermental data one must be very careful to consder the burst sze when assumng the basal producton level s zero. Furthermore, the value of the nteracton functon of the gene n the off-state (C () ;0) for the stochastc case s much larger than for the determnstc case, due to the multplcaton of Æn()æ, whch gves F() / Æn()æ ( k/(g )) Ng /(k), for large v, the effect of whch s shown n Fg., A and B. The number of repressor protens produced by the off-gene decreases as g / 0, as expected, and the probablty of the on-gene to be actve tends to be. The dependence of the effectve producton rate at whch the bfurcaton occurs on the adabatcty parameter s analogous to that of the case where g ¼ 0. FIGURE Probablty that gene s on when protens are produced n bursts of N ¼ 0 wth a basal effectve producton rate g /(k) ¼ 0.5 (A) and N ¼ 00, wth a basal effectve producton rate g /(k) ¼ 0.05 (B). Mean number of protens produced by each gene n the two cases (C and D). Symmetrc swtch; protens bnd as dmers, X eq ¼ 00, v ¼ 00. Comparson of determnstc and stochastc solutons. The probablty dstrbutons for the gene that s actve after the bfurcaton n the on- and off-states takes place are presented n Fg., A and B, for large unbndng rates from the DNA; and Fg., C and D, for small unbndng rates from the DNA. They exhbt maxma around X ad for the onstate and g /(k) for the off-state and dsplay behavor analogous to that of protens produced separately, apart from the dfferent curvature of the slopes for n, N and n. N. For small v-values the proten numbers reach a steady state before the DNA states, hence we observe bmodal probablty dstrbutons. The mechansm of competton n ths FIGURE The evoluton of the probablty dstrbuton of the gene that s on after the bfurcaton, to be on for v ¼ 00 (A and B) and v ¼ 0.5 (C) and off (D) as a functon of X ad for a swtch when protens are produced n bursts of N ¼ 0 wth a basal effectve producton rate g /(k) ¼ 0.5, X eq ¼ 00. Bfurcaton ponts at X ad ¼ 8(v ¼ 00) and X ad ¼ 6(v ¼ 0.5). Bophyscal Journal 88()

20 SCPFT of Stochastc Gene Swtches 847 nosy burst system s dfferent than n the sngle proten producton case. If the gene s n the on-state, probablty states wth hgher n-values are strongly occuped and there s hardly any probablty flux nto the lower n-states. In the offstate, however, a flux pushes the system nto the lower n-states, essentally trappng t there, hence the dfference n the slopes, as can be seen n Fg., C and D. Thssalso true for the g ¼ 0 system when protens are produced n bursts. Lmtatons of the SCPF treatment The examples presented above cover a large class of two gene swtches, all of whch are exactly solvable wthn the SCPF approxmaton. An exact soluton may be obtaned wthn ths approxmaton for systems of genetc networks and swtchng cascades. However, the SCPF approxmaton does not allow for an exact analytcal soluton of all systems. If we try to model one of the smplest natural systems where regulaton s acheved by means of a swtch,.e., the l-swtch, we encounter a problem. The genes n the l-swtch, apart from havng a toggle-lke regulaton, also exhbt autoregulaton that s, ci protens can bnd to OR3, repressng the ci gene, and the Cro protens can bnd to OR or OR, enablng the RNA polymerase from transcrbng the Cro gene (Ptashne, 99; Ptashne and Gann, 00). If we expand the master equaton (Eq. ) to account for self-regulaton we add a h n p bndng term to the P j (n ) equatons. Therefore the k th moment equaton wll dsplay a dependence on the k p th moment and the set of equatons wll not exhbt closure. One can fnd the probablty dstrbuton for a sngle self-regulatng sngle gene. However, f we consder a system lke the l-phage, where self-regulaton s also combned wth regulaton by another gene, the problem s no longer solvable exactly and demands a cutoff of the herarchy or other such approxmatons. We can nevertheless treat these systems usng the varatonal method, as proposed by Sasa and Wolynes (003). The fact that self-regulaton renders the system ncompletely solvable wthn the SCPF approxmaton s not surprsng, snce t corresponds to the exact soluton for such a system. Gene s nfluenced only by the number of protens t produces. It s ndependent of the state of the other gene. Therefore, as one would expect, the full soluton should depend on all moments of the dstrbuton of gene. However, for systems such as the l-phage, we can treat all ntergene regulaton effects exactly and truncate the self-regulaton equaton at the hghest order of the ntergene nteracton. CONCLUSIONS The self-consstent proteomc feld approxmaton for stochastc swtches reproduces many ntutve notons about ther behavor. It proves to be a very powerful tool that allows for the consderaton of all but one of the basc buldng blocks of more general swtches and networks. A swtch wth a selfrepressng/actvatng gene cannot be solved exactly wthn the SCPF approxmaton, snce, n ths case, the approxmaton s equvalent to the full soluton. Therefore the probablty dstrbuton s determned by an nfnte number of moments. The probablty dstrbutons obtaned for the systems consdered n ths artcle are not symmetrc and exhbt long tals. Ths antcpates problems for usng the varatonal prncple for fndng probablty dstrbutons when one accounts for correlatons between the two states. The possblty to expand ths method to consder networks and cascades wll allow for a more realstc treatment of complex systems wth emergent behavor at low computatonal costs. One can account for the mrna step n the system by addng a determnstc step whch, usng a determnstc knetc rate equaton, translates the number of mrna molecules nto protens produced n bursts. Ths s a vald procedure, as separately shown by Thatta and van Oudenaarden (00) and Swan et al. (00); transcrpton nose s just amplfed n the translaton process. Therefore treatng the mrna step determnstcally smply ntroduces another constant nto the dscussed case of protens produced n bursts. Therefore the presented treatment of protens produced n bursts wth a modfed effectve producton rate s a smple model of ncludng mrna n the system. Of course, the effect of mrna s much more complcated, as t also ntroduces, for example, tme delay between bndng and producton. Ths model n the present state neglects these effects. Our analyss of a large class of swtches shows how partcular elements contrbute to the emergent behavor of functonng swtches. Comparson of the stochastc and determnstc treatments of a sngle gene swtch shows convergence n the regon of fast rates of unbndng from the DNA compared to proten number fluctuatons and large effectve producton rates. For symmetrc swtches when protens are produced separately, the two solutons converge after the bfurcaton, but often dffer when defnng the regon of parameter space where the bfurcaton occurs. The agreement between the determnstc and stochastc solutons s especally good for symmetrc swtches, wth N ¼ and a nonzero basal producton rate. However, even though the mean repressor proten levels n the cell are smlar n both approxmatons, the probablty dstrbutons are broad and far from Possonan (.e., they are not completely characterzed by these means). If the adabatcty parameter s small (v, ), the proten-number state wll reach a steady state before the DNA bndng state, and we observe a bmodal probablty dstrbuton. For the symmetrc swtch, nose has a destructve effect on the regon of bstablty. Increasng the adabatcty parameter facltates the formaton of a bufferng proteomc cloud around a gene, whch leads to represson at lower effectve producton rates than for small v. As was already mentoned, the symmetrc swtch s hard to desgn and buld expermentally. The asymmetrc swtch, whch s the expermental model system, s much more susceptble to nose than the symmetrc swtch and stochastcty has not only the destructve effect on the regon of Bophyscal Journal 88()

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