Institut Non Linéaire de Nicemarcos/nicephysicsdays.org/... · id en tﬁ. I , ou r m l sap c f h actual kinetic schemes characterizing genetic systems. MacLeod (23) has recently

Université de Nice Sophia Antipolis - CNRS

Institut Non Linéaire de Nice

Jacques-Alexandre SEPULCHRE

Un grain de sel de la physique non linéaire dans la soupe moléculaire d’une cellule vivante

Université de Nice Sophia Antipolis - CNRS I N L N Université de Nice Sophia Antipolis - CNRS I N L N

The molecular soup...





DNA

Université de Nice Sophia Antipolis - CNRS I N L N

Plan

1.  Some principles of the modeling of intracellular biochemical kine:cs by coupled ODEs

2.  Op:mal response of gene:c expression under periodic s:mula:ons

3.  Onset of autonomous periodic oscilla:ons in signaling cascades


1. Some principles of the modeling…



MNK CREB

MNK +CREB! "# C! "# MNK +CREB*

P +CREB*! "# C*! "# P +CREB

Enzymatic activation (e.g. phosphorylation)

a) Kinetics associated with the law of mass action ( 3 var.)

d[CREB]dt

= ...

d[C]dt

= ..., d[C*]dt

= ...

b) Kinetics associated with the Michaelis-Menten approximation ( 1 var.)

x = [CREB*][CREB]total

, dxdt= k[MNK ]total

(1! x)K + (1! x)

! k '[P]totalx

K '+ x

Rem : an enzymatic activation involves often a double phosphorylation



CREB

n CREB* +DNA+ polymerase+ nucleotids!"! n CREB* +DNA+ polymerase+mRNAmRNA+ ribosomes+ amino# acids!"! mRNA+ ribosomes+ protein

Activation of a gene by a transcription factor

Phenomenological kinetics with a Hill function : dAdt

= !xn

" n + xn!#A

DNA Protein A

Approximation : xn

! n + xn! H (x "! )

0 0.5 1 1.5 2

0

0.5

1

x/

Hill

func

tion xn

! n + xn



CREB

n CREB* +DNA+ polymerase+ nucleotids!"! n CREB* +DNA+ polymerase+mRNAmRNA+ ribosomes+ amino# acids!"! mRNA+ ribosomes+ protein

Activation of a gene by a transcription factor

Phenomenological kinetics with a Hill function : dAdt

= !xn

" n + xn!#A

DNA Protein A

CREB DNA

Protein B

dBdt

= !" n

" n + xn!#B

0 0.5 1 1.5 2

0

0.5

1

x/

Hill

func

tion

Repression of a gene by a transcription factor:

! n

! n + xn


Plan

1.  Some principes of the modeling of intracellular biochemical kine:cs by coupled ODEs




2. Op:mal response of gene:c expression under periodic s:mula:on Context : periodic stimulations associated with the formation of memory … in the drosophila.

CREB1 DNA proteins for synapses… (memory) CREB2

5-‐HT

Smolen et al. (1998)

Yin, Tully (1995)



5-‐HT

lase gene along with suppression of cAMP-inducedtranscription by the TF YY1 (40)]. Such architecturesmay be expected to provide alternative mechanisms forgenerating dynamic phenomena such as multistabilityand oscillations.

It is unlikely that all transcriptionally regulatedgenes will exhibit the behaviors illustrated by ourmodels. However, the diversity ofTFs and their interac-tions suggest that behaviors such as these will beidentified. Indeed, our models are simplifications of theactual kinetic schemes characterizing genetic systems.MacLeod (23) has recently proposed that epigenetic,heritable changes in gene expression following expo-sure to chemicals might play a role in carcinogenesis.Such changes would correspond dynamically to pertur-bations of genetic regulatory systems from one steadystate to another.An outstanding major issue for future investigation

will be to determine whether the parameters of specificgenetic systems in vivo are permissive for specific typesof dynamic behavior. Experiments to help determinethis might include monitoring transcription of trans-fected reporter gene constructs, with defined promoterssubject to regulation by TFs, in cultured cells duringspecific patterns of hormone or neurotransmitter appli-cation. A prolactin promoter-luciferase gene constructhas been used to provide real-time quantification ofpromoter activity in cultured secretory cells (4). An-other relevant technique is polymerase chain reactionamplification and quantitation of specific mRNAs fromtissue samples (43); however, this technique does notresolve dynamics at the single cell level. We believethat, as the dynamic behaviors of gene networks areexplored empirically, the present work can provide aconceptual framework for the analysis and interpreta-tion of such experiments.

APPENDIX

Details of equations and parameters for simulations withthe model of Fig. 5 follow. We make some assumptionsconsistent with experimental results concerning the dynam-ics of a specific system with competing TFs that is thought tomediate the initial steps in LTM formation. In particular, wehave been guided by analyses of competitive interactions ofthe transcriptional activator CREB and related repressors fortheir target DNA sequences, i.e., CRE sites. It is assumedthat 1) total amounts of TF-A and TF-R remain constant (2),2) phosphorylation does not affect binding of TF-A to CREsites (37), and 3) singly phosphorylated TF-A dimers haveone-half the activity of doubly phosphorylated ones, withunphosphorylated dimers inactive (22).

We denote the concentration of free CRE sites by Gfree. Forbrevity, in equations the activator TF-A is denoted by A andthe repressor TF-R by R. [AP] is used to denote the concentra-tion of phosphorylated TF-A sites, and [AA] is used to denotethe concentration of free TF-A dimers. [AAB] denotes theconcentration of TF-A dimer bound to DNA. Atot denotes thetotal concentration of TF-A dimers. The total concentration ofrepressor dimers is Rtot. [RR] is the concentration of free,unphosphorylated TF-R dimers. RRPand RRPPdenote singleor double phosphorylation; RRPPB denotes bound, doublyphosphorylated R dimers. Because phosphorylation of TF-A isindependent of binding to DNA, the rate of change of phos-

phorylated TF-A can be described by a single differentialequation for the concentration of phosphorylated TF-A sites

d[AP]

dt

� kA,f (t)2Atot � [AP]

2Atot � [AP] � KA,ph� kA,b

[AP]

[AP] � KA,deph

(A1)

where KA,deph is the Michaelis constant for TF-Adephosphory-lation and kA,b is the backward rate constant for TF-Aphosphorylation.

A single differential equation describes the rates of changeof free and bound TF-A dimers because of conservation oftotal dimers. The association (forward) and dissociation (back-ward) rate constants are k1,f and k1,b, respectively

d[AA]

dt� �k1,f [AA] Gfree � k1,b [AAB] (A2)

where [AAB] �Atot � [AA].Separate equations are needed for the rates of change of

each species of TF-R because binding and phosphorylationare not independent. Total phosphorylation and dephosphory-lation rates are first expressed in terms of site concentrations,and then, in the differential equations, fractions of these totalrates appropriate to each molecular species are used

rR,ph � kR,f(t)2[RR] � [RRP]

2[RR] � [RRP] � KR,ph(A3)

rR,deph � kR,b

2[RRPP]�[RRP]

2[RRPP]�[RRP]�KR,deph

(A4)

d[RR]

dt� � rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

(A5)

d[RRP]

dt� rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]

(A6)

d[RRPP]

dt� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]

� k2,f [RRPP] Gfree

� k2,b [RRPPB]

(A7)

d[RRPPB]

dt� k2,f [RRPP] Gfree � k2,b [RRPPB] (A8)

C540 DYNAMICS OF GENETIC REGULATORY SYSTEMS

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It is unlikely that all transcriptionally regulatedgenes will exhibit the behaviors illustrated by ourmodels. However, the diversity ofTFs and their interac-tions suggest that behaviors such as these will beidentified. Indeed, our models are simplifications of theactual kinetic schemes characterizing genetic systems.MacLeod (23) has recently proposed that epigenetic,heritable changes in gene expression following expo-sure to chemicals might play a role in carcinogenesis.Such changes would correspond dynamically to pertur-bations of genetic regulatory systems from one steadystate to another.

An outstanding major issue for future investigationwill be to determine whether the parameters of specificgenetic systems in vivo are permissive for specific typesof dynamic behavior. Experiments to help determinethis might include monitoring transcription of trans-fected reporter gene constructs, with defined promoterssubject to regulation by TFs, in cultured cells duringspecific patterns of hormone or neurotransmitter appli-cation. A prolactin promoter-luciferase gene constructhas been used to provide real-time quantification ofpromoter activity in cultured secretory cells (4). An-other relevant technique is polymerase chain reactionamplification and quantitation of specific mRNAs fromtissue samples (43); however, this technique does notresolve dynamics at the single cell level. We believethat, as the dynamic behaviors of gene networks areexplored empirically, the present work can provide aconceptual framework for the analysis and interpreta-tion of such experiments.

APPENDIX




d[AP]

dt



[AP]

[AP] � KA,deph

(A1)



d[AA]






rR,deph � kR,b

2[RRPP]�[RRP]


(A4)

d[RR]

dt� � rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

(A5)

d[RRP]

dt� rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]

(A6)

d[RRPP]

dt� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]


� k2,b [RRPPB]

(A7)

d[RRPPB]




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It is unlikely that all transcriptionally regulatedgenes will exhibit the behaviors illustrated by ourmodels. However, the diversity ofTFs and their interac-tions suggest that behaviors such as these will beidentified. Indeed, our models are simplifications of theactual kinetic schemes characterizing genetic systems.MacLeod (23) has recently proposed that epigenetic,heritable changes in gene expression following expo-sure to chemicals might play a role in carcinogenesis.Such changes would correspond dynamically to pertur-bations of genetic regulatory systems from one steadystate to another.

An outstanding major issue for future investigationwill be to determine whether the parameters of specificgenetic systems in vivo are permissive for specific typesof dynamic behavior. Experiments to help determinethis might include monitoring transcription of trans-fected reporter gene constructs, with defined promoterssubject to regulation by TFs, in cultured cells duringspecific patterns of hormone or neurotransmitter appli-cation. A prolactin promoter-luciferase gene constructhas been used to provide real-time quantification ofpromoter activity in cultured secretory cells (4). An-other relevant technique is polymerase chain reactionamplification and quantitation of specific mRNAs fromtissue samples (43); however, this technique does notresolve dynamics at the single cell level. We believethat, as the dynamic behaviors of gene networks areexplored empirically, the present work can provide aconceptual framework for the analysis and interpreta-tion of such experiments.

APPENDIX




d[AP]

dt



[AP]

[AP] � KA,deph

(A1)



d[AA]






rR,deph � kR,b

2[RRPP]�[RRP]


(A4)

d[RR]

dt� � rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

(A5)

d[RRP]

dt� rR,ph

2[RR]

2[RR] � [RRP]

� rR,deph

[RRP]

2[RRPP] � [RRP]

� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]

(A6)

d[RRPP]

dt� rR,ph

[RRP]

2[RR] � [RRP]

� rR,deph

2[RRPP]

2[RRPP] � [RRP]


� k2,b [RRPPB]

(A7)

d[RRPPB]




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where KR,ph and KR,deph are Michaelis constants for TF-Rphosphorylation and dephosphorylation, respectively, andkR,b and k2,b are the backward rate constants for TF-Rphosphorylation and TF-R binding to DNA, respectively.

The rate of transcription of the target gene for whosepromoter region TF-A and TF-R compete (rRep) is taken asproportional to the concentration of bound TF-A dimersmultiplied by the fraction of phosphorylated TF-A sites, witha rate constant kRep

rRep � kRep

[AAB] [AP]

Atot(A9)

There is a conservation condition on the total number ofDNA binding sequences

Gfree � Gtot � [AAB] � [RRPPB] (A10)

where Gtot is the total number of CRE sites.For modeling the data of Itoh et al. (17) indicating an

optimal frequency for repression of transcription of the celladhesion molecule L1, an additional kinetic step is needed.The target gene for TF regulation is assumed to express aprotein Rep that represses transcription of the L1 gene. L1transcription is assumed to proceed at a basal rate rL1 in theabsence of Rep. L1 transcription only occurs if Rep is notbound to a promoter for the L1 gene. Rep dimers bind to thispromoter with dissociation constant KRep

d[Rep]

dt� rRep � [Rep] (A11)

d[L1]

dt�

rL1

(1 � [Rep]2/KRep2)

� [L1] (A12)

In simulations of the formation of LTM, we posited thateach stimulus immediately set kA,f (t) and kR,f (t) to maximalvalues kA,max and kR,max, respectively. After a stimulus, kA,f (t)and kR,f (t) decayed to zero with time constants �2 and �1,respectively.

All simulations used parameter values from one of the twofollowing sets. Concentrations are left dimensionless due tolack of sufficient experimental data. Parameters marked‘‘varies’’have values given in the text or in Fig. 6 or 7.

For the simulations of Fig. 6

Gtot � 0.1 Atot � 1.0 Rtot � 3.0

kA,f and kR,f vary KA,ph � 1.0 KA,deph � 1.0

kA,b � 0.1 min�1 (Fig. 6, A and B), 0.05 min�1 (Fig. 6C)

kR,b � 0.18 min�1 k1,f � 10.0 min�1 k1,b � 10.0 min�1

KR,ph � 0.5 KR,deph � 0.5 k2,f � 10.0 min�1

k2,b � 1.0 min�1 rL1 � 1.0 KRep � 0.11

kRep � 5.0 min�1

For the simulations of Fig. 6C, kA,f and kR,f were assumed toexecute small oscillations about the mean values in the figurelegend. In the absence of data to construct a kinetic model forthese oscillations, we merely assumed sinusoidal oscillationswith a frequency equal to the stimulus frequency and anamplitude of 5% of the mean value.


Gtot� 0.1 Atot � 1.0 Rtot � 3.0

KA,ph � 20.0 KA,deph � 20.0 kA,b � 0.7 min�1



k2,b � 1.0 min�1 ISI varies kA,max � 30 min�1

kR,max � 30 min�1 �1 � 4.0 min �2 � 4.0 min

We thank Pramod Dash, Ron Dror, and Shogo Endo for commentson an earlier draft of the paper and B. Ermentrout for use of his XPPprogram for simulations.

This work was supported by Office of Naval Research GrantN0014-95-1-0579, by National Institutes of Health Grants K05-MH-00649, T32-NS-07373, and R01-RR-11626, and by Texas HigherEducation Coordination Board Grant 011618-048.

Address for reprint requests: J. H. Byrne, Dept. of Neurobiologyand Anatomy, University of Texas Medical School, 6431 Fannin St.,Suite 7.046, Houston, TX 77030.

Received 30April 1997; accepted in final form 27 October 1997.

REFERENCES

1. Bagley,R. J., and L.Glass. Counting and classifying attractorsin high dimensional dynamical systems. J. Theor. Biol. 183:269–284, 1996.

2. Bartsch, D., M. Ghirardi, P. Skehel, K. Karl, S. Herder, M.Chen, C. Bailey, and E. R. Kandel. Aplysia CREB2 represseslong-term facilitation: relief of repression converts a transientfacilitation into long-term functional and structural change. Cell83: 979–992, 1995.

3. Brabant, G., K. Prank, and C. Schofl. Pulsatile patterns inhormone secretion. Trends Endocrinol. Metab. 3: 183–190, 1992.

4. Castano, J., R. Kineman, and L. S. Frawley. Dynamicmonitoring and quantification of gene expression in single, livingcells: a molecular basis for secretory cell heterogeneity. Mol.Endocrinol. 10: 599–605, 1996.

5. Doedel, E. AUTO: a program for the automatic bifurcationanalysis of autonomous systems. Congr. Num. 30: 265–284,1981.

6. Edwards,D.R. Cell signaling and the control of gene transcrip-tion. Trends Pharmacol. Sci. 15: 239–244, 1994.

7. Gear, C. The numerical integration of ordinary differentialequations. Math. Comput. 21: 146–156, 1967.

8. Ghosh, A., and M. Greenberg. Calcium signaling in neurons:molecular mechanisms and cellular consequences. Science 268:239–247, 1995.

9. Goldbeter, A. A model for circadian oscillations in the proteinPER. Proc. R. Soc. Lond. B Biol. Sci. B261: 319–324, 1995.

10. Goldbeter, A. Biochemical Oscillations and Cellular Rhythms.Cambridge, UK: Cambridge University Press, 1996.

11. Goldbeter, A., and D. Koshland. An amplified sensitivityarising from covalent modification in biological systems. Proc.Natl. Acad. Sci. USA 78: 6840–6844, 1991.

12. Gonzalez, G., P. Menzel, J. Leonard, W. Fischer, and R.Montminy. Characterization of motifs which are critical foractivity of the cyclic AMP-responsive transcription factor CREB.Mol. Cell. Biol. 11: 1306–1312, 1991.

13. Guckenheimer, J., and P. Holmes. Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields. Heidel-berg, Germany: Springer-Verlag, 1983.

14. Hai, T., and T. Curran. Cross-family dimerization of transcrip-tion factors Fos, Jun andATF/CREB alters DNAbinding specific-ity. Proc. Natl. Acad. Sci. USA 88: 3720–3724, 1991.

15. Howard, P., and R. Maurer. A composite Ets/Pit-1 binding sitein the prolactin gene can mediate transcriptional responses tomultiple signal transduction pathways. J. Biol. Chem. 270:20930–20936, 1995.

C541DYNAMICS OF GENETIC REGULATORY SYSTEMS


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rRep � kRep

[AAB] [AP]

Atot(A9)





d[Rep]


d[L1]

dt�

rL1

(1 � [Rep]2/KRep2)

� [L1] (A12)




Gtot � 0.1 Atot � 1.0 Rtot � 3.0





k2,b � 1.0 min�1 rL1 � 1.0 KRep � 0.11




Gtot� 0.1 Atot � 1.0 Rtot � 3.0





kR,max � 30 min�1 �1 � 4.0 min �2 � 4.0 min





REFERENCES


















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rRep � kRep

[AAB] [AP]

Atot(A9)





d[Rep]


d[L1]

dt�

rL1

(1 � [Rep]2/KRep2)

� [L1] (A12)




Gtot � 0.1 Atot � 1.0 Rtot � 3.0





k2,b � 1.0 min�1 rL1 � 1.0 KRep � 0.11




Gtot� 0.1 Atot � 1.0 Rtot � 3.0





kR,max � 30 min�1 �1 � 4.0 min �2 � 4.0 min





REFERENCES


















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of 6 stimuli at 10 Hz repeated every minute werereported to yield much more transcription than evenlyspaced 0.1-Hz stimuli or bursts of 12 stimuli at 10 Hzrepeated every 2 min. However, our current modelcannot simulate these results. If it is assumed, asabove, that velocities ofTF phosphorylation are approxi-mately proportional to stimulus frequency, the stimu-lus paradigms would be expected to yield, on average,the same velocities of phosphorylation of the TFs, andthe same transcription rates, because the number ofstimuli averaged over time is the same in all para-digms. A more detailed model of stimulus coupling tonuclear events, considering nonlinear kinetics of par-ticular second messenger systems, may be required.

To explain why massed stimulus presentations areless effective than spaced presentations in producingLTM in Drosophila, Yin et al. (49) proposed the samegenericmechanism that is considered here.An interme-diate intensity or frequency of stimulation phosphory-lates and activates a TF that activates transcription ofgenes essential for LTM formation, while a higherfrequency of stimulation activates also a second TFthat counteracts the effect of the first. However, ratherthan assuming fixed average values for phosphoryla-tion velocities, Yin et al. (49) assumed that the netdephosphorylation rate for the repressor TF-R is fasterthan that of the activator TF-A during ISIs. Then,during spaced stimuli, the net difference (phosphory-lated activator � phosphorylated repressor) becomeslarge during the long ISIs, but during massed stimuliTF-A phosphorylation is always approximately can-celed out by TF-R phosphorylation. The kinetic schemeof Fig. 5 is again used to test this hypothesis. Becausethe ISIs are now on the order of minutes rather thanseconds, we assume that each stimulus abruptly resetsthe phosphorylation rate constants to maximal valuesthat decay exponentially.As Fig. 7 demonstrates, our model predicts an opti-

mal stimulus frequency for transcription, and by infer-ence for LTM formation, when maximal velocities forTF dephosphorylation are chosen in accordance withthe hypothesis of Yin et al. (49). We also found (notshown) that qualitatively similar results are obtained ifboth TFs are dephosphorylated at identical rates and itis assumed instead that the phosphorylation rate ofTF-R is slower than that of TF-A during exposure to astimulus. Then TF-R is again only able to becomehighly phosphorylated during massed stimuli. In addi-tion, alternative kinetic schemes utilizing only one TFwere also found to give an optimal stimulus frequencyfor transcription. One such model variant postulatesboth activating and inhibiting phosphorylation sites onTF-A, with the inhibiting site only becoming signifi-cantly phosphorylated by massed stimuli. Anothermodel variant relies on competing kinase and phospha-tase activities. In principle, these model variants couldalso explain the aspects of L1 transcriptional regula-tion simulated above (Fig. 6C). It may be inferred thatthe existence of two competing processes, such asactivator and repressor phosphorylation, that havedifferent sensitivity to stimuli and opposing effects on

transcription of a specific gene could provide a generalmechanism for tuning the response of a genetic systemto an optimum stimulus frequency.

DISCUSSION

Biochemical nonlinearities such as dimerization, feed-back loops, and time delays are common in geneticregulatory systems (10). Our results indicate thatincorporating these features into models of relativelysimple genetic regulatory systems can give rise tocomplex dynamic activity and nonmonotonic depen-dence of response strength on stimulus. Thus thedynamic principles illustrated are likely to be impor-tant in phenomena in which regulation of transcription

Fig. 7. Nonmonotonic dependence of transcription rate on stimulusfrequency in model of Fig. 5 with kinetic parameters consistent withhypothesis of Yin et al. (49) for explaining greater efficacy of spacedstimuli in formation of LTM (APPENDIX, 2nd set of parameter values).A: time course of transcription rate (in units of min�1) during spacedstimuli. B: massed stimuli are used, and in all other respectssimulation and graph are as in A, including time scale. Comparing Aand B demonstrates that, over 100 min, 100 massed stimuli [inter-stimulus interval (ISI) � 1 min] produce considerably less transcrip-tion, and by inference less LTM formation, than 8 spaced stimuli(ISI � 15 min). C: dependence of transcription rate on ISI for 2 cases.Top curve, dephosphorylation rate constant for TF-A (kA,b; 1.0min�1)�kR,b (7.0 min�1), as in hypothesis of Yin et al. (49). Bottom curve,dephosphorylation rate constants are equal (1.0 min�1).



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Result : « frequency selectivity » In the gene response

Qualitative modeling (Smolen et al, , Am J Physiol Cell Physiol, 1998) of 6 stimuli at 10 Hz repeated every minute werereported to yield much more transcription than evenlyspaced 0.1-Hz stimuli or bursts of 12 stimuli at 10 Hzrepeated every 2 min. However, our current modelcannot simulate these results. If it is assumed, asabove, that velocities ofTF phosphorylation are approxi-mately proportional to stimulus frequency, the stimu-lus paradigms would be expected to yield, on average,the same velocities of phosphorylation of the TFs, andthe same transcription rates, because the number ofstimuli averaged over time is the same in all para-digms. A more detailed model of stimulus coupling tonuclear events, considering nonlinear kinetics of par-ticular second messenger systems, may be required.

To explain why massed stimulus presentations areless effective than spaced presentations in producingLTM in Drosophila, Yin et al. (49) proposed the samegenericmechanism that is considered here.An interme-diate intensity or frequency of stimulation phosphory-lates and activates a TF that activates transcription ofgenes essential for LTM formation, while a higherfrequency of stimulation activates also a second TFthat counteracts the effect of the first. However, ratherthan assuming fixed average values for phosphoryla-tion velocities, Yin et al. (49) assumed that the netdephosphorylation rate for the repressor TF-R is fasterthan that of the activator TF-A during ISIs. Then,during spaced stimuli, the net difference (phosphory-lated activator � phosphorylated repressor) becomeslarge during the long ISIs, but during massed stimuliTF-A phosphorylation is always approximately can-celed out by TF-R phosphorylation. The kinetic schemeof Fig. 5 is again used to test this hypothesis. Becausethe ISIs are now on the order of minutes rather thanseconds, we assume that each stimulus abruptly resetsthe phosphorylation rate constants to maximal valuesthat decay exponentially.

As Fig. 7 demonstrates, our model predicts an opti-mal stimulus frequency for transcription, and by infer-ence for LTM formation, when maximal velocities forTF dephosphorylation are chosen in accordance withthe hypothesis of Yin et al. (49). We also found (notshown) that qualitatively similar results are obtained ifboth TFs are dephosphorylated at identical rates and itis assumed instead that the phosphorylation rate ofTF-R is slower than that of TF-A during exposure to astimulus. Then TF-R is again only able to becomehighly phosphorylated during massed stimuli. In addi-tion, alternative kinetic schemes utilizing only one TFwere also found to give an optimal stimulus frequencyfor transcription. One such model variant postulatesboth activating and inhibiting phosphorylation sites onTF-A, with the inhibiting site only becoming signifi-cantly phosphorylated by massed stimuli. Anothermodel variant relies on competing kinase and phospha-tase activities. In principle, these model variants couldalso explain the aspects of L1 transcriptional regula-tion simulated above (Fig. 6C). It may be inferred thatthe existence of two competing processes, such asactivator and repressor phosphorylation, that havedifferent sensitivity to stimuli and opposing effects on

transcription of a specific gene could provide a generalmechanism for tuning the response of a genetic systemto an optimum stimulus frequency.

DISCUSSION

Biochemical nonlinearities such as dimerization, feed-back loops, and time delays are common in geneticregulatory systems (10). Our results indicate thatincorporating these features into models of relativelysimple genetic regulatory systems can give rise tocomplex dynamic activity and nonmonotonic depen-dence of response strength on stimulus. Thus thedynamic principles illustrated are likely to be impor-tant in phenomena in which regulation of transcription

Fig. 7. Nonmonotonic dependence of transcription rate on stimulusfrequency in model of Fig. 5 with kinetic parameters consistent withhypothesis of Yin et al. (49) for explaining greater efficacy of spacedstimuli in formation of LTM (APPENDIX, 2nd set of parameter values).A: time course of transcription rate (in units of min�1) during spacedstimuli. B: massed stimuli are used, and in all other respectssimulation and graph are as in A, including time scale. Comparing Aand B demonstrates that, over 100 min, 100 massed stimuli [inter-stimulus interval (ISI) � 1 min] produce considerably less transcrip-tion, and by inference less LTM formation, than 8 spaced stimuli(ISI � 15 min). C: dependence of transcription rate on ISI for 2 cases.Top curve, dephosphorylation rate constant for TF-A (kA,b; 1.0min�1)�kR,b (7.0 min�1), as in hypothesis of Yin et al. (49). Bottom curve,dephosphorylation rate constants are equal (1.0 min�1).



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5-‐HT

Result : interspike selectivity In the gene response

!x = k S(t)(xtot ! x)! "k x!y = k S(t)(ytot ! y)! "k y!z = ! H (x !"1)H ("2 ! y)!# z

Qualitative modeling (A. Cournac and JAS, BMC Systems Biol., 2009)

Some benefits of the simplified modeling: -  Analytical calculations are made possible

-  Highlight the role of repressor, (and a closed link with the IFFL of U. Alon)

-  Highlight the role of the interspike rather than the frequency of the simulus

fixed τ

S(t)

< z >T


Plan

1.  Some principes of the modeling of intracellular biochemical kine:cs by coupled ODEs




3. Periodic oscilla:ons in signaling cascades


Autonomous oscilla:ons in MAPK cascade

82% 9% 9%

Ras

Raf

Mek

Erk

marker without a nuclear-localization signal showed relativelylower ERK–GFP fusion protein/mRFP ratio in the nucleusbefore EGF addition (Supplementary Figure S1). The additionof EGF caused a specific elevation of the ERK–GFP fusionprotein/mRFP ratio in the nucleus and a simultaneousdecrease in the ERK–GFP fusion protein/mRFP ratio in thecytoplasm with a periodicity of 12–15min (SupplementaryFigure S1). Immunofluorescence of fixed cells using antibodiesagainst total ERK and phosphorylated ERK showed an increasein the levels of both species in the cell nucleus after EGFtreatment, in agreement with previous studies (Supplemen-tary Figure S2; Lenormand et al, 1993).As shown in Figure 2A, oscillations required the continuous

presence of EGF. After removal of EGF and blocking the EGFRwith antagonistic mAb 225, cells underwent only a singleadditional nuclear translocation cycle. We then monitored thechange in nuclear ERK–GFP fusion protein level over time in afield of cells grown at low density (Figure 2B). Each celldisplayed a rapid oscillation pattern for the entire duration ofthe experiment (Supplementary Movie S4), and this patternwas sustained for more than 40 cycles (Figure 2C), at whichpoint the experiment was terminated. There did not seem to beany relationship between the expression level of the ERK–GFPfusion protein and the oscillation pattern, and there was a lowdegree of synchrony between the cells in the population

(Supplementary Movie S4). During this long experiment(410 h), several cells underwent mitosis, allowing us todetermine whether oscillations persisted throughout the cellcycle. Although there was a brief interruption in oscillationsduring mitosis itself when the nuclear membrane disappeared,on reformation of the nuclei, oscillations could once again beobserved (Supplementary Figure S3). There was no obvioussynchrony between the oscillations of the daughter cells. Aswe were following an asynchronous population of cells,continuous oscillations in all the cells suggest that ERKoscillations persist throughout the cell cycle.

Oscillations are independent of ERK–GFPexpression levels

It has been suggested that the oscillations of NFkB, which havebeen observed after cell activation are a result of highexpression levels of the GFP–NFkB fusion protein (Barkenet al, 2005). To determine whether ERK oscillations could bedue to high expression levels of our ERK–GFP construct, weused flow cytometry to sort out cells into ‘low’ and ‘high’expression sets. The relative levels of ERK–GFP fusion proteinversus endogenous ERK expression was then estimated byquantitative western blots (Figure 3A). This showed that theaverage level of ERK–GFP fusion protein in the population

0

400

450

500

+EGF +225 mAb

0

400

600

Cell #1

Cell #2

Cell #3

Time (min)

Nuc

lear

ER

K-G

FPN

ucle

ar E

RK-

GFP

Time (min)

300200100

600400200

#1

#3

#2A

C

B

Figure 2 ERK oscillations are persistent and require the continuous presence of EGF. (A) Effect of termination of EGF binding on ERK–GFP dynamics. The nuclearlevel of ERK–GFP fusion protein in a representative cell was followed as described in Figure 1C. EGF (1 ng/ml) was added at the indicated time (84 min). At 255min, thecells were rinsed twice with EGF-free medium, and 10 mg/ml of the antagonistic anti-EGFR mAb antibody 225 was added to prevent additional ligand binding. (B) Cellsexpressing both ERK–GFP and mRFPnuc were imaged at 371C using a ! 20 objective. The indicated cells were chosen for analysis because of their varying ERK–GFPexpression levels. (C) The fluorescence intensities of ERK–GFP co-localized in the nucleus of the cells indicated in (B) were measured at 1-min intervals continuously forover 10 h. The uncorrected mean pixel intensity levels of nuclear ERK–GFP fusion protein are shown. Source data is available for this figure at www.nature.com/msb.

Sustained nuclear–cytoplasmic ERK oscillationsH Shankaran et al

& 2009 EMBO and Macmillan Publishers Limited Molecular Systems Biology 2009 3

Rapid and sustained nuclear–cytoplasmic ERK oscillations induced by epidermal growth factor Shankaran et al. Mol. Syst. Biol. 2009.



82% 9% 9%

Ras

Raf

Mek

Erk

marker without a nuclear-localization signal showed relativelylower ERK–GFP fusion protein/mRFP ratio in the nucleusbefore EGF addition (Supplementary Figure S1). The additionof EGF caused a specific elevation of the ERK–GFP fusionprotein/mRFP ratio in the nucleus and a simultaneousdecrease in the ERK–GFP fusion protein/mRFP ratio in thecytoplasm with a periodicity of 12–15min (SupplementaryFigure S1). Immunofluorescence of fixed cells using antibodiesagainst total ERK and phosphorylated ERK showed an increasein the levels of both species in the cell nucleus after EGFtreatment, in agreement with previous studies (Supplemen-tary Figure S2; Lenormand et al, 1993).As shown in Figure 2A, oscillations required the continuous

presence of EGF. After removal of EGF and blocking the EGFRwith antagonistic mAb 225, cells underwent only a singleadditional nuclear translocation cycle. We then monitored thechange in nuclear ERK–GFP fusion protein level over time in afield of cells grown at low density (Figure 2B). Each celldisplayed a rapid oscillation pattern for the entire duration ofthe experiment (Supplementary Movie S4), and this patternwas sustained for more than 40 cycles (Figure 2C), at whichpoint the experiment was terminated. There did not seem to beany relationship between the expression level of the ERK–GFPfusion protein and the oscillation pattern, and there was a lowdegree of synchrony between the cells in the population

(Supplementary Movie S4). During this long experiment(410 h), several cells underwent mitosis, allowing us todetermine whether oscillations persisted throughout the cellcycle. Although there was a brief interruption in oscillationsduring mitosis itself when the nuclear membrane disappeared,on reformation of the nuclei, oscillations could once again beobserved (Supplementary Figure S3). There was no obvioussynchrony between the oscillations of the daughter cells. Aswe were following an asynchronous population of cells,continuous oscillations in all the cells suggest that ERKoscillations persist throughout the cell cycle.

Oscillations are independent of ERK–GFPexpression levels

It has been suggested that the oscillations of NFkB, which havebeen observed after cell activation are a result of highexpression levels of the GFP–NFkB fusion protein (Barkenet al, 2005). To determine whether ERK oscillations could bedue to high expression levels of our ERK–GFP construct, weused flow cytometry to sort out cells into ‘low’ and ‘high’expression sets. The relative levels of ERK–GFP fusion proteinversus endogenous ERK expression was then estimated byquantitative western blots (Figure 3A). This showed that theaverage level of ERK–GFP fusion protein in the population

0

400

450

500

+EGF +225 mAb

0

400

600

Cell #1

Cell #2

Cell #3

Time (min)

Nuc

lear

ER

K-G

FPN

ucle

ar E

RK-

GFP

Time (min)

300200100

600400200

#1

#3

#2A

C

B

Figure 2 ERK oscillations are persistent and require the continuous presence of EGF. (A) Effect of termination of EGF binding on ERK–GFP dynamics. The nuclearlevel of ERK–GFP fusion protein in a representative cell was followed as described in Figure 1C. EGF (1 ng/ml) was added at the indicated time (84 min). At 255min, thecells were rinsed twice with EGF-free medium, and 10 mg/ml of the antagonistic anti-EGFR mAb antibody 225 was added to prevent additional ligand binding. (B) Cellsexpressing both ERK–GFP and mRFPnuc were imaged at 371C using a ! 20 objective. The indicated cells were chosen for analysis because of their varying ERK–GFPexpression levels. (C) The fluorescence intensities of ERK–GFP co-localized in the nucleus of the cells indicated in (B) were measured at 1-min intervals continuously forover 10 h. The uncorrected mean pixel intensity levels of nuclear ERK–GFP fusion protein are shown. Source data is available for this figure at www.nature.com/msb.

Sustained nuclear–cytoplasmic ERK oscillationsH Shankaran et al

& 2009 EMBO and Macmillan Publishers Limited Molecular Systems Biology 2009 3

Rapid and sustained nuclear–cytoplasmic ERK oscillations induced by epidermal growth factor Shankaran et al. Mol. Syst. Biol. 2009.

(negative feedback loop)



Qiao et al., (PLoS sept. 2007)

82% 9% 9%

Ras

Raf

Mek

Erk



(e.g. Kholodenko,2000, Angeli et al, 2004, Giuraniuc et al, 2007)

!y1* =V1

y1K1 + y1

! "V1y1*

"K1 + y1*

!y2* =V2y1

* y2K2 + y2

! "V2y2*

"K2 + y2*

!y3* =V3y2

* y3K3 + y3

! "V3y3*

"K3 + y3*

with yi =1! yi*


Autonomous oscilla:ons in MAPK cascade Results from a rigorous perturbation scheme (Ventura, JAS, PLoS CB 2008)

!x1 =V1E1y1

K1 + y1! "V1

x1"K1(1+ y2 /K2 )+ x1

!x2 =V2x1y2

K2 + y2! "V2

x2"K2 (1+ y3 /K3)+ x2

!x3 =V3x2y3

K3 + y3! "V3

x3"K3 + x3

with yi =1! xi and xi = yi*(1+ yi!1 /Ki!1)

There is a intrinsic negative feedback


Autonomous oscilla:ons in MAPK cascades

y1* y1

y2* y2

E1

y2**

0 0.5 1

0

0.5

1

x1

y 2** !x1 = 0

!y**2 = 0, y2 = 0

« Nullclines »



y1* y1

y2* y2

E1

y2**

0 0.5 1

0

0.5

1

x1

y 2**

!

˙ x 1 < 0

!

˙ x 1 > 0

!x1 = 0



y1* y1

y2* y2

E1

y2**

0 0.5 1

0

0.5

1

x1

y 2** !

˙ x 1 < 0

!

˙ x 1 > 0

0 10 20 30 40 500

0.5

1

time

y 2**

Limit cycle !

Relaxation oscillations

Bifurcations: - Supecritical Hopf bifurcation - Saddle-node bifurcation on a cycle


Conclusion : via 2 study cases, we saw that 1.  Periodic pulsa:le s:mula:ons of gene network mo:fs can lead to op:mal

response in the produc:on of proteins. (Possible applica:ons to memory forma:on)

2.  A nega:ve retroac:vity exists in signaling cacades. It leads to the possibility of autonomous oscilla:ons in (MAPK) signaling pathways.

We conclude that the « relevant » level of modeling is not absolute:

–  Looking for a « minimal » model allows one to beXer understand (and control) the underlying mechanisms

–  This form of reduc:onism can typically be brought up by physicists !


Study case 1 Axel COURNAC, LPTMC (postdoc), Univ. Pierre et Marie Curie Study case 2 Alejandra VENTURA, University of Buenos Aires, Argentina (main collaboration) Sofia MERAJVER, Cancer Center, University of Michigan

Acknowledgements

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Institut Non Linéaire de Nicemarcos/nicephysicsdays.org/... · id en tﬁ. I , ou r m l sap c f h actual kinetic schemes characterizing genetic systems. MacLeod (23) has recently