Models of Circadian Rhythms

Daniel Forger

~24-hour Clocks time important events

Circadian Clocks important for

• Bipolar Disorder• FASPS (and other Sleep Disorders)• Alzheimer’s Disease (institutionalization)• Leukemia (chemotherapy most effective at

night, direct link to cell cycle and cancer)• Cholesterol (4-fold increase in biosynthesis at

night controlled by liver pacemakers)• The Blind (can’t synchronize)• Schedules/Missions/Shiftwork

movies

Tyson-Hong-Thron Novak model of circadian rhythms in

Drosophila

Leloup-Goldbeter Model

Equations Transcription Translation Degradation Phosphorylation Binding Nuclear Transport 

L = Light LeveldGRv/dt = binRv*RvnRvn *(1 - GRv) - unbinRv*GRvdMnRo/dt = trRo*(1 - G)*(1 - GRv)3 - tmc*MnRodMcRo/dt = tmc*MnRo - umR*McRodMnRt/dt = trRt*(1 - G) - tmc*MnRtdMcRt/dt = tmc*MnRt - umR*McRtdMnPo/dt = trPo*(1 - G)5 + L - tmc*MnPodMcPo/dt = tmc*MnPo - umPo*McPodMnPt/dt = trPt*(1 - G)5 + L - tmc*MnPtdMcPt/dt = tmc*MnPt - umPt*McPtdMnRv/dt = trRv*(1 - G)3 - tmc*MnRvdMcRv/dt = tmc*MnRv - umRv*McRvdRv/dt = tlrv*McRv - 2*arv*Rv*Rv + 2*drv*RvRv - nl*Rv + ne*Rvn - uRv*RvdRvn/dt = - 2*Nf*arv*Rvn*Rvn + 2*drv*RvnRvn + nl*Rv - ne*Rvn - uRv*RvndRvRv/dt = arv*Rv*Rv - drv*RvRv - nl*RvRv + ne*RvnRvn - 2*uRv*RvRvdRvnRvn/dt = Nf*arv*Rvn*Rvn - drv*RvnRvn + nl*RvRv - ne*RvnRvn - 2*uRv*RvnRvndPo/dt = tlp*McPo - ac*Po*C + dc*PoC - upu*PodPt/dt = tlp*McPt - ac*Pt*C + dc*PtC - upu*PodPoC/dt = ac*Po*C - dc*PoC - hoo*PoCdPtC/dt = ac*Pt*C - dc*PtC - hot*PtCdPopC/dt = hoo*PoC + ac*Pop*C - dc*PopC - up*PopC - ht*PopC - nl*PopC + ne*PonpCn - ar*PopC*Ro + dr*PopCRo - ar*PopC*Rt + dr*PopCRtdPtpC/dt = hot*PtC + ac*Ptp*C - dc*PtpC - up*PtpC - ht*PtpC - nl*PtpC + ne*PtnpCn - ar*PtpC*Ro + dr*PtpCRo - ar*PtpC*Rt + dr*PtpCRtdPop/dt = - ac*Pop*C+ dc*PopC - up*Pop - ar*Pop*Ro + dr*PopRo - ar*Pop*Rt + dr*PopRt - nl*Pop + ne*PonpdPtp/dt = - ac*Ptp*C+ dc*PtpC - up*Ptp - ar*Ptp*Ro + dr*PtpRo - ar*Ptp*Rt + dr*PtpRt - nl*Ptp + ne*PtnpdPoppC/dt = ht*PopC - up*PoppC + ac*Popp*C- dc*PoppC + ne*PonppCn - ar*PoppC*Ro + dr*PoppCRo - ar*PoppC*Rt + dr*PoppCRtdPtppC/dt = ht*PtpC - up*PtppC + ac*Ptpp*C- dc*PtppC + ne*PtnppCn - ar*PtppC*Ro + dr*PtppCRo - ar*PtppC*Rt + dr*PtppCRtdPopRo/dt = ar*Pop*Ro - dr*PopRo - ac*PopRo*C + dc*PopCRo - nl*PopRo + ne*PonpRondPtpRo/dt = ar*Ptp*Ro - dr*PtpRo - ac*PtpRo*C+ dc*PtpCRo - nl*PtpRo + ne*PtnpRondPopRt/dt = ar*Pop*Rt - dr*PopRt - ac*PopRt*C+ dc*PopCRt - nl*PopRt + ne*PonpRtndPtpRt/dt = ar*Ptp*Rt - dr*PtpRt - ac*PtpRt*C + dc*PtpCRt - nl*PtpRt + ne*PtnpRtndPoppRo/dt = ar*Popp*Ro - dr*PoppRo - ac*PoppRo*C + dc*PoppCRo + ne*PonppRondPoppRt/dt = ar*Popp*Rt - dr*PoppRt - ac*PoppRt*C + dc*PoppCRt + ne*PonppRtndPtppRo/dt = ar*Ptpp*Ro - dr*PtppRo - ac*PtppRo*C + dc*PtppCRo + ne*PtnppRondPtppRt/dt = ar*Ptpp*Rt - dr*PtppRt - ac*PtppRt*C + dc*PtppCRt + ne*PtnppRtndPopp/dt = - ac*Popp*C + dc*PoppC + ne*Ponpp - ar*Popp*Ro + dr*PoppRo – ar*Popp*Rt + dr*PoppRt - up*PoppdPtpp/dt = - ac*Ptpp*C+ dc*PtppC + ne*Ptnpp - ar*Ptpp*Ro + dr*PtppRo - ar*Ptpp*Rt + dr*PtppRt - up*PtppdPopCRo/dt = ar*PopC*Ro - dr*PopCRo + ac*PopRo*C- dc*PopCRo - nl*PopCRo + ne*PonpCnRon - ht*PopCRodPtpCRo/dt = ar*PtpC*Ro - dr*PtpCRo + ac*PtpRo*C - dc*PtpCRo - nl*PtpCRo + ne*PtnpCnRon - ht*PtpCRodPopCRt/dt = ar*PopC*Rt - dr*PopCRt + ac*PopRt*C- dc*PopCRt - nl*PopCRt + ne*PonpCnRtn - ht*PopCRtdPtpCRt/dt = ar*PtpC*Rt - dr*PtpCRt + ac*PtpRt*C - dc*PtpCRt - nl*PtpCRt + ne*PtnpCnRtn - ht*PtpCRtdPoppCRo/dt = ar*PoppC*Ro - dr*PoppCRo + ac*PoppRo*C- dc*PoppCRo + ne*PonppCnRon + ht*PopCRo

Equations Cont. Transcription Translation Degradation Phosphorylation Binding Nuclear Transport

dPtppCRo/dt = ar*PtppC*Ro - dr*PtppCRo + ac*PtppRo*C - dc*PtppCRo + ne*PtnppCnRon + ht*PtpCRodPoppCRt/dt = ar*PoppC*Rt - dr*PoppCRt + ac*PoppRt*C - dc*PoppCRt + ne*PonppCnRtn + ht*PopCRtdPtppCRt/dt = ar*PtppC*Rt - dr*PtppCRt + ac*PtppRt*C - dc*PtppCRt + ne*PtnppCnRtn + ht*PtpCRtdRo/dt = - ar*Ro*Pop - ar*Ro*Popp - ar*Ro*PopC - ar*Ro*PoppC + dr*PopRo + dr*PoppRo + dr*PopCRo + dr*PoppCRo - ar*Ro*Ptp - ar*Ro*Ptpp - ar*Ro*PtpC - ar*Ro*PtppC + dr*PtpRo + dr*PtppRo + dr*PtpCRo + dr*PtppCRo + tlr*McRo - uro*RodRt/dt = - ar*Rt*Pop - ar*Rt*Popp - ar*Rt*PopC - ar*Rt*PoppC + dr*PopRt + dr*PoppRt + dr*PopCRt + dr*PoppCRt - ar*Rt*Ptp - ar*Rt*Ptpp - ar*Rt*PtpC - ar*Rt*PtppC + dr*PtpRt + dr*PtppRt + dr*PtpCRt + dr*PtppCRt + tlr*McRt -

urt*RtdPonpCn/dt = ac*Nf*Ponp*Cn - dc*PonpCn - ht*PonpCn + nl*PopC - ne*PonpCn - ar*Nf*PonpCn*Ron + dr*PonpCnRon - ar*Nf*PonpCn*Rtn + dr*PonpCnRtn - upn*PonpCndPtnpCn/dt = ac*Nf*Ptnp*Cn - dc*PtnpCn - ht*PtnpCn + nl*PtpC - ne*PtnpCn - ar*Nf*PtnpCn*Ron + dr*PtnpCnRon - ar*Nf*PtnpCn*Rtn + dr*PtnpCnRtn - upn*PtnpCndPonp/dt = - ac*Nf*Ponp*Cn + dc*PonpCn - ar*Nf*Ponp*Ron + dr*PonpRon - ar*Nf*Ponp*Rtn + dr*PonpRtn + nl*Pop - ne*Ponp - upn*PonpdPtnp/dt = - ac*Nf*Ptnp*Cn + dc*PtnpCn - ar*Nf*Ptnp*Ron + dr*PtnpRon - ar*Nf*Ptnp*Rtn + dr*PtnpRtn + nl*Ptp - ne*Ptnp - upn*PtnpdPonppCn/dt = ht*PonpCn + ac*Nf*Ponpp*Cn - dc*PonppCn - ne*PonppCn - ar*Nf*PonppCn*Ron + dr*PonppCnRon – ar*Nf*PonppCn*Rtn + dr*PonppCnRtn - upn*PonppCndPtnppCn/dt = ht*PtnpCn + ac*Nf*Ptnpp*Cn - dc*PtnppCn - ne*PtnppCn - ar*Nf*PtnppCn*Ron + dr*PtnppCnRon - ar*Nf*PtnppCn*Rtn + dr*PtnppCnRtn - upn*PtnppCndPonpRon/dt = ar*Nf*Ponp*Ron - dr*PonpRon - ac*Nf*PonpRon*Cn + dc*PonpCnRon + nl*PopRo - ne*PonpRondPtnpRon/dt = ar*Nf*Ptnp*Ron - dr*PtnpRon - ac*Nf*PtnpRon*Cn + dc*PtnpCnRon + nl*PtpRo - ne*PtnpRondPonpRtn/dt = ar*Nf*Ponp*Rtn - dr*PonpRtn - ac*Nf*PonpRtn*Cn + dc*PonpCnRtn + nl*PopRt - ne*PonpRtndPtnpRtn/dt = ar*Nf*Ptnp*Rtn - dr*PtnpRtn - ac*Nf*PtnpRtn*Cn + dc*PtnpCnRtn + nl*PtpRt - ne*PtnpRtndPonppRon/dt = ar*Nf*Ponpp*Ron - dr*PonppRon - ac*Nf*PonppRon*Cn + dc*PonppCnRon - ne*PonppRondPtnppRon/dt = ar*Nf*Ptnpp*Ron - dr*PtnppRon - ac*Nf*PtnppRon*Cn + dc*PtnppCnRon - ne*PtnppRondPonppRtn/dt = ar*Nf*Ponpp*Rtn - dr*PonppRtn - ac*Nf*PonppRtn*Cn + dc*PonppCnRtn - ne*PonppRtndPtnppRtn/dt = ar*Nf*Ptnpp*Rtn - dr*PtnppRtn - ac*Nf*PtnppRtn*Cn + dc*PtnppCnRtn - ne*PtnppRtndPonpp/dt = - ac*Nf*Ponpp*Cn + dc*PonppCn - ne*Ponpp - ar*Nf*Ponpp*Ron + dr*PonppRon - ar*Nf*Ponpp*Rtn + dr*PonppRtn - upn*PonppdPtnpp/dt = - ac*Nf*Ptnpp*Cn + dc*PtnppCn - ne*Ptnpp - ar*Nf*Ptnpp*Ron + dr*PtnppRon - ar*Nf*Ptnpp*Rtn + dr*PtnppRtn - upn*PtnppdPonpCnRon/dt = ar*Nf*PonpCn*Ron - dr*PonpCnRon + ac*Nf*PonpRon*Cn - dc*PonpCnRon + nl*PopCRo - ne*PonpCnRon - ht*PonpCnRondPtnpCnRon/dt = ar*Nf*PtnpCn*Ron - dr*PtnpCnRon + ac*Nf*PtnpRon*Cn - dc*PtnpCnRon + nl*PtpCRo - ne*PtnpCnRon - ht*PtnpCnRondPonpCnRtn/dt = ar*Nf*PonpCn*Rtn - dr*PonpCnRtn + ac*Nf*PonpRtn*Cn - dc*PonpCnRtn + nl*PopCRt - ne*PonpCnRtn - ht*PonpCnRtndPtnpCnRtn/dt = ar*Nf*PtnpCn*Rtn - dr*PtnpCnRtn + ac*Nf*PtnpRtn*Cn - dc*PtnpCnRtn + nl*PtpCRt - ne*PtnpCnRtn - ht*PtnpCnRtndPonppCnRon/dt = ar*Nf*PonppCn*Ron - dr*PonppCnRon + ac*Nf*PonppRon*Cn - dc*PonppCnRon - ne*PonppCnRon + ht*PonpCnRondPtnppCnRon/dt = ar*Nf*PtnppCn*Ron - dr*PtnppCnRon + ac*Nf*PtnppRon*Cn - dc*PtnppCnRon - ne*PtnppCnRon + ht*PtnpCnRondPonppCnRtn/dt = ar*Nf*PonppCn*Rtn - dr*PonppCnRtn + ac*Nf*PonppRtn*Cn - dc*PonppCnRtn - ne*PonppCnRtn + ht*PonpCnRtndPtnppCnRtn/dt = ar*Nf*PtnppCn*Rtn - dr*PtnppCnRtn + ac*Nf*PtnppRtn*Cn - dc*PtnppCnRtn - ne*PtnppCnRtn + ht*PtnpCnRtndRon/dt = - ar*Nf*Ron*Ponp - ar*Nf*Ron*Ponpp - ar*Nf*Ron*PonpCn - ar*Nf*Ron*PonppCn + dr*PonpRon + dr*PonppRon + dr*PonpCnRon + dr*PonppCnRon - ar*Nf*Ron*Ptnp - ar*Nf*Ron*Ptnpp - ar*Nf*Ron*PtnpCn -

ar*Nf*Ron*PtnppCn + dr*PtnpRon + dr*PtnppRon + dr*PtnpCnRon + dr*PtnppCnRon - uro*RondRtn/dt = - ar*Nf*Rtn*Ponp - ar*Nf*Rtn*Ponpp - ar*Nf*Rtn*PonpCn - ar*Nf*Rtn*PonppCn + dr*PonpRtn + dr*PonppRtn + dr*PonpCnRtn + dr*PonppCnRtn - ar*Nf*Rtn*Ptnp - ar*Nf*Rtn*Ptnpp - ar*Nf*Rtn*PtnpCn - ar*Nf*Rtn*PtnppCn

+ dr*PtnpRtn + dr*PtnppRtn + dr*PtnpCnRtn + dr*PtnppCnRtn - urt*RtndCn/dt = - ac*Nf*Cn*Ponp - ac*Nf*Cn*Ponpp - ac*Nf*Cn*PonpRon - ac*Nf*Cn*PonppRon + dc*PonpCn + dc*PonppCn + dc*PonpCnRon + dc*PonppCnRon - ac*Nf*Cn*Ptnp - ac*Nf*Cn*Ptnpp - ac*Nf*Cn*PtnpRon -

ac*Nf*Cn*PtnppRon + dc*PtnpCn + dc*PtnppCn + dc*PtnpCnRon + dc*PtnppCnRon - ac*Nf*Cn*PonpRtn - ac*Nf*Cn*PonppRtn + dc*PonpCnRtn + dc*PonppCnRtn - ac*Nf*Cn*PtnpRtn - ac*Nf*Cn*PtnppRtn + dc*PtnpCnRtn + dc*PtnppCnRtn + upn*PonpCn + upn*PonppCn + upn*PtnpCn + upn*PtnppCn

  RelationsC = Ct - (Ron + PonpRon + PonppRon + PonpCnRon + PonppCnRon + PtnpRon + PtnppRon + PtnpCnRon + PtnppCnRon + Rtn + PonpRtn + PonppRtn + PonpCnRtn + PonppCnRtn + PtnpRtn + PtnppRtn + PtnpCnRtn +

PtnppCnRtn)Rn = (Ron + PonpRon + PonppRon + PonpCnRon + PonppCnRon + PtnpRon + PtnppRon + PtnpCnRon + PtnppCnRon + Rtn + PonpRtn + PonppRtn + PonpCnRtn + PonppCnRtn + PtnpRtn + PtnppRtn + PtnpCnRtn + PtnppCnRtn

Tau mutation

• First discovered mutation in mammals– Subject of the widest study (many science papers)

• Missense mutation in CKI which decreases kinase activity

• Similar mutations in Drosophila• Similar mutations in man (FASPS, CKI-T44A)

cause sleep disorders• Phenotype:

– 22 hour period in heterozygotes– 20 hour period in homozygotes

Lowrey et al. Science 288 483

Lowrey et al.

Xu et al. Nature 434, 640

Xu et al.

Tau mutant predictions• WT period 24.2991• Half primary phosphorylation rate of PER1

– 24.43

• Half secondary phosphorylation rate of PER1– 26.73

• Half primary phosphorylation rate of PER2– 24.33

• Half binding rate to kinases– 26.62

• Decreased BMAL activation– 24.42

80 100 120 140Percent Rate

24.25

24.3

24.35

24.4

Period

80 100 120 140Percent Rate

23.9

24.1

24.2

24.3

24.4

24.5

Period

80 100 120 140Percent Rate

24.5

25

25.5

Period

80 100 120 140Percent Rate

24

25

26

Period

80 100 120 140Percent Rate

24.2

24.3

24.4

Period

80 100 120 140Percent Rate

24.27

24.28

24.29

24.31

24.32

24.33Period

Simpler model

jnjj

nnnnn

jjjjjjj

m

mmmmm

n

ppbkpdt

dpp

pbpgdt

dp

kppgbpgdt

dp

pgpbtrmdt

dp

mamcdt

dm

mcmapfdt

dm

)(

...

)(

...

...

)(

1

11

11

11111

11

11111

+

−−

−−

−−

−=

−=

−+−=

−−=

−=

−−=

Period Estimate

...

)(

...

)(

...

11

11

jjjjjj

jjjjjj

PkgbPgiwP

McaMciwM

++−=

+−=

−−

−−

€

...(iw + a j + c j )...(iw + b j + g j + k)...F(pn )

Pnc1...cm−1trg1...gn−1

=1

Decreasing gj, k or bj (because of a lower phosphorylation

rate) increases the phase of (iw + bj+gj+k). To balance this, w

must decrease and the period lengthens.

Nothing in the Model can explain tau!

(get help from friends)

Expression of CKItau but not inactive CKI decreases the abundance of PER proteins

CKI Tau K38A CKI Tau K38A

WB: Myc

WB: Actin

WB: HA

HA-Kinase

PER1 PER2

Immunoblot of PER levels after co-expression with CKI

PER

0 3

Tau K38ACKI

1.5 5 0 31.5 5 0 31.5 5 0 31.5 5CHX(hrs)

PER2

HA-CKI

Actin

Tau CKI shortens PER2 half-life

MG

UbUbUbUb

Phosphorylation-dependent mPER2 proteolysis

26S Proteasome

Per2

P P

PP

+ +TrCP

Proteasome inhibition prevents PER degradation induced by

CKItau expression

0 5 0 5 0 5

CKI Tau

CHX(h)

Vehicle

0 5 0 5 0 5

CKI Tau

MG132

PER1

Actin

MG

Dominant negative TrCP blocks Tau-induced degradation of PER2

Kinase

DN TrCP

Tau KA- Tau KA-

- - - + + +

PER2

Actin

MG

HA-Kinase Tau KA- CKI Tau KA- CKI

WT-PER2 PER2(S477A/G479A)

PER2

Actin

Mutation of TrCP binding site blocks Tau-induced degradation (but not phosphorylation) of PER2

NLSCKICLD Cry

450 755

PAS

MG

The tau mutation in CKI increases activity in vivo

• Is this really increased kinase activity?

• Is it specific to PER and circadian rhythm?

0

0.5

1

1.5

2

2.5

3

Empty WT CKIe Tau K38A

Fold

act

ivati

on

CKItau is inactive in Wnt signaling in vivo

Em

pt

y LRP6

DN W

T-

CK

ITa

u K3

8A

HA-CKI

Non specific

Phosphopeptide mapping of in vivo metabolically labeled PER2

8

1

2a2b

2c

3a3b 4a

4b

5

6

7

CKItau

1

2a2b

2c

3a

3b4a

4b

5

6

7

vector

700 cpm/TLC plate, 96 hr autoradiographic exposure EJE

Can Simulate these Directly(Gillespie’s SSA Method)

• Choose randomly among all possible reactions (weighted by their rates)

• Find the time until the next reaction:

• For speedup, do the above for reaction classes, and then choose individual reaction

• # of molecules given by experiments (100s to 1000s)• Measure period by several upcrossings (mean, 80% of

mean, 120% of mean etc.)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00

1ln

1

raτ

PopCn Popn + Cn dc*PopCn PoppCn Poppn + Cn dc*PoppCn PtpCn Ptpn + Cn dc*PtpCn PtppCn Ptppn + Cn dc*PtppCn PopCRon PopRon + Cn dc*PopCRon PoppCRon PoppRon + Cn dc*PoppCRon PopCRtn PopRtn + Cn dc*PopCRtn PoppCRtn PoppRtn + Cn dc*PoppCRtn PtpCRon PtpRon + Cn dc*PtpCRon PtppCRon PtppRon + Cn dc*PtppCRon PtpCRtn PtpRtn + Cn dc*PtpCRtn PtppCRtn PtppRtn + Cn dc*PtppCRtn PoC PopC hoo*PoC PopC PoppC hto*PopC PtC PtpC hot*PtC PtpC PtppC ht*PtpC PopCn PoppCn hto*PopCn PtpCn PtppCn ht*PtpCn PopCRo PoppCRo hto*PopCRo PopCRt PoppCRt hto*PopCRt PtpCRo PtppCRo ht*PtpCRo PtpCRt PtppCRt ht*PtpCRt PopCRon PoppCRon hto*PopCRon PopCRtn PoppCRtn hto*PopCRtn PtpCRon PtppCRon ht*PtpCRon PtpCRtn PtppCRtn ht*PtpCRtn Pop + Ro PopRo ar*Pop*Ro Popp + Ro PoppRo ar*Popp*Ro Pop + Rt PopRt ar*Pop*Rt Popp + Rt PoppRt ar*Popp*Rt Ptp + Ro PtpRo ar*Ptp*Ro Ptpp + Ro PtppRo ar*Ptpp*Ro Ptp + Rt PtpRt ar*Ptp*Rt Ptpp + Rt PtppRt ar*Ptpp*Rt PopC + Ro PopCRo ar*PopC*Ro PoppC + Ro PoppCRo ar*PoppC*Ro PopC + Rt PopCRt ar*PopC*Rt PoppC + Rt PoppCRt ar*PoppC*Rt PtpC + Ro PtpCRo ar*PtpC*Ro PtppC + Ro PtppCRo ar*PtppC*Ro PtpC + Rt PtpCRt ar*PtpC*Rt PtppC + Rt PtppCRt ar*PtppC*Rt Popn + Ron PopRon ar*N*Popn*Ron Poppn + Ron PoppRon ar*N*Poppn*Ron Popn + Rtn PopRtn ar*N*Popn*Rtn Poppn + Rtn PoppRtn ar*N*Poppn*Rtn Ptpn + Ron PtpRon ar*N*Ptpn*Ron

Reaction RatePo + C PoC ac*Po*C Pop + C PopC ac*Pop*C

Popp + C PoppC ac*Popp*C

Pt + C PtC ac*Pt*C

Ptp + C PtpC ac*Ptp*C

Ptpp + C PtppC ac*Ptpp*C

PopRo + C PopCRo ac*PopRo*C PoppRo + C -> PoppCRoac*PoppRo*C PopRt + C -> PopCRt ac*PopRt*CPoppRt + C -> PoppCRt ac*PoppRt*C PtpRo + C PtpCRo ac*PtpRo*C PtppRo + C PtppCRo ac*PtppRo*C PtpRt + C PtpCRt ac*PtpRt*C PtppRt + C PtppCRt ac*PtppRt*C Popn + Cn PopCn ac*N*Popn*Cn Poppn + Cn PoppCn ac*N*Poppn*Cn Ptpn + Cn PtpCn ac*N*Ptpn*Cn Ptppn + Cn PtppCn ac*N*Ptppn*Cn PopRon + Cn PopCRon ac*N*PopRon*Cn PoppRon + Cn PoppCRon

ac*N*PoppRon*C PopRtn + Cn PopCRtn ac*N*PopRtn*Cn PoppRtn + Cn PoppCRtn

ac*N*PoppRtn*Cn PtpRon + Cn PtpCRon ac*N*PtpRon*Cn PtppRon + Cn PtppCRon

ac*N*PtppRon*Cn PtpRtn + Cn PtpCRtn ac*N*PtpRtn*Cn PtppRtn + Cn PtppCRtn ac*N*PtppRtn*Cn PoC Po + C dc*PoC PopC Pop + C dc*PopC

PoppC Popp + C dc*Popp PtC Pt + C dc*PtC

PtpC Ptp + C dc*PtpC

PtppC Ptpp + C dc*PtppC

PopCRo PopRo + C dc*PopCRo PoppCRo PoppRo + C dc*PoppCRo PopCRt PopRt + C dc*PopCRt PoppCRt PoppRt + C dc*PoppCRt PtpCRo PtpRo + C dc*PtpCRo PtppCRo PtppRo + C dc*PtppCRo PtpCRt PtpRt + C dc*PtpCRt PtppCRt PtppRt + C dc*PtppCRt

PopRo PopRon nl*PopRo PopRt PopRtn nl*PopRt PtpRo PtpRon nl*PtpRo PtpRt PtpRtn nl*PtpRt PopCRo PopCRon nl*PopCRo PopCRt PopCRtn nl*PopCRt PtpCRo PtpCRon nl*PtpCRo PtpCRt PtpCRtn nl*PtpCRt Popn Pop ne*Popn Ptpn Ptp ne*Ptpn PopCn PopC ne*PopCn PtpCn PtpC ne*PtpCn PopRon PopRo ne*PopRon PopRtn PopRt ne*PopRtn PtpRon PtpRo ne*PtpRon PtpRtn PtpRt ne*PtpRtn PopCRon PopCRo ne*PopCRon PopCRtn PopCRt ne*PopCRtn PtpCRon PtpCRo ne*PtpCRon PtpCRtn PtpCRt ne*PtpCRtn Poppn Popp ne*Poppn Ptppn Ptpp ne*Ptppn PoppCn PoppC ne*PoppCn PtppCn PtppC ne*PtppCn PoppRon PoppRo ne*PoppRon PoppRtn PoppRt ne*PoppRtn PtppRon PtppRo ne*PtppRon PtppRtn PtppRt ne*PtppRtn PoppCRon PoppCRo ne*PoppCRon PoppCRtn PoppCRt ne*PoppCRtn PtppCRon PtppCRo ne*PtppCRon PtppCRtn PtppCRt ne*PtppCRtn Pop up*Pop Popp up*Popp Ptp up*Ptp Ptpp up*Ptpp PopC C up*PopC PoppC C up*PoppC PtpC C up*PtpC PtppC C up*PtppC Popn up*Popn Poppn up*Poppn Ptpn up*Ptpn Ptppn up*Ptppn PopCn Cn up*PopCn PoppCn Cn up*PoppCn PtpCn Cn up*PtpCn PtppCn Cn up*PtppCn Ro uro*Ro

Rt urt*Rt Ron uro*Ron Rtn urt*Rtn G (1.0-G)*bin*(Ron+Rtn+PopRon+PoppRon+

PtpRon+PtppRon+PopRtn+PoppRtn+PtpRtn+PtppRtn+PopCRon+PoppCRon+PtpCRon+PtppCRon+PopCRtn+PoppCRtn+PtpCRtn+PtppCRtn)

G unbin*G MnPo trPo*(1.0-G)5

MnPo McPo tmc*MnPo McPo umPo*McPo Po tlpo*McPo Pt tlpt*McPt Ro tlro*McRo Rt tlrt*McRt Po upu*Po Pt upu*Pt PoC C upu*PoC PtC C upu*PtC MnRv trRv*(1.0-G)3

MnRv McRv tmc*MnRv McRv umRv*McRv Rv tlRv*McRv Rv + Rv RvRv arv*Rv*Rv RvRv Rv + Rv drv*RvRv Rv Rvn nl*Rv Rvn Rv ne*Rvn RvRv RvRvn nl*RvRv RvRvn RvRv ne*RvRvn Rvn + Rvn RvRvn N*arv*Rvn*Rvn RvRvn Rvn + Rvn drv*RvRvn Rv uRv*Rv Rvn uRv*Rvn RvRv 2.0*uRv*RvRv RvRvn 2.0*uRv*RvRvn MnPt trPt*(1.0-G)5

MnPt McPt tmc*MnPt McPt umPt*McPt MnRo trRo*(1.0-G)*(1.0-GRv)3

MnRo McRo tmc*MnRo McRo umR*McRo MnRt trRt*(1.0-G) MnRt McRt tmc*MnRt McRt umR*McRt GRv binRv*(1.0-GRv)*RvRvn GRv unBinRv*GRv

Ptppn + Ron PtppRon ar*N*Ptppn*Ron Ptpn + Rtn PtpRtn ar*N*Ptpn*Rtn Ptppn + Rtn PtppRtn ar*N*Ptppn*Rtn PopCn + Ron PopCRon ar*N*PopCn*Ron PoppCn + Ron PoppCRon ar*N*PoppCn*Ron PopCn + Rtn PopCRtn ar*N*PopCn*Rtn PoppCn + Rtn PoppCRtn ar*N*PoppCn*Rtn PtpCn + Ron PtpCRon ar*N*PtpCn*Ron PtppCn + Ron PtppCRon ar*N*PtppCn*Ron PtpCn + Rtn PtpCRtn ar*N*PtpCn*Rtn PtppCn + Rtn PtppCRtn ar*N*PtppCn*Rtn PopRo Pop + Ro dr*PopRo PoppRo Popp + Ro dr*PoppRo PopRt Pop + Rt dr*PopRt PoppRt Popp + Rt dr*PoppRt PtpRo Ptp + Ro dr*PtpRo PtppRo Ptpp + Ro dr*PtppRo PtpRt Ptp + Rt dr*PtpRt PtppRt Ptpp + Rt dr*PtppRt PopCRo PopC + Ro dr*PopCRo PoppCRo PoppC + Ro dr*PoppCRo PopCRt PopC + Rt dr*PopCRt PoppCRt PoppC + Rt dr*PoppCRt PtpCRo PtpC + Ro dr*PtpCRo PtppCRo PtppC + Ro dr*PtppCRo PtpCRt PtpC + Rt dr*PtpCRt PtppCRt PtppC + Rt dr*PtppCRt PopRon Popn + Ron dr*PopRon PoppRon Poppn + Ron dr*PoppRon PopRtn Popn + Rtn dr*PopRtn PoppRtn Poppn + Rtn dr*PoppRtn PtpRon Ptpn + Ron dr*PtpRon PtppRon Ptppn + Ron dr*PtppRon PtpRtn Ptpn + Rtn dr*PtpRtn PtppRtn Ptppn + Rtn dr*PtppRtn PopCRon PopCn + Ron dr*PopCRon PoppCRon PoppCn + Ron dr*PoppCRon PopCRtn PopCn + Rtn dr*PopCRtn PoppCRtn PoppCn + Rtn dr*PoppCRtn PtpCRon PtpCn + Ron dr*PtpCRon PtppCRon PtppCn + Ron dr*PtppCRon PtpCRtn PtpCn + Rtn dr*PtpCRtn PtppCRtn PtppCn + Rtn dr*PtppCRtn Pop Popn nl*Pop Ptp Ptpn nl*Ptp PopC PopCn nl*PopC PtpC PtpCn nl*PtpC

Stochastic Simulation of Mutations

1/n1/2 Behavior

What is Temperature Compensation?

Chemical reaction speed up as temperature increases• Circadian rhythms have an approximately constant

period regardless of external temperature• Not Temperature Independence (Rhythms can entrain to temperature)

• Q10 = (Period at T)/(Period at T+10)

(Q10 of circadian rhythms is typically 0.95-1.05,

Q10 is typically 2-4 in chemical reactions)

Pittendrigh

Hastings and Sweeney (1957)

• Some parts of the circadian clock increase period with increasing temperature (TCE)

• The rest decrease the period with increasing temperature

• Temperature compensation is achieved by balancing these processes.

Ruoff’s Theory

• All biochemical oscillators have TCEs• Consider the period of the oscillator

• From kinetics and assumption

for some I

• Approximate numerically (assume constant) and choose to achieve the balance

0=⎟⎠

⎞⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=∑ Tk

kP

dTdP i

i

0>∂∂Tki 0<

∂∂

ikP

ik

P

∂∂

T

ki∂∂

Modeling the SCN

Coupled SCNQuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

http://dsm.bwh.harvard.edu/bmu/cpss

The Circadian Performance Simulation Software