Network Transformations of Switches and Oscillators

Transcription

1 Network Transformations of Switches and Oscillators Luca Cardelli Microsoft Research Cambridge

2 Motivation Building snthetic (DNA) oscillators DSD simulation 2

3 The Trammel of Archimedes A device to draw ellipses o Two interconnected switches. o Note that amplitude is kept constant b mechanical constraints. o When one switch is on (off) it flips the other switch on (off). When the other switch is on (off) it flips the first switch off (on) en.wikipedia.org/wiki/trammel_of_archimedes 3

4 The Shishi Odoshi A Japanese scarecrow (scare-deer) o Used b Bela Novak to illustrate the cell ccle switch. up dn em pt full tap empt + tap tap + full up + full full + dn full + dn dn + empt dn + empt empt + up To make it into a full trammel (dotted line), we could make the up position mechanicall open the tap (i.e. take up = tap) 4

5 Feedback speeds o fast (post-translational) The Cell Ccle o slow (transcriptional) Some feedbacks ma be missing Switches are asmmetric o One switch is usuall simpler than that, just causing a negative feedback o One switch is usuall more sophisticated than that, because of biochemical constraints 5

6 Outline Questions that nature has answered o Building good bistable sstems o Building switches (switchable bistable sstem) o Building switches with hsteress (needed for good oscillators) o Building limit-ccle oscillators o Building robust oscillators that resist parameter variations Engineering solutions to the same problems o Are the related? o In nature there are chemical constraints Not all reactions can be easil implemented Not all molecules can perform all functions we want them to From the point of view of network structure o Transforming a network and preserve some function o Program transformations 6

7 Switches

8 The Cell Ccle Switch Wh this network structure? Double positive feedback on Double negative feedback on No feedback on Wh on earth...?? 8

9 A Bad Algorithm Direct - competition o catalzes the transformation of into o catalzes the transformation of into This sstem is bistable, but o Convergence to a stable state is slow (a random walk). o An perturbation of a stable state can initiate a random walk to the other stable state. o With 100 molecules of and, convergence is quick, but with molecules, even at the same concentration, ou will wait for a long time. directive sample directive plot (); (); b() val rt = 10.0 new cat@rt:chan new cat@rt:chan let () = do!cat; () or?cat; () and () = do!cat; () or?cat; () run 100 of () run 100 of () 9

10 A Ver Good Algorithm Approimate Majorit o Decide which of two populations is in majorit A fundamental population protocol o Agents in a population start in state or state. o A pair of agents is chosen randoml at each step, the interact ("collide") and change state. o The whole population must eventuall agree on a majorit value (all or all ) with probabilit 1. 10

11 Properties [Angluin et al.] Parallel time is the number of steps divided b the number of agents. Hence the algorithm terminates with high probabilit in O(log n) steps per agent. N.B. this bound holds even if the, populations are initiall of equal size! 11

12 Chemical Implementation + + b + + b b + + b + + b Alternatives: b c This too is a bistable sstem, but: It converges slowl, b a random walk, hence O(n 2 ). It is unstable: an random fluctuation from an all- or all- state can send it (b a random walk) to the other state. This one gives no significant improvement over the above. 12

13 Majorit of > directive sample directive plot (); (); b() val r = 0.1 new b@r:chan new b@r:chan let () = do?; b() or!; () or!b; () and () = do!; () or?; b() or!b; () and b() = do?b; () or?b; () run of () run of () 2000k molecules 1100k 900k + + b + + b b + + b + + Eventuall: all no no b Gillespie simulation of the chemical reactions in SPiM. All rates are equal. 13

14 Majorit of = (!!) directive sample directive plot (); (); b() val r = 0.1 new b@r:chan new b@r:chan let () = do?; b() or!; () or!b; () and () = do!; () or?; b() or!b; () and b() = do?b; () or?b; () run of () run of () 2000k molecules Gillespie simulation of the chemical reactions in SPiM. + + b + + b b + + b + + Eventuall either: all all no no no b no b All rates are equal. The final majorit is robust (insensitive to possible noise) because a significant majorit alwas stas a majorit: N.B. a deterministic (ODE) simulation with = would not converge ever! 14

15 A Digression about Other Switches The AM network is an optimal switch in a computational sense. How does it compare with other switches? Let us first compare the kernel of AM without feedbacks (i.e. double phosphorlation ) with the Goldbeter-Koshland switch And then compare the full AM network with GK plus the same feedbacks as AM 15

16 Double-Phosphorlation Switch Ultrasensitive (but no hsteresis) + E E + b b + E E + + F F + b b + F F + AM without feedbacks E b kinase/ phosphatase E=3000 F E=100 E=100 E=0 E=0 directive sample directive plot (); (); b(); F();E() (*;Time() *) val a = new e@a:chan new f@a:chan new kille:chan new killf: chan new time:chan F=100 let E() = do!e; E() or?kille; () and F() = do!f; F() or?killf; () and () =?e; b() and () =?f; b() and b() = do?e; () or?f; () let clock(p:proc(int), t:float) = (* Produce one p(m) ever t sec with precision dt, with m incremented from 0 *) (val dt= run step(p, 0, t, dt, dt)) and step(p:proc(int), m:int, t:float, n:float, dt:float) = if n<=0.0 then (p(m) step(p,m+1,t,dt,dt)) else dela@dt/t; step(p,m,t,n-1.0,dt) let Time() =?time; () let schedule(n:int) = (Time(); if n<1000 then () else if n<4000 then E() else if n<8000 then!kille else () ) Initiall 10000, no, 100 F, no E. E growing from 0 (t=100) to 3000 (t=400) then back to 0 (t=800) run of () run 100 of F() run clock(schedule,10.0) 16

17 The Goldbeter-Koshland Switch Ultrasensitive (but no hsteresis) S + E d a SE k P + E P + F d a PF k S + F S E enzmatic P F S P S E=2000 E=1000 E=1000 E=0 E=0 directive sample directive plot S();P();F();E();ES();FP() (*;Time() *) val a = 1.0 val d = 1.0 val k = 1.0 new es@a:chan new fp@a:chan new kille:chan new killf: chan new time:chan F=1000 (* S + E <-> SE -> P + E P + F <-> PF -> S + F *) let S() =?es; () and E() = do!es; ES() or?kille; () and ES() = do dela@d; (S() E()) or dela@k; (P() E()) and P() =?fp; () and F() = do!fp; FP() or?killf; () and FP() = do dela@d; (P() F()) or dela@k; (S() F()) let clock(p:proc(int), t:float) = (* Produce one p(m) ever t sec with precision dt, with m incremented from 0 *) (val dt= run step(p, 0, t, dt, dt)) and step(p:proc(int), m:int, t:float, n:float, dt:float) = if n<=0.0 then (p(m) step(p,m+1,t,dt,dt)) else dela@dt/t; step(p,m,t,n-1.0,dt) let Time() =?time; () let schedule(n:int) = (Time(); if n<1000 then () else if n<3000 then E() else if n<6000 then!kille else () ) run of S() run 1000 of F() run clock(schedule,0.1) Initiall S, no P, 1000 F, no E. E growing from 0 (t=100) to 2000 (t=300) then back to 0 (t=500) The first switch happens at t=200, the second at t=400. E/F ratio can be lower: GK is a better more sensitive switch. 17

18 Can GK do majorit switching? directive sample GK in "AM configuration" enzmatic directive plot S();P();PS();SP() val a = 1.0 val d = 1.0 val k = 1.0 new sp@a:chan new ps@a:chan let S() = do?ps; () or!sp; SP() and PS() = do dela@d; (P() S()) or dela@k; (P() P()) and P() = do?sp; () or!ps; PS() and SP() = do dela@d; (S() P()) or dela@k; (S() S()) S + P d a PS k P + P P + S d a SP k S + S S P run of S() run of P() S > P S = P GK in "AM configuration" does not compute a majorit. - The initial minorit goes down to 0 - The initial majorit goes down to maj t=0 min t=0 - When maj t=0 ~ min t=0 the sstem cannot decide. 18

19 Problem ma be that the feedbacks put GK outside of zero-order regime. Hence, should check to see if GK works in the case of + w d a w k + w + r d a r k + r w s z p + d a p k r + r + t d a rt k p + t p r w + s d a ws k z + s z + d a z k w + t 19

20 Double phosphorlation motif is ke AM enzmatic autocataltic GK mass action b b = c It split-am b c is not just a non-linearit of the - transition mechanism that matters: it is the 'double phosphorlation' network structure of AM, with a common 'undecided' state. 20

21 Chemical Constraints The AM circuit is chemicall demanding o It requires molecules to be net to molecules beacause the interact directl o It requires both and to be catalsts, and in fact autocatalsts, and in fact each-other s autocatalst! b 21

22 Network Transformations An eample of relaing those constraints o This circuit works just as well as the original, but it no longer requires to be net to. The no longer interact directl. Instead, the interact through an additional 0-0 equilibrium. 0 b = val r = 10.0 new 0@r:chan new 0@r:chan new 00@r:chan new 00@r:chan new b@r:chan new b@r:chan let () = do?0; b() or!b; () or!00; () and () = do?0; b() or!b; () or!00; () directive sample directive plot (); (); b() 0 and 0() = do!0(); 0() or?00; 0() and b() = do?b; () or?b; () and 0() = do!0(); 0() or?00; 0() run 5000 of () run 5000 of 0() run 5000 of () run 5000 of 0() cf. 22

23 Network Transformations Another eample of relaing constraints o Build an Approimate Majorit network that requires onl to be a catalst. How? o Enter the Cell Ccle switches 23

24 Some Notation Cataltic reaction z + z z + = Double kinase-phosphatase reactions z z = b w w 24

25 Zero-Input Switches Zero-input switch = majorit circuit: just working off the initial conditions, with no other inputs. Step 1: the original AM Network = b 25

26 Zero-Input Switches Step 2: remove auto-catalsis o B introducing intermediate species w, r. o Here w breaks the auto-catalsis, and r breaks the auto-catalsis, while preserving the feedbacks. o w and r need to rela back (to z and t) when the are not catalzed: s and t provide the back pressure. s w z p r t 26

27 can simplif? p t r w s z (it appears just sligthl noisier/slower) 27

28 no, it gets stuck! Equal-size initial conditions directive sample directive plot (); (); b() (*; z(); w(); r(); s(); t(); p(); q() *) val rt = 10.0 new cat@rt:chan new cat@rt:chan new wcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan let () = do!cat; () or?wcat; b() and () = do!cat; () or?rcat; b() and b() = do?rcat; () or?wcat; () and z() =?cat; w() and r() = do!rcat; r() or?tcat; p() and s() =!scat(); s() and w() = do!wcat; w() or?scat; z() and t() =!tcat; t() and p() =?cat; r() run of s() run of t() run of w() run of z() run of p() run of r() run of () run of b() run of () 28

29 Step 3: transform a double-positive loop on into a double-negative loop on. o Instead of (activel) activating itself through w, we have z activating (which is passive). To counteract, now has to switch from inhibiting to inhibiting z. p t Zero-Input Switches r w s z So that no longer catalzes anthing o All species have one active and one inactive form w p s t z r 29

30 Zero-Input Switches Still an AM circuit s s w p t z r = (The equal-likelihood outcome here is around 4500 vs 5500, and can be adjusted b s/t ratio) w p t u q z r b directive sample directive plot (); (); b() (* z(); w(); r(); s(); t(); p(); q() *) val rt = 10.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan let () = do!cat; () or?zcat; b() and () =?rcat; b() and b() = do?rcat; () or?zcat; () and z() = do!zcat; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() All rates are equal. run 1000 of s() run 1000 of t() run of p() run of z() run 4500 of () run 5500 of () 30

31 Equal-size initial conditions s directive sample directive plot (); (); b() (* z(); w(); r(); s(); t(); p(); q() *) val rt = 10.0 w u z new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan p t q r b let () = do!cat; () or?zcat; b() and () =?rcat; b() and b() = do?rcat; () or?zcat; () and z() = do!zcat; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() run of s() run of t() All initial species = Probabilit of win seems to be ==50% Note that when wins, the sstem does not terminate because has active competition from s. run of w() run of u() run of z() run of p() run of q() run of r() run of () run of b() run of () 31

32 AM Equal-size initial conditions directive sample directive plot (); (); b() val r = 0.1 new b@r:chan new b@r:chan let () = do?; b() or!; () or!b; () and () = do!; () or?; b() or!b; () and b() = do?b; () or?b; () run of () run of b() run of () All initial species =

33 The Cell Ccle Switch w s z r w p s z r t t p (Some of the bistable states can be enzmatic rather than AM.) 33

34 can simplif? s w z directive p r t It works better than the original?!? sample directive plot (); (); b() (* z(); w(); r(); s(); t(); p(); q() *) val rt = 10.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan let () = do!cat; () or?zcat; b() and () =?rcat; b() and b() = do?rcat; () or?zcat; () and z() = do!zcat; z() or?cat; w() and r() = do!rcat; r() or?tcat; p() and s() =!scat(); s() and w() =?scat; z() and t() =!tcat; t() and p() =?cat; r() run 1000 of s() run 1000 of t() run of p() run of z() run 4500 of () run 5500 of () 34

35 no, it gets stuck! Equal-size initial conditions directive sample directive plot (); (); b() (* z(); w(); r(); s(); t(); p(); q() *) val rt = 10.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan let () = do!cat; () or?zcat; b() and () =?rcat; b() and b() = do?rcat; () or?zcat; () and z() = do!zcat; z() or?cat; w() and r() = do!rcat; r() or?tcat; p() and s() =!scat(); s() and w() =?scat; z() and t() =!tcat; t() and p() =?cat; r() run of s() run of t() run of w() run of z() run of p() run of r() run of () run of b() run of () 35

36 More Zero-Input Switches Other designs o A version with no eternal bias (s,t) where is still non-cataltic and and z are self-cataltic. o Both and z have an inactive form, and w, although the both are double catalsts. directive sample directive plot (); (); b() (*; z(); w(); s() *) val rt = 10.0 w z new cat@rt:chan new zcat@rt:chan let () = do!cat; () or?zcat; b() and () =?cat; b() and b() = do?cat; () or?zcat; () and z() = do!zcat; z() or?cat; u() and w() =?zcat; u() and u() = do?zcat; z() or?cat; w() run 5000 of z() run 5000 of w() run 5000 of () run 5000 of () 36

37 Equal-size intial conditions directive sample directive plot (); (); b() (*; z(); w(); s() *) val rt = 10.0 new cat@rt:chan new zcat@rt:chan let () = do!cat; () or?zcat; b() and () =?cat; b() and b() = do?cat; () or?zcat; () and z() = do!zcat; z() or?cat; u() and w() =?zcat; u() and u() = do?zcat; z() or?cat; w() run of z() run of w() run of () run of () run of b() run of u() 37

38 One-Input Switches Ultrasensitivit (none) and hsteresis (none) in trivial majorit s r s r=5.0 s=1000 directive sample directive plot (); (); s(); s() (* b(); *) val rt = 10.0 val r = 5.0 new cat@r:chan new cat@rt:chan new scat@rt:chan new skill:chan new scat@rt:chan new skill:chan let () = do!cat; () or?cat; () or?scat; () and () = do!cat; () or?cat; () or?scat; () and s() = do!scat; s() or?skill; () and s() = do!scat; s() or?skill; () run of () run 1000 of s() let clock(p:proc(int), t:float) = (* Produce one p(m) ever t sec with precision dt, with m incremented from 0 *) (val dt= run step(p, 0, t, dt, dt)) and step(p:proc(int), m:int, t:float, n:float, dt:float) = if n<=0.0 then (p(m) step(p,m+1,t,dt,dt)) else dela@dt/t; step(p,m,t,n-1.0,dt) let schedule(n:int) = if n < then s() else if n < then!skill;() else () run clock(schedule, )

39 One-Input Switches Hsteresis in unbiased AM-like switches. o All rates are equal; constant amount of s is sufficient for switch-back. s directive sample directive plot (); (); s(); s() (* b(); *) val rt = 10.0 val r = 10.0 new cat@r:chan new cat@rt:chan new scat@rt:chan new skill:chan new scat@rt:chan new skill:chan s let () = do!cat; () or?cat; b() or?scat; b() and () = do!cat; () or?cat; b() or?scat; b() and b() = do?cat; () or?scat; () or?cat; () or?scat; () and s() = do!scat; s() or?skill; () and s() = do!scat; s() or?skill; () run of () run 2000 of s() let clock(p:proc(int), t:float) = (* Produce one p(m) ever t sec with precision dt, with m incremented from 0 *) (val dt= run step(p, 0, t, dt, dt)) and step(p:proc(int), m:int, t:float, n:float, dt:float) = if n<=0.0 then (p(m) step(p,m+1,t,dt,dt)) else dela@dt/t; step(p,m,t,n-1.0,dt) let schedule(n:int) = if n < then s() else if n < then!skill;() else () run clock(schedule, )

40 One-Input Switches Hsteresis in biased AM-like switches s r s r=5.0 s=1000 directive sample directive plot (); (); s(); s() (* b(); *) val rt = 10.0 val r = 5.0 new cat@r:chan new cat@rt:chan new scat@rt:chan new skill:chan new scat@rt:chan new skill:chan let () = do!cat; () or?cat; b() or?scat; b() and () = do!cat; () or?cat; b() or?scat; b() and b() = do?cat; () or?scat; () or?cat; () or?scat; () r=0.1 s=100 and s() = do!scat; s() or?skill; () and s() = do!scat; s() or?skill; () run of () run 1000 of s() let clock(p:proc(int), t:float) = (* Produce one p(m) ever t sec with precision dt, with m incremented from 0 *) (val dt= run step(p, 0, t, dt, dt)) and step(p:proc(int), m:int, t:float, n:float, dt:float) = if n<=0.0 then (p(m) step(p,m+1,t,dt,dt)) else dela@dt/t; step(p,m,t,n-1.0,dt) let schedule(n:int) = if n < then s() else if n < then!skill;() else () run clock(schedule, )

41 One-Input Switches Hsteresis vs. feedbacks in cell ccle switch reference sstem all rates are equal. p=r=0 Without pos-pos feedback 2s, ½t, ½s w=z=0 Without neg-neg feedback w p s t z r s s initial conditions: 1000 of 1000 of z 1000 of p 1000 of t 200 of s 100 of s directive sample directive plot (); (); s(); s() (* ; b(); z(); w(); u(); s(); p(); q(); r(); t() *) val rt = new cat@rt:chan new zcat@rt:chan new scat@rt:chan new tcat@rt:chan new rcat@rt:chan new scat@rt:chan new skill:chan new scat@rt:channew skill:chan let () = do!cat; () or?zcat; b() or?scat; b() and () = do?rcat; b() or?scat; b() and b() = do?rcat; () or?scat; () or?zcat; () or?scat; () and z() = do!zcat; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and w() =?scat; u() and u() = do?cat; w() or?scat; z() and s() =!scat; s() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() and s() = do!scat; s() or?skill; () and s() = do!scat; s() or?skill; () run 1000 of () run 1000 of z() run 1000 of p() run 200 of s() run 1000 of t() run 100 of s() let clock(ps:proc(int), tm:float) = (* Produce one ps(m) ever tm sec with precision dt, with m incremented from 0 *) (val dt= run step(ps, 0, tm, dt, dt)) and step(ps:proc(int), m:int, tm:float, n:float, dt:float) = if n<=0.0 then (ps(m) step(ps,m+1,tm,dt,dt)) else dela@dt/tm; step(ps,m,tm,n-1.0,dt) let schedule(n:int) = if n < 1000 then s() else if n < 2000 then!skill;() else () run clock(schedule, ) varing s 0 to 1000 to 0 41

42 s w u z s b p t q r s 42

43 Ferrell oscillator Novak-Tson oscillator l h v f s w p t z r c h k v f i s w p t z r c 43

44 Two-input Switches I had rediscovered (but not analzed so well) the same sstem, while looking for a memor circuit. The point here was not computing majorit, but switching easil and quickl and stabl. s s A + B -> B + C B + A -> A + C C + A -> A + A C + B -> B + B 44

45 Oscillators

46 The Trammel of Archimedes A device to draw ellipses o Two interconnected switches. o When one switch is on (off) it flips the other switch on (off). When the other switch is on (off) it flips the first switch off (on) en.wikipedia.org/wiki/trammel_of_archimedes 46

47 The Shishi Odoshi A Japanese scarecrow (scare-deer) o Used b Bela Novak to illustrate the cell ccle switch. up dn em pt full tap empt + tap tap + full up + full full + dn full + dn dn + empt dn + empt empt + up To make it into a full trammel (dotted line), we could make the up position mechanicall open the tap (i.e. take up = tap) 47

48 The 2AM Limit-Ccle Oscillator Two AM switches in a Trammel pattern directive sample directive plot 1(); 1(); b1(); 2(); 2(); b2() val r = 10.0 new 1cat@r:chan new 1cat@r:chan new 2cat@r:chan new 2cat@r:chan val s = 8.0 new 1cat2@s:chan new 1cat2@s:chan new 2cat1@s:chan new 2cat1@s:chan The red reactions need to be slower (even slightl) than the black reactions, but otherwise the oscillation is robust. Oscillation stops at 10 vs. 10 and 1 vs. 10. Here the rates are 8(red) vs 10(black) top, and 2 vs 10, bottom. (Simple limit-ccle oscillators in the literature have ver critical rate ranges.) let 1() = do!1cat; 1() or!1cat2; 1() or?1cat; b1() or?2cat1; b1() and 1() = do!1cat; 1() or!1cat2; 1() or?1cat; b1() or?2cat1; b1() and b1() = do?1cat; 1() or?2cat1; 1() or?1cat; 1() or?2cat1; 1() let 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?1cat2; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?1cat2; b2() and b2() = do?2cat; 2() or?1cat2; 2() or?2cat; 2() or?1cat2; 2() run 3666 of 1() run 3000 of 1() run 3333 of b1() run 3333 of 2() run 3333 of 2() run 3333 of b2() 48

49 The Switch Module inxy caty outx outy catx inyx This is flipped! outy2 outy1 2 2 outx2 outx

50 Replacing Switch Modules Replace inxy outx outy inyx With w outy s z inxy Or inxy outx w s outy z Etc.. p r inyx p r t outx inyx t 50

51 Modified Oscillator 1 2cat zcat1 directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val rt = 10.0 val st = 8.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan new 2cat@rt:chan new 2cat@rt:chan new zcat1@st:chan new rcat1@st:chan new 2cat1@st:chan new 2cat1@st:chan 2cat cat 2cat1 s w z let () = do!cat; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?2cat1; b() and b() = do?rcat; () or?2cat1; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or!zcat1; z() or?cat; u() and r() = do!rcat; r() or!rcat1; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?rcat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?zcat1; b2() and b2() = do?2cat; 2() or?zcat1; 2() or?2cat; 2() or?rcat1; 2() run 3000 of s() run 3000 of t() run of p() run of z() run 1000 of () run 6666 of () run 2333 of b() run 3333 of 2() run 3333 of 2() run 3333 of b2() rcat1 p t r directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val rt = 10.0 val st = 8.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan new 2cat@rt:chan new 2cat@rt:chan new zcat1@st:chan new rcat1@st:chan new 2cat1@st:chan new 2cat1@st:chan let () = do!cat; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?2cat1; b() and b() = do?rcat; () or?2cat1; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or!zcat1; z() or?cat; u() and r() = do!rcat; r() or!rcat1; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?rcat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?zcat1; b2() and b2() = do?2cat; 2() or?zcat1; 2() or?2cat; 2() or?rcat1; 2() run 3333 of s() run 3333 of t() run 3333 of p() run 3333 of q() run 3333 of r() run 3333 of z() run 3333 of u() run 3333 of w() run 3333 of () run 3333 of b() run 3333 of () run 3333 of 2() run 3333 of 2() run 3333 of b2() 51

52 Modified Oscillator 2 2cat zcat1 outy directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val rt = 10.0 val st = 8.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan new 2cat@rt:chan new 2cat@rt:chan 2cat cat cat1 outx 2cat1 s w p z r inxy new zcat1@st:chan new cat1@st:chan new 2cat1@st:chan new 2cat1@st:chan let () = do!cat; () or!cat1; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?2cat1; b() and b() = do?rcat; () or?2cat1; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or!zcat1; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?cat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?zcat1; b2() and b2() = do?2cat; 2() or?zcat1; 2() or?2cat; 2() or?cat1; 2() run 3000 of s() run 3000 of t() run of p() run of z() run 1000 of () run 6666 of () run 2333 of b() run 3333 of 2() run 3333 of 2() run 3333 of b2() directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val rt = 10.0 val st = 8.0 t inyx new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan new 2cat@rt:chan new 2cat@rt:chan new zcat1@st:chan new cat1@st:chan new 2cat1@st:chan new 2cat1@st:chan let () = do!cat; () or!cat1; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?2cat1; b() and b() = do?rcat; () or?2cat1; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or!zcat1; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?cat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?zcat1; b2() and b2() = do?2cat; 2() or?zcat1; 2() or?2cat; 2() or?cat1; 2() run 3333 of s() run 3333 of t() run 3333 of p() run 3333 of q() run 3333 of r() run 3333 of z() run 3333 of u() run 3333 of w() run 3333 of () run 3333 of b() run 3333 of () run 3333 of 2() run of 2() run 3333 of b2()

53 Modified Oscillator 3 2cat s scat 0.7 directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val ri = 1.0 val re = 0.7 new cat@ri:chan new zcat@ri:chan new rcat@ri:chan new scat@ri:chan new tcat@ri:chan new 2cat@ri:chan new 2cat@ri:chan 2 2 2cat cat1 outx 2cat1 s w p z r inxy new cat1@re:chan new 2cat1@re:chan new scat@re:chan new scat@re:chan let s() =!scat; s() and s() =!scat; s() let () = do!cat; () or!cat1; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?scat; b() and b() = do?rcat; () or?scat; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or?cat; u() and r() = do!rcat; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or?2cat; b2() or?cat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?scat; b2() and b2() = do?2cat; 2() or?scat; 2() or?2cat; 2() or?cat1; 2() run of s() run of t() run of p() run of q() run of r() run of z() run of u() run of w() run of () run of b() run of () run of 2() run of 2() run of b2() run of s() run of s() t scat s

54 Constant-Influ Oscillator 2 2 As in the Shishi Odoshi (and the cell ccle) 1 1 directive sample directive plot 1(); 1(); b1(); 2(); 2(); b2() val r = 10.0 new 1cat@r:chan new 1cat@r:chan new 2cat@r:chan new 2cat@r:chan val s = 8.0 new 1cat2@s:chan new 1cat2@s:chan new 2cat1@s:chan new 2cat1@s:chan let 1() = do!1cat; 1() or!1cat2; 1() or?1cat; b1() or?2cat1; b1() and 1() = do!1cat; 1() or!1cat2; 1() or?1cat; b1() or?2cat1; b1() and b1() = do?1cat; 1() or?2cat1; 1() or?1cat; 1() or?2cat1; 1() let 2() = do!2cat; 2() or?2cat; b2() or?1cat2; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?1cat2; b2() and b2() = do?2cat; 2() or?1cat2; 2() or?2cat; 2() or?1cat2; 2() let c() =!2cat1; c() run 3666 of 1() run 3000 of 1() run 3333 of b1() run 3333 of 2() run 3333 of 2() run 3333 of b2() run 2000 of c() 54

55 Constant influ 2cat zcat cat1 2cat1 2cat s w z rcat1 p t r Still working fine with the replaced switch. directive sample directive plot (); (); b(); 2(); 2(); b2(); z(); r() val rt = 10.0 val st = 8.0 new cat@rt:chan new zcat@rt:chan new rcat@rt:chan new scat@rt:chan new tcat@rt:chan new 2cat@rt:chan new 2cat@rt:chan new zcat1@st:chan new rcat1@st:chan new 2cat1@st:chan new 2cat1@st:chan let () = do!cat; () or?zcat; b() or?2cat1; b() and () = do?rcat; b() or?2cat1; b() and b() = do?rcat; () or?2cat1; () or?zcat; () or?2cat1; () and z() = do!zcat; z() or!zcat1; z() or?cat; u() and r() = do!rcat; r() or!rcat1; r() or?tcat; q() and s() =!scat(); s() and w() =?scat; u() and u() = do?scat; z() or?cat; w() and t() =!tcat; t() and p() =?cat; q() and q() = do?cat; r() or?tcat; p() let 2() = do!2cat; 2() or?2cat; b2() or?rcat1; b2() and 2() = do!2cat; 2() or!2cat1; 2() or?2cat; b2() or?zcat1; b2() and b2() = do?2cat; 2() or?zcat1; 2() or?2cat; 2() or?rcat1; 2() let c() =!2cat1; c() run 3000 of s() run 4200 of t() run of p() run of z() run 1000 of () run 6666 of () run 2333 of b() run 3333 of 2() run 3333 of 2() run 3333 of b2() run 3000 of c() 55

56 The Novak-Tson Oscillator First switch o Is the transformed AM switch in one-input configuration (driven b constant influ of cclin). Second switch o Is a simple two-stage switch working as a dela (the first switch is so good in terms of hsteresis that the second switch is not ver critical for oscillation). Connection o The feedback from second to first switch is a bit comple, since both and are repressed b degrading cclin. And there are more details still. h k v l f i s w p t z r MPF CAK cclin cdc2 56

57 One of Ferrell s Oscillators Second switch o Replaced b a one-stage switch. The oscillation still works, but is it harder to obtain (parameter tuning). h v f w s z MPF cdc2 p r CAK cclin t 57

58 Conclusions

59 Conclusions A vast literature on cell ccle switching o Ferrell et.al., Novak-Tson et.al., etc. Mostl ODE based analsis, plus noise o Man bistable transitions have different implementations in different cell ccle phases and organisms (phosphorlation, enzmes, snthesis/degradation, etc.) o We focused on a mechanism that can onl be seen stochasticall (quick majorit switching with =) A range of network transformation o Can eplain the structure of some natural networks o From some non-trivial underling algorithms o Discovering the transformation can elucidate the structure and function of the networks o But how can we sa that these transformations preserve (essential) behavior? 59

60 Acknowledgements David Soloveichik Attila Csikasz-Nag 60