Review Turing-Gierer-Meinhardt models Local excitation, global inhibition
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1 viw Turing-Girr-Minhart mol Local xcitation global inhibition a a ra ka γ aa t i i i kia γ ii i t x a a x a: concntration activator i: concntration inhibitor t: tim x: poition r a : baal activator ynthi rat k a k i : rat contant for ynthi γ a γ i : cay rat a i : iffuion contant variabl contant (paramtr
2 x i i a k t i x a a i a k r t a i i i a a a a γ γ choo imnionl variabl normaliz variabl ( P Q only on fix point inc both an >0 ( homognou olution 0 / / t
3 homognou olution / / t 0 3
4 tability of homognou olution Q Q Q Q ( ( trac < 0 t > 0 0 > < Q Q or in gnral ral part of ignvalu > 0 ( ( ( ( inhomognou olution:
5 inhomognou olution ( ( ( ( ( 5
6 ( ( P Q Q ( ( ( ( co( ˆ( ( co( ˆ( ( l l trial olution: 6
7 7 co( ˆ( ( co( ˆ( ( l l ( ( ( ( (
8 P Q Q ˆ ˆ ( ˆ ˆ ( ˆ ˆ l l co( ˆ( ( co( ˆ( ( l l 0 0 < > l l l l P Q Q P Q tability inhomognou olution > P Q 8
9 homognou tability: tability againt patial itrubanc: Q > Q P > ( if P < ( i < a ytm i alway tabl againt any prturbation both patial an tmporal 9
10 homognouly tabl: rlax back to prviou valu aftr mall uniform iturbanc rlax back to tabl againt patial aftr mall patial iturbanc: iturbanc 0
11 ntroucing th molcul: - FtZ function: mbly of a polymric ring of th tubulin-lik GTPa FtZ (Z ring. Th Z-ring i localiz to th cntr by th action of th MinC Min an Min protin. - MinC inhibit th initiation of th Z ring. MinC colocaliz with Min. n wil-typ (WT cll MinC/ form a polar pattrn that ocillat btwn th pol kping th cntr fr for initiation of cll iviion. Thu virtually all of MinC/ ynamically ambl on th mmbran in th hap of a tt tub covring th mmbran from on pol up to approximatly micll.
12 Mot of Min accumulat at th rim of thi tub in th hap of a ring (th ring. Th rim of th MinC/ tub an aociat ring mov from a cntral poition to th cll pol until both th tub an ring vanih. Manwhil a nw MinC/ tub an aociat ring form in th oppoit cll half an th proc rpat rulting in a pol-to-pol ocillation cycl of th iviion inhibitor. full cycl tak about 50. mag rmov u to copyright coniration. 3
13 How o thi work? moling ffort: Minhart an Bor PNS 98 0 (00; Howar t al. Phy. v. Lt (00; Kru Biophy. J (00; Huang Mir an Wingrn PNS 00 7 (003.
14 Summary of main function of protin: FtZ polymriz in a contractil Z-ring that initiat ptum formation MinC inhibit formation of Z-ring Min mmbran aociat protin that rcruit minc an min to mmbran Min jct minc/min from mmbran into cytoplam 5
15 Howar t al. mol (PL min min min min 3 mmbran cytoplam in wor: - firt orr raction for own pci - inhibit mmbran aociation of (MM - nhanc mmbran iociation of (linar - nhanc mmbran aociation of (rcruitmnt linar - inhibit mmbran iociation of (MM - an o not iffu - an iffu 6
16 Howar t al. mol (PL min min mmbran cytoplam aociation of cytoplamic min with mmbran i inhibit by min in mmbran MM tak car of ingularity a min go to zro. biological intrprtation: min 3 min in mmbran patially block mmbran for min imilar to minc blocking FtZ aociation with mmbran min 7
17 Howar t al. mol (PL min min mmbran cytoplam iociation of mmbran min i timulat by min in mmbran aftr min i jct min tay in mmbran biological intrprtation: min 3 bining of min to min lowr affinity of min with mmbran but mmbran affinity of min rmain unchang min 8
18 Howar t al. mol (PL min min mmbran cytoplam iociation of mmbran min i inhibit by min in cytoplam MM tak car of ingularity biological intrprtation: min? 3 min 9
19 Howar t al. mol (PL min min mmbran cytoplam aociation of cytoplamic min with mmbran i timulat by min in cytoplam aftr livry of min to th mmbran min iv back in th cytoplam min biological intrprtation: min 3 min-min complx ha high affinity to mmbran inc th iffuion of thi complx on t appar in th mol it houl b vry fat. 0
20 ytm of quation: t x t t x t 3 3
21 tability analyi. fin fix point (.g. numrically: how_homog.m t x 0 0 iffrnt ranom initial conition rlax to am fix point rult: on fix point:
22 . fin tability matrix (Jacobian ( 0 ( ( 0 ( 0 3
23 3. tt tability of fluctuation aroun homognou olution δ( x t ˆ( tco( qx δ( x t ˆ( tco( qx δ( x t δ( x t ˆ ( tco( qx ˆ( tco( qx δ(xt x
24 3. tt tability of fluctuation aroun homognou olution q q ( 0 ( ( 0 ( 0 ˆ 5
25 . - trmin ignvalu of tability matrix - fin ral part of ignvalu - plot th largt a a function of q. (.g. how_ig.m q.5 (µm - Max(al(ignvalu / q λ π/q. µm q.3 (µm - λ π/q.7 µm 6
26 Howar t al.: ult mag rmov u to copyright coniration. 7
27 Huang Mir an Wingrn PNS 00 7 (003. main iffrnc: - TP cycl - vru 3 (projct on mag rmov u to copyright coniration. 8
28 mag rmov u to copyright coniration. : mmbran boun min:tp complx : mmbran boun min:min:tp complx :P : concntration cytoplamic min boun to P :TP :concntration cytoplamic min boun to TP : concntration cytoplamic min only min-tp can aociat with mmbran min only bin min-tp oligomr in mmbran 9 only min-min-tpcomplx can iociat from mmbran
29 action : min-tp bin both linarly an autocatalytically to min-tp in mmbran mag rmov u to copyright coniration. min form polymr in mmbran t t t :P :TP t t x x : x : P TP P TP P TP :P :P [ ( ] : TP [ ( ] : TP 30
30 action : min bin min-tp in mmbran ~ [min]*[min] mag rmov u to copyright coniration. t t t :P :TP t t x x : x : P TP P TP P TP :P :P [ ( ] : TP [ ( ] : TP 3
31 action 3: min-min-tp complx iaociat from mmbran hyrolyzing TP ~ [min] mag rmov u to copyright coniration. t t t :P :TP t t x x : x : P TP P TP P TP :P :P [ ( ] : TP [ ( ] : TP 3
32 action : charging of min in cytoplam from P to TP boun mag rmov u to copyright coniration. t t t :P :TP t t x x : x : P TP P TP P TP :P :P [ ( ] : TP [ ( ] : TP 33
33 t t t t x P ( TP : x : P : TP : TP : ( ( TP t x ( t x : P : P P : P 3
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