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1 ( [1, 15]) 3 2 6 2.1 -.... 6 2.1.1............ 7 2.1.2... 7 2.1.3................. 8 2.1.4 (Enzyme Kinetics)......... 9 2.1.5 (quasi-steady state approximation) 12 2.1.6..... 14 2.1.7 (equilibrium approxmation)... 15 2.2................. 16 2.2.1 (competitive inhibition)... 17 2.2.2 (noncompetitive inhibition)... 18 2.2.3 (substrate inhibition)... 20 2.2.4 (uncompetitive inhibition)... 21 2.2.5... 22 2.2.6 (Hill equation)... 26 3 29 3.1 [6, 7, 15 17]... 31 3.1.1 1 - - [15]... 34 3.1.2 2 - - [8]... 38 3.1.3 2... 41 3.1.4... 41 3.1.5... 42 3.1.6.................... 45 3.1.7... 46 3.1.8 Hopf... 52 3.2 2... 55 3.2.1 Ca 2+ [13].................... 55 3.2.2 (ubstrate-depletion oscillation) [6] 58 1
3.2.3 - (Activetor-inhibiotr oscillation) [6] 64 3.3 3... 67 3.4............ 70 4 75 4.1................. 75 4.2 -Hodgkin-Huxley -......... 75 4.2.1... 75 4.2.2... 77 4.2.3 Hodgkin-Huxley............ 79 4.2.4 (Voltage-clamp method)... 81 4.2.5..................... 83 4.2.6 Hodgkin-Huxley... 86 4.3 4 Hodgkin-Huxley 2 -FitzHugh- Nagumo -........................ 90 4.3.1 Hodgkin-Huxley........ 90 4.3.2 FitzHugh-Nagumo... 92 2
1 ( [1,15]) ( ) 4 0 1 1 1+4 1=5 2 5+4 5 k n k n k = n k 1 +4n k 1 1 1 α n k n k 1 = αn k 1 1 t n t = n k n k 1 = αn k 1 t 0 dn dt = αn (1.1) n 0 t =0 n(0) = n 0 n = n 0 e αt (1.2) 3
1 (1.1) dn dt =(a bn)n (1.3) α = a bn a b (logistic equation) (1.1) n(0) = n 0 a bn 0 0 n(t) = n max 1+( nmax n 0 1)e at (1.4) n max = a/b (1.4) t n = n max n 0 n max (1.3) n dn dt =0 (a bn)n =0 n =0 n = n max n 0 (1.4) n = n max n max n 0 =0 (1.4) n =0 0 1 Malthus: 4
0 ( 1-1) (1.1), (1.3) (1.2), (1.4) n(0) = n 0 (1) 0 <n 0 < a 2b, (2) a 2b <n 0 < a b, (3) n 0 > a b t n n (1.3) (1.1) 5
2 2.1 - (AB)* エネルギー A+B Ea C 分子の状態 2.1: - (Michaelis-Menten equation) 1 1 [2] [3] 4 chapter 1 6
2.1.1 A+B k C (2.1) 2.1 A+B C (AB) (activation energy)e a 2 k = A exp( E a /RT ) (2.2) k (rate constant) (arrhenius equation) T A (frequency factor) 3 E a (2.2) ln k =lna E a R 1 T (2.3) ( ) A E a 2.1.2 (2.1) A, B [A], [B] C [C] k [A], [B] ( (law of mass action) ) 4 d[c] = k[a][b] (2.4) dt 2 [4] 1 3 3 [4] T (2.2) T 1/2 A 4 7
( 2-1) P (1) A+B+C k P (2) (3) (4) A+B k P+C A k P+C 2A k P 2.1.3 (reversible reaction) A+B k + k C (2.5) A d[a] dt =( ) (2.6) (equilibrium state) A 5 d[a] dt =0 (2.6) A,B,C [A] eq, [B] eq, [C] eq K k + k = [C] eq [A] eq [B] eq (2.7) 5 (steady state) 8
K (2.2) K ( 2-2) A P [A] eq (1) (2) (3) (4) A+B+C k + k P A+B k + k P+C A k + k P+C 2A k + k P ( 2-3) (2.2) 2.7?? 2.1.4 (Enzyme Kinetics) (ubstrate) (Enzyme)E (catalyst) (Product)P 2.2 [] ( [E] ) V( P ) 9
2.2:... (1) [] [] (2) [], E P +E k P+E V = d[p] dt = k[][e] [] [E] Michaelis Menten(1913) ( (Complex))C +E k +1 C k 2 P+E (2.8) k 1 [] (2.8) [] E 10
E - (Michaelis-Menten equation)(2.18) 2.1.7 Briggs Haldane (1925) (2.8) P 6 [] 0 [E] 0 C E 7 E C P (quasi-steady state) 8 2.1.5 - s =[],e=[e],c=[c],p=[p] ds dt = (2.9) de dt = (2.10) dc dt = (2.11) dp dt = (2.12) ( 2-4) V s 9 V = dp dt 6 7 2.1.6 [C] [] [C] [] 8 9 s ( s 0 ) 11
(2.12) c (2.10) (2.11) de dt + dc =0 dt (2.13) e + c = const e 0 (2.14) e 0 e c s 2 2.1.5 (quasi-steady state approximation) Briggs Haldane C C (2.11) dc dt =0 (2.15) (2.14) e c = e 0s s + K m (2.16) K m k 1 + k 2 k +1 (2.17) K m (Michaelis constant) V (2.12) (2.16) V = dp dt = k 2c = V maxs s + K m (2.18) V max = k 2 e 0 (2.18) - (Michaelis-Menten equation) 2.3(a) 2.1.4 V max s K m ( ) K m 12
Lineweaver-Burk ( 2.3(b)) V max, K m 10 1/V = s + K m V max s (2.19) =1/V max + K m V max 1/s (2.20) ( 2-4) - (2.18) - s = const., s, s (2.18) ( 2-5) - (2.18) Lineweaver-Burk (1/s 1/V ) Eadie-Hofstee (V/s V ) Hanes-woolf (s s/v ) (2.20) 2.3(b) 3 V 1/V V max V /2 max 1/V max K m/vmax K m s -1/K m 1/s 2.3: - (2.18) s V (a), 1/s 1/V (b). 10 13
2.1.6 (2.9) (2.12) 2 ds dt = k 1c k +1 s(e 0 c) (2.21) dc dt = k 1c k 2 c + k +1 s(e 0 c) (2.22) s e s 0 e 0 s 0 e 0 (2.23) (2.22) 11 12 s = s 0 σ c = e 0 χ k +1 e 0 t = τ (2.24) e 0 s 0 ϵ (2.25) α, κ ( 2-6 ) dσ dτ ϵ dχ dτ = αχ σ(1 χ) (2.26) = κχ + σ(1 χ) (2.27) (2.23) ϵ 0 (2.27) 0 χ c ( 2-6) α κ k 1, k +1, k 2, s 0 (2.26, 2.27) 11 s, c 12 k 1, k +1, k 2 14
2.1.7 (equilibrium approxmation) - E C ( (2.8) ) (2.7) K s = k 1 /k +1 13 se K s c =0 (2.28) c (2.14) V = dp dt = k 2c = V maxs s + K s (2.29) (2.18) K s K m 13 2.1.3 15
2.2 (inhibitor) (a) (c) (d) I P P E E E (b) E E I E E E I 2.4:. (a), (b), (c), (d). ( ) (competitive inhibition) (active site) (noncompetitive inhibition) (substrate inhibition) - (uncompetitive inhibition) - 16
2.2.1 (competitive inhibition) I +E k +1 k C 1 2 P+E k 1 I+E k +3 k 3 C 2 (2.30) s =[],e=[e],i=[i],c 1 =[C 1 ],c 2 =[C 2 ],p=[p] de dt = k 1c 1 k +1 se + k 2 c 1 k +3 ei + k 3 c 2 (2.31) ds dt = k 1c 1 k +1 se (2.32) di dt = k 3c 2 k +3 ie (2.33) dc 1 dt = k +1se k 1 c 1 k 2 c 1 (2.34) dc 2 dt = k +3ei k 3 c 2 (2.35) (free enzyme)e C 1 C 2 (2.31,2.34,2.35) de dt + dc 1 dt + dc 2 dt e 0 =0 (2.36) e + c 1 + c 2 = e 0 (2.37) C 1 C 2 dc 1 /dt = 0, dc 2 /dt =0 (2.34), (2.35), (2.37) e K i e 0 s c 1 = K m i + K i s + K m K i (2.38) K m e 0 i c 2 = K m i + K i s + K m K i (2.39) K m k 1 + k 2 k +1 K i k 3 k +3 17
P V = dp dt = k 2c 1 V max s = s + K m (1 + i/k i ) (2.40) V max k 2 e 0 ( 2-5) (2.40) s V 1/s 1/V ( 2-6) K m = k 1 /k +1 2.40 2.2.2 (noncompetitive inhibition) I +E k +1 E k 2 P+E k 1 I+E k +3 k 3 EI +EI k +1 k 1 EI (2.41) I+E k +3 k 3 EI E, EI, EI s =[],e=[e],i=[i],x=[e],y =[EI],z =[EI],p=[P] ė = k +1 se + k 1 x + k 2 x k +3 ie + k 3 y (2.42) ṡ = k +1 se + k 1 x k +1 sy + k 1 z (2.43) i = k +3 ie + k 3 y k +3 ix + k 3 z (2.44) ẋ = k +1 se k 1 x k 2 x k +3 ix + k 3 z (2.45) ẏ = k +3 ie k 3 y k +1 sy + k 1 z (2.46) ż = k +1 sy k 1 z + k +3 ix k 3 z (2.47) 18
e 0 e + x + y + z = e 0 (2.48) (2.41) ( 2.5) E K i EI K s E K i K s EI 2.5:. E.K s, K i. EI E EI E EI E. E es K s x =0 (2.49) ei K i y =0 (2.50) ys K s z =0 (2.51) xi K i z =0 (2.52) K s k 1 k +1 K i k 3 k +3 19
e x, y, z s, i x = e 0K i K i + i s K s + s p V max = k s e 0 (2.53) V = dp dt = k 2x V max = (1 + i/k i ) s s + K s V max = (1 + i/k i ) s/k s 1+s/K s (2.54) ( 2-6) (2.54) 2.3 2.2.3 (substrate inhibition) E E +E k +1 E k 2 P+E k 1 +E k +3 k 3 E V = K s k 1 k +1 K ss k 3 k +3 V max s K s + s + s 2 /K ss (2.55) 20
( 2-7) s V 2.2.4 (uncompetitive inhibition) I E E +E k +1 E k 2 P+E k 1 I+E k +3 k 3 EI ( 2-8) K s k 1 k +1 K m = k 1 + k 2 k +1 K i k 3 k +3 s V 21
2.2.5 2.3-2.6 ( ) - ( ) (allosteric effector) (allosteric modifier) ( 2.2.2) 14 n n ( ) (allosteric efffect) (cooperative) (activator) (inhibitor) ( 2.7 ) 14 allo stere 22
V V max s 2.6:. 23
2.7:.(a). 1 1 2.(b).. (,, ),.(c).. ( ) ( )E C 1 C 2 +E k +1 k C 1 2 P+E k 1 (2.56) k +3 k +C 1 C 2 4 P+C1 k 3 ( 2.2.3) 1 24
1 (E) (k ±1 ) 2 (C 1 ) (k ±3 ) s =[],e=[e],c 1 =[C 1 ],c 2 =[C 2 ],p=[p] V =ṗ = k 2 c 1 + k 4 c 2 = f(s) 2 c 1 = k +1 se k 1 c 1 k 2 c 1 k +3 sc 1 + k 3 c 2 + k 4 c 2 (2.57) c 2 = k +3 sc 1 k 3 c 2 k 4 c 2 (2.58) e 0 e + c 1 + c 2 = e 0 (2.59) C 1,C 2 c 1 = 0, c 2 =0 c 1 = c 2 = K 2 e 0 s K 1 K 2 + K 2 s + s 2 (2.60) e 0 s 2 K 1 K 2 + K 2 s + s 2 (2.61) K 1 k 1 + k 2 k +1 K 2 k 3 + k 4 k +3 V = dp dt = k 2c 1 + k 4 c 2 (2.62) = (k 2K 2 + k 4 s)e 0 s K 1 K 2 + K 2 s + s 2 (2.63) E C 1 k +1 =2k +3 2k + C 1 1 C 2 2 25
2k 1 = k 3 2k 2k 2 = k 4 K 1 = k + k 2 2k + 1 2 K K 2 = 2k +2k 2 k + 2K K K +K 2 K + (2.63) V = 2V maxs s + K m 2 - ( 2-9) (2.2.5) C 1 ( E), C 2 ( E) 2.8 K 1 K 2 E E E 2.8: (2.2.5) K 1, K 2 E,E (2.63) 2.2.6 (Hill equation) k +1 0( ), k +3 ( ) K 1, K 2 0 K 1 K 2 = const K 2 m 26
(2.63) K 2 0 V = k 4se 0 s K 1 K 2 + s 2 = V maxs 2 K 2 m + s 2 = V max(s/k m ) 2 1+(s/K m ) 2 (2.64) V max k 4 e 0 ( 2-10) 3 1 n (n )K 1 K n 2 K 1 K n = const K n 0, K 1 V = V maxs n K n m + s n = V max(s/k m ) n 1+(s/K m ) n (2.65) (Hill equation) K n m = K 1 K 2 K n K 2 K n 1 n i=1 K i ( 2-11) (Hill ) n 27
(2.54) V = V max 1+(i/K i ) n (s/k s ) m 1+(s/K s ) m (2.66) n m 15 ( 2-12) K 1 K 2 E EI IEI K 3 K 4 K 5 AE K 6 AEI AIEI K 7 2.9: E I A 2.9 E I A AE AEI (2.65) n = 2, m =1 ( 2-14) 15 n m Hill n m 28
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