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2 1 ( [1, 15]) (Enzyme Kinetics) (quasi-steady state approximation) (equilibrium approxmation) (competitive inhibition) (noncompetitive inhibition) (substrate inhibition) (uncompetitive inhibition) (Hill equation) [6, 7, 15 17] [15] [8] Hopf Ca 2+ [13] (ubstrate-depletion oscillation) [6] 58 1
3 (Activetor-inhibiotr oscillation) [6] Hodgkin-Huxley Hodgkin-Huxley (Voltage-clamp method) Hodgkin-Huxley Hodgkin-Huxley 2 -FitzHugh- Nagumo Hodgkin-Huxley FitzHugh-Nagumo
4 1 ( [1,15]) ( ) = k n k n k = n k 1 +4n k 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
5 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
6 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
7 (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
8 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) A, B [A], [B] C [C] k [A], [B] ( (law of mass action) ) 4 d[c] = k[a][b] (2.4) dt 2 [4] [4] T (2.2) T 1/2 A 4 7
9 ( 2-1) P (1) A+B+C k P (2) (3) (4) A+B k P+C A k P+C 2A k P (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
10 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?? (Enzyme Kinetics) (ubstrate) (Enzyme)E (catalyst) (Product)P 2.2 [] ( [E] ) V( P ) 9
11 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
12 E - (Michaelis-Menten equation)(2.18) Briggs Haldane (1925) (2.8) P 6 [] 0 [E] 0 C E 7 E C P (quasi-steady state) 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 [C] [] [C] [] 8 9 s ( s 0 ) 11
13 (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 (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) V max s K m ( ) K m 12
14 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)
15 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) 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
16 2.1.7 (equilibrium approxmation) - E C ( (2.8) ) (2.7) K s = k 1 /k 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
17 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
18 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
19 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 (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
20 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
21 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) (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
22 ( 2-7) s V (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
23 ( ) - ( ) (allosteric effector) (allosteric modifier) ( 2.2.2) 14 n n ( ) (allosteric efffect) (cooperative) (activator) (inhibitor) ( 2.7 ) 14 allo stere 22
24 V V max s 2.6:. 23
25 2.7:.(a) (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
26 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
27 2k 1 = k 3 2k 2k 2 = k 4 K 1 = k + k 2 2k 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) (Hill equation) k +1 0( ), k +3 ( ) K 1, K 2 0 K 1 K 2 = const K 2 m 26
28 (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
29 (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
30 [1],,,,,, 2007 [2],, 8,, 1996 [3] J. Keener and J. neyd, Mathematical Physiology, pringer, 1991 [4],,, [5] A. Bruce et al.,,, Essential 2,, [6] C. P. Fall, E.. Marland, J. M. Wagner, J.J. Tyson, Computational Cell Biology, pringer, [7] J. D. Murray, Mathematical Biology I, 3rd Ed., pringer, 2001 [8],,, [9],,,, 1980 [10],, GC 33,, 2004 [11],,, 2015 [12],, 1988 [13] A. Goldbeter, G. Dupont, and M.J.Berridge (1990) Mimimal model for signal-induced Ca 2+ Oscillations and for their frequency encoding through protein phosphorylation. Proc. Natl. Acad. ci. UA, 87, [14] N. MacDonald (1989), Biological delay systems: linear stability theory, Cambridge University Press. [15] L. Edelsterin-Keshet, Mathematical Models in Biology, IAM,
31 [16] G. D. Vries, T. Hillen, M. Lewis, J. Muller, B. chonfish, A Course in Mathematical Biology, IAM, 2006 [17] D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, pringer, 1995 [18] A. L. Hodgkin and A. F. Huxley (1952) Quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117,
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6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
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63 3.2,.,.,. (2.6.38a), (2.6.38b), V + V V + Φ + fk V = 0 (3.2.1)., Φ = gh, f.,. (2.6.40), Φ + V Φ + Φ V = 0 (3.2.2). T = L/C (3.2.3), C. C V, T = L/V
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