Chapter 8: Modeling chains of ODEs d R d t = [ r (1 R/K ) bN ] R , - - PowerPoint PPT Presentation

Chapter 8: Modeling chains of ODEs

dR dt = [r(1 − R/K) − bN]R , dN dt = [bR − d − cM]N and dM dt = [cN − e]M ,

¯ R = d b and ¯ N = r b ✓ 1 − d bK ◆ = r b ✓ 1 − 1 R0 ◆

¯ R = K

n=1 n=2 1 2 1

Modeling chains of ODEs

dR dt = [r(1 − R/K) − bN]R , dN dt = [bR − d − cM]N and dM dt = [cN − e]M ,

¯ R = d b and ¯ N = r b ✓ 1 − d bK ◆ = r b ✓ 1 − 1 R0 ◆

¯ R = K

¯ N = e c , ¯ R = K ✓ 1 − be cr ◆ and ¯ M = b ¯ R − d c

n=1 n=2 n=3 For odd chain lengths R depends on K 1 2 3

R0

0 = cr

be

Modeling chains of ODEs

dR dt = [r(1 − R/K) − bN]R , dN dt = [bR − d − cM]N and dM dt = [cN − e]M , K

(a)

¯ R

KNKM

d b

n=1 n=3 n=2

(b)

¯ N

KNKM

e c

n=2 n=3

(c)

¯ M

KM

n=3

Kaunzinger & Morin, Nature, 1998

Prey alone Prey with predator Predator Bacterial food chain

Modeling chains with saturated interaction terms

dR dt =  r ⇣ 1 R K ⌘

• bN

hR + R

• R ,

dN dt =  bR hR + R d cM hN + N

• N , and

dM dt =  cN hN + N e

• M ,

fR: R and N fN: in absence of M no N fM: no M

dR dt =  r ⇣ 1 R K ⌘

• bN

hR + R + N

• R ,

dN dt =  bR hR + R + N d cM hN + N + M

• N , and

dM dt =  cN hN + N + M e

• M

Per capita function always depends on variable itself.

aXY ' aXY 1 + X/k + Y/k when k is large

Modeling chains with Beddington interaction terms

(a)

¯ R

KNKM

d b

n=1 n=3 n=2

(b)

¯ N

KNKM

e c

n=2 n=3

(c)

¯ M

KM

n=3

(d)

K ¯ R

KN KM

n=1 n=2

n=3

(e)

K ¯ N

KN KM

n=2 n=3

(f)

K ¯ M

KM

n=3

Other famous chains don’t suffer from this problem

dS dt = s − dS − βSI , dE dt = βSI − (d + γ)E , dI dt = γE − (δ + r)I and dR dt = rI − dR

SEIR model:

¯ R = r d ¯ I , ¯ I = γ δ + r ¯ E , ¯ S = (d + γ)(δ + r) γβ , ¯ E = s d + γ − d(δ + r) γβ

R and I are proportional to previous level _ _

dN0 dt = s − (p + d)N0 , dNi dt = 2pNi−1 − (p + d)Ni and dNn dt = 2pNn−1 − dNn ,

¯ N0 = s p + d , ¯ Ni = 2p p + d ¯ Ni−1 and ¯ Nn = 2p d ¯ Nn−1

J =        −(p + d) . . . . . . 2p −(p + d) . . . . . . 2p −(p + d) . . . . . . . . . . . . . . . 2p −d       

(J00 − λ)(J11 − λ)(J22 − λ) . . . (Jnn − λ) = 0

Cascade of cell divisions

Cascade of cell divisions

dN0 dt = s − (p + d)N0 , dNi dt = 2pNi−1 − (p + d)Ni and dNn dt = 2pNn−1 − dNn ,

¯ N0 = s p + d , ¯ Ni = 2p p + d ¯ Ni−1 and ¯ Nn = 2p d ¯ Nn−1

¯ N0 = s p + d , ¯ Ni = 2ipis (p + d)i+1 and ¯ Nn = s d ✓ 2p p + d ◆n

dQ dt = −aQ − dQQ + d X fiNi and s = aQ

Kinetic proofreading

Kinetic proofreading:

¯ Cn = RL Km + L ✓ k2 k−1 + k2 ◆n

dC0 dt = k1FL − (k−1 + k2)C0 , dCi dt = k2Ci−1 − (k−1 + k2)Ci and dCn dt = k2Cn−1 − k−1Cn (8.15)

F + L

k1

⌦

k−1 C

• r

dC dt = k1FL − k−1C with F = R − C gives C = RL Km + L

Michaelis Menten:

F + L

k1

⌦

k−1 C0 ,

Ci−1

k2

→ Ci and Ci

k−1

→ F

F = R −

n

X

i

Ci

with gives

Kinetic proofreading

log[k−1] log[L]

n=0 n=10 n=100 k∗

−1

High affinity Low affinity

¯ Cn = RL Km + L ✓ k2 k−1 + k2 ◆n

Cn>θ

F + L

k1

⌦

k−1 C0 ,

Ci−1

k2

→ Ci and Ci

k−1

→ F ,

F = R − C ,

where

dC0 dt = k1FL − (k−1 + k2)C0 , dCi dt = k2Ci−1 − (k−1 + k2)Ci and dCn dt = k2Cn−1 − k−1Cn (8.15)