Computational Methods for Systems and Synthetic Biology

(1)

20/01/2017 François Fages C2-19 MPRI 1

Computational Methods for Systems and Synthetic Biology

François Fages

Inria Saclay – Ile de France

Lifeware project-team

http://lifeware.inria.fr/

(2)

Chemical Master Equation (CME)

d

dt p

^(t)

(x) = X

j:x vj 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt p(t) = ~ p(t)Q(~ ~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c2

C !

^c³

E + P R

_j

(0, t) = Y

_j

✓Z

_t

0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt E [X (t)] = X

j

v

_j

E [↵

_j

(X (t))]

d

dt E [X (t)] ⇡ X

j

v

_j

↵

_j

(E [X (t)]) p

_i

(x, t) =

@

@ t p(x, t) + @

@ x p(x, t)R(x) = p(x, t)Q(x)

probability of being in state x at time t

Differential equation as the probability of entering in state x minus the probability of leaving state x

x is the vector of molecule numbers is propensity of jth reaction is change vector of jth reaction

The movie shows the distributions of the enzyme and the product In one enzymatic reaction

over time

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt E [X (t)] = X

j

v

_j

E [↵

_j

(X (t))]

d

dt E [X (t)] ⇡ X

j

v

_j

↵

_j

(E [X (t)]) p

_i

(x, t) =

@

@ t p(x, t) + @

@ x p(x, t)R(x) = p(x, t)Q(x)

d

dt p ^(t) (x) = X

j :x v

_j

0 ↵ _j (x v _j ) · p ^(t) (x ~ v _j ) X

j

↵ _j (x) · p ^(t) (x) p ^(t) (x) = P ( X ~ ^(t) = ~ x)

k · [S ₁ ](t) · [S ₂ ](t) R(t) = k · Y

i

([S _i ](t)) ^l

ⁱ

d

dt [S _i ](t) = X

j

(m _j l _j ) · R _j (t) d

dt x _i (t) = X

j :R

_j

kont.

v _ij r _ij (~ x(t)) + X

k 6 =i

x _k (t) p _k (t)q _ki (~ x(t)) p _i (t)

d

dt ~ p(t) = p(t)Q(~ ~ x(t))

x _i (t) := E [X (t) | M (t) = i]

X _i (t + h) = X _i (t) + X

j

R _j (t, t + h)

| {z } · v _ij

|{z}

E + S ) ^c *

¹

c

₂

C ! ^c

³

E + P R _j (0, t) = Y _j

✓Z t 0

↵ _j (X (s))

◆ X ₃ (t + h) = X ₃ (t) + R ₁ (t, t + h) · (+1) +R ₂ (t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵ _j (x v _j )p(x v _j , t) ↵ _j (x)p(x, t) d

dt E [X (t)] = X

j

v _j E [↵ _j (X (t))]

d

dt E [X (t)] ⇡ X

j

v _j ↵ _j (E [X (t)]) p _i (x, t) =

@

@ t p(x, t) + @

@x p(x, t)R(x) = p(x, t)Q(x)

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

1

](t) · [S

2

](t) R(t) = k · Y

i

([S

i

](t))

^lⁱ

d

dt [S

i

](t) = X

j

(m

j

l

j

) · R

j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

k

(t)q

ki

(~ x(t)) p

i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t)) x

i

(t) := E [X (t) | M (t) = i]

X

i

(t + h) = X

i

(t) + X

j

R

j

(t, t + h)

| {z } · v

ij

|{z}

E + S )

^c

*

¹

c2

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

3

(t + h) = X

3

(t) + R

1

(t, t + h) · (+1) +R

2

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

j

(x v

j

)p(x v

j

, t) ↵

j

(x)p(x, t) d

dt E [X (t)] = X

j

v

j

E [↵

j

(X (t))]

d

dt E [X (t)] ⇡ X

j

v

j

↵

j

(E [X (t)]) p

i

(x, t) =

@

@ t p(x, t) + @

@x p(x, t)R(x) = p(x, t)Q(x)

(3)

Moment Equation for the Mean

Consider the time evolution of the mean:

by posing x’=x+vj

involves higher moments if jth reaction is at least bimolecular,

e.g.

↵

_j

(X ) = c

_j

· X

₁

· X

₂

d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:Rj kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t)) x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt E [X (t)] = X

j

v

_j

E [↵

_j

(X (t))]

next: replace by a Taylor series about the mean

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = p(t)Q(~ ~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt E [X (t)] = X

j

v

_j

E[↵

_j

(X (t))]

d

dt E [X (t)] ⇡ X

j

v

_j

↵

_j

(E [X (t)]) p

_i

(x, t) =

@

@ t p(x, t) + @

@ x p(x, t)R(x) = p(x, t)Q(x)

↵

_j

(X ) = c

_j

· X

₁

· X

₂

d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x vj 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:Rj kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c2

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt µ(t) = X

j

v

_j

E [↵

_j

(X (t))]

↵ _j (X ) = c _j · X ₁ · X ₂ d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p ^(t) (x) = X

x

x · d

dt p ^(t) (x) d

dt p ^(t) (x) = X

j :x v

_j

0 ↵ _j (x v _j ) · p ^(t) (x ~ v _j ) X

j

↵ _j (x) · p ^(t) (x) p ^(t) (x) = P ( X ~ ^(t) = ~ x)

k · [S ₁ ](t) · [S ₂ ](t) R(t) = k · Y

i

([S _i ](t)) ^l

ⁱ

d

dt [S _i ](t) = X

j

(m _j l _j ) · R _j (t) d

dt x i (t) = X

j :R

_j

kont.

v ij r ij (~ x(t)) + X

k 6 =i

x k (t) p _k (t)q _ki (~ x(t)) p _i (t)

d

dt ~ p(t) = p(t)Q(~ ~ x(t))

x _i (t) := E [X (t) | M (t) = i]

X _i (t + h) = X _i (t) + X

j

R _j (t, t + h)

| {z } · v _ij

|{z}

E + S ) ^c *

¹

c

₂

C ^c !

³

E + P R _j (0, t) = Y _j

✓Z t 0

↵ _j (X (s))

◆ X ₃ (t + h) = X ₃ (t) + R ₁ (t, t + h) · (+1) +R ₂ (t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵ _j (x v _j )p(x v _j , t) ↵ _j (x)p(x, t) d

dt µ(t) = X

j

v _j E [↵ _j (X (t))]

After Verena Wolf, U. Saarbrucken

(4)

Moment Closures for the Mean

time evolution of the mean:

Taylor series (one-dimensional case) about the mean:

and thus

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

1

](t) · [S

2

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t)) x

_i

(t) := E [X (t) | M (t) = i]

X

i

(t + h) = X

i

(t) + X

j

R

j

(t, t + h)

| {z } · v

ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

3

(t + h) = X

3

(t) + R

1

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

j

(x v

j

)p(x v

j

, t) ↵

j

(x)p(x, t) E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

j

(x) = ↵

j

(µ) + x µ

1! · @

@ x ↵

j

(µ) + (x µ)

²

2! · @

²

@x

²

↵

j

(µ) + . . . d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

i

](t) = X

j

(m

j

l

j

) · R

j

(t) d

dt x

_i

(t) = X

j:Rj kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t)) x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C !

^c³

E + P R

_j

(0, t) = Y

_j

✓Z

_t

0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

2

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t)

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

=0 in general: (co-)variances appear here and give more detailed description of the time evolution of the mean

if law of mass action and at most bimolecular reactions, terms of order >2 are zero

order 1 approx

(5)

Michaelis-Menten Example

Example:

d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = ~ p(t)Q(~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt µ(t) = X

j

v

_j

E [↵

_j

(X (t))]

d

dt µ(t) ⇡ X

j

v

_j

↵

_j

(E [X (t)]) p

_i

(x, t) =

@

@ t p(x, t) + @

@ x p(x, t)R(x) = p(x, t)Q(x) X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E [(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

_j

E [↵

_j

(X ) · X ] + v

_j²

E [↵

_j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E[(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

_j

E [↵

_j

(X ) · X ] + v

_j²

E [↵

_j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E[(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t)

order 1 approximation:

X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

2

(X ) = c

2

X

C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E [(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

_j

E [↵

_j

(X ) · X ] + v

_j²

E [↵

_j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

d

dt p

^(t)

(x) = X

j:x vj 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t)

d

dt

µ

_E

= c

₁

µ

_E

µ

_S

+ (c

₂

+ c

₃

)µ

_C

d

dt

µ

_S

= c

₁

µ

_E

µ

_S

+ c

₂

µ

_C

d

dt

µ

_C

= +c

₁

µ

_E

µ

_S

(c

₂

+ c

₃

)µ

_C

d

dt

µ

_S

= + c

₃

µ

_C

X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E [(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

_j

E [↵

_j

(X ) · X ] + v

_j²

E [↵

_j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

(6)

Michaelis-Menten 2 ^nd order approximation

Example:

d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = p(t)Q(~ ~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

d

dt p(x, t) = X

j

↵

_j

(x v

_j

)p(x v

_j

, t) ↵

_j

(x)p(x, t) d

dt µ(t) = X

j

v

_j

E [↵

_j

(X (t))]

d

dt µ(t) ⇡ X

j

v

_j

↵

_j

(E [X (t)]) p

_i

(x, t) =

@

@ t p( x, t) + @

@ x p(x, t)R(x) = p(x, t)Q(x)

X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E[(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

_j

E [↵

_j

(X ) · X ] + v

_j²

E [↵

_j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E[(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

d

dt p

^(t)

(x) = X

j:x vj 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t)

X = (X

_E

, X

_S

, X

_C

, X

_P

)

↵

₁

(X ) = c

₁

X

_E

X

_S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

_j

(X )] ⇡ ↵

_j

(µ)

f (x) = ↵

_j

(x) · x f (x) = f (µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E [(X µ)]

1! · @

@ x f (µ) + E [(X µ)

²

]

2! · @

²

@ x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

j

E [↵

j

(X ) · X ] + v

_j²

E [↵

j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt

2

= d

dt (E [X

²

] µ

²

) = d

dt E [X

²

] d dt µ

²

d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t)

order-2 approximation:

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

E [↵

_j

(X )] = ↵

_j

(µ) + E [(X µ)]

1! · @

@ x ↵

_j

(µ) + E [(X µ)

²

]

2! · @

²

@ x

²

↵

_j

(µ) + . . .

↵

_j

(x) = ↵

_j

(µ) + x µ

1! · @

@ x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . d

dt µ(t) = d

dt E [X (t)] = d dt

X

x

x · p

^(t)

(x) = X

x

x · d

dt p

^(t)

(x) d

dt p

^(t)

(x) = X

j:x v_j 0

↵

_j

(x v

_j

) · p

^(t)

(x ~ v

_j

) X

j

↵

_j

(x) · p

^(t)

(x) p

^(t)

(x) = P ( X ~

^(t)

= ~ x)

k · [S

₁

](t) · [S

₂

](t) R(t) = k · Y

i

([S

_i

](t))

^lⁱ

d

dt [S

_i

](t) = X

j

(m

_j

l

_j

) · R

_j

(t) d

dt x

_i

(t) = X

j:R_j kont.

v

_ij

r

_ij

(~ x(t)) + X

k6=i

x

_k

(t) p

_k

(t)q

_ki

(~ x(t)) p

_i

(t)

d

dt ~ p(t) = p(t)Q(~ ~ x(t))

x

_i

(t) := E [X (t) | M (t) = i]

X

_i

(t + h) = X

_i

(t) + X

j

R

_j

(t, t + h)

| {z } · v

_ij

|{z}

E + S )

^c

*

¹

c₂

C

^c

!

³

E + P R

_j

(0, t) = Y

_j

✓Z

t 0

↵

_j

(X (s))

◆ X

₃

(t + h) = X

₃

(t) + R

₁

(t, t + h) · (+1) +R

₂

(t, t + h) · ( 1)

exact time evolution of means:

d

dtµE = c1µEµS + (c2 + c3)µC d

dtµS = c1µEµS + c2µC d

dtµC = +c1µEµS (c2 + c3)µC d

dtµS = + c3µC

X = (XE, XS, XC, XP)

↵1(X) = c1XEXS

↵2(X) = c2XC

↵3(X) = c3XC

E[↵1(X)] = ↵1(µ) + E[(XE µE)(XS µS)]

2! · @²

@xExS

↵1(µ)

f(x) = ↵j(x) · x f(x) = f(µ) + x µ

1! · @

@x↵j(µ) + (x µ)²

2! · @²

@x²↵j(µ) + . . . E[f(X)] = f(µ) + E[(X µ)]

1! · @

@xf(µ) + E[(X µ)²]

2! · @²

@x²f(µ) + . . . d

dtµ(t) = X

j

vj↵j(µ(t))

2µ d dtµ d

dtE[X²] = X

j

2vjE[↵j(X) · X] + v_j²E[↵j(X)]

d

dtµ = X

j

vjE[↵j(X)]

E[↵j(X)] = ↵j(µ) + E[(X µ)]

1! · @

@x↵j(µ) + E[(X µ)²]

2! · @²

@x²↵j(µ) +. . .

↵j(x) = ↵j(µ) + x µ 1! · @

@x↵j(µ) + (x µ)²

2! · @²

@x²↵j(µ) + . . . d

dt

2 = d

dt(E[X²] µ²) = d

dtE[X²] d dtµ²

=> approximate time evolution of covariance

(7)

Michaelis-Menten 2 ^nd order approximation

Close after 2nd

moment => approximation because we set all higher centered moments to zero Equations for rate functions that are at most quadratic:

d

dt µ

_i

= X

j

v

_ji

↵

_j

(µ) + X

k,l

@

²

↵

j

(µ)

@x

_k

@x

_l

kl

2!

!

d

dt

^ir

= X

j

v

_ji

X

k

@↵

j

(µ)

@ x

_k

kr

1! + v

_jr

X

l

@↵

j

(µ)

@x

_l

il

1!

!

+ X

j

v

_ji

v

_jr

↵

_j

(µ) + X

k,l

@

²

↵

j

(µ)

@x

_k

@x

_l

kl

2!

!

d

dt

µ

E

= c

1

µ

E

µ

S

+ (c

2

+ c

3

)µ

C d

dt

µ

_S

= c

₁

µ

_E

µ

_S

+ c

₂

µ

_C

d

dt

µ

_C

= +c

₁

µ

_E

µ

_S

(c

₂

+ c

₃

)µ

_C

d

dt

µ

_S

= + c

₃

µ

_C

X = (X

E

, X

S

, X

C

, X

P

)

↵

1

(X ) = c

1

X

E

X

S

↵

₂

(X ) = c

₂

X

_C

↵

₃

(X ) = c

₃

X

_C

E [↵

₁

(X )] = ↵

₁

(µ) + E [(X

E

µ

E

)(X

S

µ

S

)]

2! · @

²

@ x

_E

x

_S

↵

₁

(µ) f (x) = ↵

_j

(x) · x

f (x) = f (µ) + x µ 1! · @

@x ↵

_j

(µ) + (x µ)

²

2! · @

²

@ x

²

↵

_j

(µ) + . . . E [f (X )] = f (µ) + E [(X µ)]

1! · @

@x f (µ) + E [(X µ)

²

]

2! · @

²

@x

²

f (µ) + . . . d

dt µ(t) = X

j

v

_j

↵

_j

(µ(t))

2µ d dt µ d

dt E [X

²

] = X

j

2v

j

E [↵

j

(X ) · X ] + v

_j²

E [↵

j

(X )]

d

dt µ = X

j

v

_j

E [↵

_j

(X )]

There are many examples where means + covariances give a very good approximation since

distributions are often similar to multivariate normal distribution (even when populations are small)

(8)

Michaelis-Menten 2 ^nd Order Closures

time evolution of means (including approximation of the covariances) and covariances

(9)

Switch Example

Species

Proteins: X1, X2

Promoter state: DNA, DNA.X1, DNA.X2 Exclusive Switch (Loinger et al., Phy. Rev. E, 2007)

Reactions

DNA ⟶ DNA+ Xi Xi ⟶ 0

DNA+Xi ⟷ DNA.Xi

DNA.Xi ⟶ DNA.Xi + Xi

(10)

State Distributions

Exclusive Switch (Loinger et al., Phy. Rev. E, 2007)

0 50 100

P1

P2

0 50 100

P1

P2

X1 X2

X1

X1 X2

⟵ low rate for binding to the promoter

high rate for binding to the promoter

⟶

(11)

TD 4: Toggle Switchactions

1. Connect to http://lifeware.inria.fr/biocham4 2. Open examples/MPRI

3. Run the notebook TD4_toggle_switch.ipynb

MPRI C2-19 January 2017 François Fages

(12)

Switch at Low Binding Rate

X1 low binding

rate

(13)

Switch at High Binding Rate

X1 high binding rate

order 4 (normal

pdf uses only mean and covariance matrix)

(14)

2 ^nd Order Moment Closure for Mean

X1 X2

X2

⟵ low rate for binding to the promoter

high rate for binding to the promoter ⟶

how good is moment closure?

means

means order 2 approximation

(15)

2 ^nd Order Moment Closure for Variance

X1

⟵ low rate for binding to the promoter

high rate for binding to the promoter ⟶

(co)variances

(16)

4 ^th Order Moment Closure for Variance

Exclusive Switch:

X1 Covariances

(high binding rate)

order 2

order 4

(17)

20/01/2017 François Fages C2-19 MPRI 17

Lotka-Volterra Prey-Predator Model

0.3[RA] for _ =[RA]=> RA. 0.15[RB] for RB => _.

0.3[RA][RB] for RA =[RB]=> RB. present(RA,0.5). present(RB,0.5).

Continuous oscillations Stochastic extinctions