October 20, 2010

Daniela Calvetti

Accounting for variability and uncertainty in a brain metabolism model via a probabilistic framework

Brain is a complex biochemical device

• Neurons and astrocytes are linked via a complex metabolic coupling

• Cycling of neurotransmitters, essential for transmission of action potentials along axon bundles, requires energy

• Metabolic processes provide what is needed to support neural activity (energy and more)

model parameters concentrations input

functionplus side constraints .

( )( ( ), , ( ))

dC tF C t I t

dt

Basic mathematical model of metabolism

A lot of uncertainty and variability in this picture

Mass balances in capillary blood

, ,[ ]

([ ] [ ])b art A b ECS A ECS bd A Q

V A A J Jdt F

[ ] [ ] (1 )[ ]venous artA F A F A

where

22 2

2

([ ] )[ ] [0 ] 4 [ ]

([ ] )

nfree

tot free rbcn n

free

OO Hct Hb

K O

2 2

22 , 2 , ,

[ ]([ ] [ ] )

totb a tot tot O b ECS O ECS bd O Q

V O O J Jdt F

A closer look at oxygen

max[ ]( )

( )[ ]( )

VK

A tt

A t

max( ) [ ]( )

( )( ) [ ]( )

Vr t A t

tt AKr t

Express reaction flux as

Reaction type

A +E B+FReaction type

A B

( )( )

( )

E tr t

F t

, ( ) ([ ] ( ) [ ] ( ))A x y x yJ t A t A t

Diffusion based (passive) transport rate:

ma, x[ ] ( )

( )[ ] ( )

xA x y

x

AT

tJ t

AM t

is modelled in Michaelis-Menten form

Carrier facilitated transport rate

Ax+X AX Ay+X

Transport rates

We assume that maximum transport rate of glutamate and affinity increase concomitantly with increase activity

,[ ] ( )

( ) ( )( ) [ ] ( )

nGlu n c

n

Glu tJ t T t

M t Glu t

0( ) (1 2 ( ))T t uT t 0( ) (1 ( ) / 2)MM t u t

where

Biochemical model of activity

( ) ( )lowQ t Q u t Q

max ([ ]

( )[ ]

[ ] )n

ATPase base cn

f GATP

t VK AT

luP

2

2

[ ]([ ]

[)

]

n

Gluc

n

Glu

K Glf lu

uG

where

regulated by glutamate concentration in synaptic cleft:

increase in blood flow

Concurrently with activity we have

and postsynaptic ATP hydrolysis

1/

,

1 ( )k

b b

low b low

dV Q t V

dt Q V

, ,( ) [ ]( )

A Aart A b ECS A ECS b

b

dm mQ t A J J

dt V t

where κ = 0.4 and τ = 2 seconds

[ ]A b bm A VMass balance in blood to account for variable volume :

Differential equation for capillary blood volume

Hemoglobin and BOLD signal

Concentrations of oxy- and deoxy-hemoglobin are of interest for the coupling between neuronal activity, cerebral hemodynamics and metabolic rate

2 1j

j

kj j

kO Hb Hb

1 4j

Binding process of free O2 to the heme group implies that at equilibrium the conditions

1 2[ ][ ] [ ]j free j jHb O K Hb j

jj

kK

k

4

0j tot

jHb Hb

determine uniquely saturation states of hemoglobin.

S MJ QKC + Vddt

In summary, the changes in the mass of a prototypical metabolite or intermediates can be expressed by a differential equation of the form

Φ= Reaction fluxesJ = Transport ratesQK = Blood flow term

S = stoichiometric matrixM = matrix of transport relations

• These large systems of coupled, nonlinear stiff differential equations which depend on lots of parameters

• Heterogeneous data – various individuals, conditions, laboratories• Possible inconsistencies between data and constraints • Solution to parameter estimation problem needed to specify model

may not exist, or may be very sensitive to constraints• Need to quantify the expected variability of model predictions over

population under given assumptions

Challenges for a deterministic approach

Recasting problem in probabilistic terms

All unknown parameters are regarded as random variables Randomness is an expression of our ignorance, not a property of the unknowns (if they were known they would not be random variables)

The type of distribution conveys our beliefs or knowledge about parameters

The stationary or steady state case

The derivatives of the masses are constant or zeros

The reaction fluxes and transport rates constant

Reaction fluxes and transport rates, which are the unknowns of primary interest, are regarded as random variables

Constraints need to be added to ensure that solution has physiological sense

Deterministic Flux Balance Analysis

Linear problem

plus inequality constraints

Linear programming

0 S MJ QK +

Needs an objective functionSolution is sensitive to boundsDoes not tell how probable the solution isMay fail if there are inconsistencies

Bayesian Flux Balance Analysis

Assume that an ensemble of fluxes and rates can support a steady state, which may be more or less strict

Supplement data and steady state assumption with belief about constituents (preferred direction, limited range, target value) recasting problem in probabilistic terms

Solution is a distribution of flux values, whose shape is an indication of expected variability over a population. Some fluxes can be tightly estimated, other very loosely.

The range of possible solutions mimics the variability observed in measured data in lab experiments and allows further investigation (correlation analysis)

A

B

C

24

5

13

3 4 = 5 (= a +error)+

Add some information about 3

And here is what we have if the bounds are wrong

GLC GLC

GLC

GLC

GLC

O2

O2

O2

O2

O2

PYR

PYR

CO2

LAC

CO2

CO2

CO2

CO2

LAC

LAC

H2O

H2O

ATP ADP+Pi

AD

P+

Pi

ATP

PCR CR

PCR CR

Gln GlnGLU

GLU

GLU

Gln

Toy brain model: 5 compartments

,( / )( ) 0

0 0

0 0

0

0

b b a b b ECS ECS b

ECS b ECS ECS b ECS a n ECS a ECS

c n c c n a c c a

n n n n c c n n ECS ECS n

a a a a c c a a

C Q F C C J J

C J J J J Jd

V C J J J Jdt

C S J J J J

C S J J J

ECS ECS aJ

Strict steady state

;

n

au

J

/ ( )

0

0

0

0

art bQ F C C

r

A= matrix of stoichiometry and

transport information

1( ) 0V Au r

Relaxed steady state

random variable Γ=cov(w)

11( | ) ~ exp

2

Tr u Au r V V Au r

Likelihood density of r conditional on u

Au r Vw

We want to solve inverse problem: we have r and we want u

( | ) ~ ( ) ( | )prioru r u r u Bayes’ formula

Input for Bayesian FBA

Arterial concentration= assumed constant(known)

Use literature values of CMR of LAC and GLC to compute venous concentrations

Prior density: what we believe is true

Posterior: we experience it in sampled form

Generate a large sample using MCMC as implemented in METABOLICA http://filer.case.edu/ejs49/Metabolica

, ,/ , ( ,( ) )A A ar At ar vtb AACMR Q F C C Q C C

From a distribution of steady state configurations to a family of kinetic models

In kinetic model the random variables are the parameters

max[ ]( )

( )[ ]( )

VK

A tt

A t

max

( ) [ ]( )( )

( ) [ ]( )V

r t A tt

t AKr t

ma, x[ ] ( )

( )[ ] ( )

xA x y

x

AT

tJ t

AM t

maxC

VK C

Saturation levels

Time constants

Concentration info

A priori estimates

Idea: adjust parameter values systematically so that they satisfy near steady state and model is in agreement with basic understanding

Glucose (GLC) transport across BBB

Assume symmetry: BBB b ECS ECS bJ J J

max,[ ]

[ ]

bb ECS BBB

BBB b

GLCJ T

M GLC

max,

[ ]

[ ]

ECS

ECS

ECS b BBBBBB

GLCJ T

M GLC

Literature: max rate ~ 2-4 times unidirectional flux ~ 5 times net flux

Translate into saturation levels

0

[ ] 1

[ ] 2

b

bBBB

GLCs

M GLC

1

[ ] 1

[ ] 5

ECS

ECSBBB

GLCs

M GLC

Now solve first for 0

11 [ ]BBB bM GLC

s

0

1

1/ 1[ ] [ ]

1/ 1ECS b

sGLC GLC

s

then for

From the sample value of the net flux at near steady state:

0 1

max, [ ]BBB

BBB bT GJ

LCs s

Parameters of glucose transport into astrocyte/neuron

GLUT-1

max, max,5000n BBBT Tmax, max,5000a BBBT T

GLUT-3

Literature: time constants for transports

150 2.5minn s 60 0.001mina ms GLUT-3

GLUT-1

Not rate limiting: [GLC] smaller than affinity

2max,

max,[ ] [ ]

[ ][ ]

nn

n n n

GLC T GLCT GLC O

M GLC M M

Link time constants and Michaelis-Menten parameters

max,

nn

n

M

T

max,

aa

a

M

T

Solve for the affinity

max,n n nM T max,a a aM T

and from the flux realization and the concentration in ECS compute the initial GLC concentration

[ ]1

n nGLC M

max,

[ ]

[ ]

ECS n

n ECS n

GLC J

M GLC T

where

In the case of more complicated fluxes:

max1/

(1/] )

[ ]

[

Phos

Pho

GLC RedoxV

K GLC ds Re ox

[ ]

[ ]

ATPPhos

ADP

[ ]

[ ]

NADRedox

NADH

[ ]

[ ]GLC

GLCs

K GLC

Saturation level: non-equilibrium=1/2

More saturated=1/2

Less saturated=1/10

Solve for µ and ν. From steady state

maxGLC Phos Redoxs s s

V

In summary

For each realization in the ensemble of Bayesian steady state configurations we compute, conditional on first principles, parameters of a kinetic models

Ensemble of kinetic models provide ensemble of predictions

Organize predictions into p% predictive output envelopes

Note: can add uncertainty to literature statements by treating values as random variables (hierarchical model)