October 20, 2010 Daniela Calvetti Accounting for variability and uncertainty in a brain metabolism...
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Transcript of October 20, 2010 Daniela Calvetti Accounting for variability and uncertainty in a brain metabolism...
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)