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 function plus side constraints. Basic mathematical model of metabolism

A lot of uncertainty and variability in this picture

Mass balances in capillary blood where

A closer look at oxygen

Express reaction flux as Reaction type A +E B+FReaction type A B

Diffusion based (passive) transport rate: 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 where Biochemical model of activity

where regulated by glutamate concentration in synaptic cleft: increase in blood flow Concurrently with activity we have and postsynaptic ATP hydrolysis

where κ = 0.4 and τ = 2 seconds Mass 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 Binding process of free O2 to the heme group implies that at equilibrium the conditions determine uniquely saturation states of hemoglobin.

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 fluxes J = Transport rates QK = Blood flow term S = stoichiometric matrix M = 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 Needs an objective function Solution is sensitive to bounds Does not tell how probable the solution is May 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 =(= a +error) +

Add some information about

And here is what we have if the bounds are wrong

GLC O2O2 O2O2 O2O2 O2O2 O2O2 PYR CO 2 LAC CO 2 LAC H 2O ATPADP+Pi ATP PCR CR PCRCR Gln GLU Gln

Toy brain model: 5 compartments Strict steady state A= matrix of stoichiometry and transport information

Relaxed steady state random variable Γ=cov(w) Likelihood density of r conditional on u We want to solve inverse problem: we have r and we want 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

From a distribution of steady state configurations to a family of kinetic models In kinetic model the random variables are the parameters

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: Literature: max rate ~ 2-4 times unidirectional flux ~ 5 times net flux Translate into saturation levels

Now solve first for then for From the sample value of the net flux at near steady state:

Parameters of glucose transport into astrocyte/neuron GLUT-1 GLUT-3 Literature: time constants for transports GLUT-3 GLUT-1 Not rate limiting: [GLC] smaller than affinity

Link time constants and Michaelis-Menten parameters Solve for the affinity and from the flux realization and the concentration in ECS compute the initial GLC concentration where

In the case of more complicated fluxes: Saturation level: non- equilibrium=1/2 More saturated=1/2 Less saturated=1/10 Solve for µ and ν. From steady state

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)