## Abstract

Organisms have evolved a variety of mechanisms to cope with the unpredictability of environmental conditions, and yet mainstream models of metabolic regulation are typically based on strict optimality principles that do not account for uncertainty. This paper introduces a dynamic metabolic modelling framework that is a synthesis of recent ideas on resource allocation and the powerful optimal control formulation of Ramkrishna and colleagues. In particular, their work is extended based on the hypothesis that cellular resources are allocated among elementary flux modes according to the principle of maximum entropy. These concepts both generalise and unify prior approaches to dynamic metabolic modelling by establishing a smooth interpolation between dynamic flux balance analysis and dynamic metabolic models without regulation. The resulting theory is successful in describing ‘bet-hedging’ strategies employed by cell populations dealing with uncertainty in a fluctuating environment, including heterogenous resource investment, accumulation of reserves in growth-limiting conditions, and the observed behaviour of yeast growing in batch and continuous cultures. The maximum entropy principle is also shown to yield an optimal control law consistent with partitioning resources between elementary flux mode families, which has important practical implications for model reduction, selection, and simulation.

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## Acknowledgements

This work benefited from advice from JD Young, W Liebermeister, and P Dixit on metabolic modelling, and discussions with JS O’Neill and HC Causton on yeast metabolism. Thanks are extended to JD Young for also sharing their PhD Thesis, and to D Foley and M Dean for highlighting relevance of the maximum entropy principle in behavioural economics. DS Tourigny is a Simons Foundation Fellow of the Life Sciences Research Foundation.

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## Appendices

### A Derivation of maximum entropy control

To solve the optimal control problem (14) one introduces the Hamiltonian

where \(\varvec{\lambda }\) is a co-state vector the same dimension as \(\mathbf {X}\). Applying Pontryagin’s maximum principle implies maximisation of \(\mathcal {H}\), or equivalently the functional

with respect to \(\mathbf {u}\) subject to the constraint

This results in the *K* first-order conditions

where \(\alpha \ge 0\) is a Lagrange multiplier and \(\mathbf {B}^k\) is the *k*th column of \(\mathbf {B}\). The general solution of these equations takes the form

and *Q* is determined from the above constraint such that

The value of the co-state vector \(\varvec{\lambda }\) is obtained as in Young and Ramkrishna (2007) by solving the boundary value problem

whose solution is

This expression for \(\varvec{\lambda }\) is substituted into the above expression for \(u_k\) and because, ultimately, only the optimal control input at the current time *t* is of interest, one can set \(\tau = 0\) as in Young and Ramkrishna (2007), which yields the maximum entropy control (15).

### B First-order correction to return-on-investment

Expressed in terms of the average zeroth-order return-on-investment

the vector \(\mathbf {q}^T\mathbf {A}\) is

The first-order expansion of \(\mathbf {e}^{\mathbf {A} \Delta t}\) takes the form \(\mathbf {I} + \Delta t \mathbf {A}\), where \(\mathbf {I}\) is the identity matrix, and therefore the effective return-on-investment \(\mathcal {R}^k_{\Delta t}(\mathbf {m}_s)\) in (16) is approximated to first-order by

Taking the coefficient of \(x \Delta t\) and substituting for \(\mathbf {q}^T\mathbf {A}\) gives the first-order correction to return-on-investment

which can be written as

using the definition of \(Y_k(\mathbf {m}_s)\) provided in (20).

### C Expressing \(\mathcal {F}(\mathbf {u})\) in terms of EFM families

Using (17) to substitute for \(\mathbf {B}^k\) in the effective return-on-investment (16) means that \(\mathcal {F}(\mathbf {u})\) takes the form

The sum over *k* in the first term can be written as

where the second equality follows from the identity \(\tilde{u}_j = u_j /U_J\) for \(j \in F_J\), which implies

Using the composition property (24) of entropy \(H(\mathbf {u})\), one has

which is (25) with

the objective functional \(\mathcal {F}\) restricted to the *J*th family. Notice that the weighting (22) defines the effective return-on-investment for the *J*th family derived directly from system (9) to be

and therefore \(\mathcal {F}_J(\tilde{\mathbf {u}}_J)\) includes an entropic correction that is only zero when \(\tilde{u}_J\) describes an FBA/Bang–Bang policy.

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Tourigny, D.S. Dynamic metabolic resource allocation based on the maximum entropy principle.
*J. Math. Biol.* **80**, 2395–2430 (2020). https://doi.org/10.1007/s00285-020-01499-6

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DOI: https://doi.org/10.1007/s00285-020-01499-6