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Message Passing and Metabolism - PubMed

  • ️Fri Jan 01 2021

Message Passing and Metabolism

Thomas Parr. Entropy (Basel). 2021.

Abstract

Active inference is an increasingly prominent paradigm in theoretical biology. It frames the dynamics of living systems as if they were solving an inference problem. This rests upon their flow towards some (non-equilibrium) steady state-or equivalently, their maximisation of the Bayesian model evidence for an implicit probabilistic model. For many models, these self-evidencing dynamics manifest as messages passed among elements of a system. Such messages resemble synaptic communication at a neuronal network level but could also apply to other network structures. This paper attempts to apply the same formulation to biochemical networks. The chemical computation that occurs in regulation of metabolism relies upon sparse interactions between coupled reactions, where enzymes induce conditional dependencies between reactants. We will see that these reactions may be viewed as the movement of probability mass between alternative categorical states. When framed in this way, the master equations describing such systems can be reformulated in terms of their steady-state distribution. This distribution plays the role of a generative model, affording an inferential interpretation of the underlying biochemistry. Finally, we see that-in analogy with computational neurology and psychiatry-metabolic disorders may be characterized as false inference under aberrant prior beliefs.

Keywords: Bayesian; master equations; message passing; metabolism; non-equilibrium; stochastic.

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Conflict of interest statement

The author declares no conflict of interest.

Figures

Figure 1
Figure 1

Solenoidal and dissipative dynamics in categorical systems. This figure provides a numerical example of a (three-dimensional) system consistent with Equation (5), and its decomposition as in Equation (6), starting from a series of random initial states. Each trajectory is shown in white. In addition, it illustrates the free energy landscape (in 2 dimensions) to demonstrate the interpretation given in Equation (7). On the left, we see the combination of the dissipative and solenoidal flows that tend towards the free energy minimum. In the centre, the dissipative part of the flow has been suppressed, leading to trajectories around the free energy contours. Such trajectories conserve free energy (but not probability) so do not find its minimum. On the right, the purely dissipative trajectories find the free energy minimum, but take subtly different paths compared to those supplemented with the solenoidal flow.

Figure 2
Figure 2

Sparse models and messages. This figure illustrates a generative model using normal (Forney) factor graph. Here, we have 8 different variables. The y variables are indicated by the small squares at the bottom of the factor graph. Dependencies between variables, represented on the edges of the graph, are indicated by the square factor nodes. The Markov blanket of a variable is determined by identifying those variables that share a factor (i.e., any edges connected to the associated square nodes). Not every variable is conditionally dependent upon every other; implying this generative model has a degree of sparsity. This lets us simplify the mean-field dynamics such that the rate of change of each marginal distribution depends only upon its Markov blanket. The result has the appearance of message passing, as indicated by the arrows. Each arrow represents a message coming from a factor. Where they meet, they each contribute to the local steady state.

Figure 3
Figure 3

A chemical reaction. This figure illustrates the solution to the generative model outlined in Equation (17), under the dynamics given in Equation (20). The upper-left plot shows the rate of change of the substrates and products. The two substrates have equal concentrations to one another, as do the two products. Under this model, with α = ¼, the substrates are converted into products until the substrates are at a quarter of their maximum concentration, with the remainder converted to the products. The same information is presented in probabilistic form in the lower right. Here, black indicates a probability of 1, white of 0, and intermediate shades represent intermediate probabilities. The plot of free energy over time shows that, despite the mean-field approximation and the constraints applied to the transition rate matrix, the reaction still evolves towards a free energy minimum—as in Figure 1. Note that, in the absence of an external input to this system, the free energy reduces to a Kullback–Leibler divergence between the current state and the steady state.

Figure 4
Figure 4

Reaction networks. This schematic illustrates the factor graph associated with a system comprising a pair of coupled reversible reactions (i.e., four reactions in total). The factors are specified in the blue panel. These are chosen to enforce conservation of mass, in the sense that the marginal of S1 or of S2 plus the marginal of S3 plus the marginal of S4 or of S5 is one.

Figure 5
Figure 5

Enzymes, Markov blankets, and chemical inference. This figure illustrates several points. The plots on the left use the same formats as in Figure 3 to show the evolution of the reaction in terms of concentration and probability. The plots on the right exploit the Markov blanket structure implicit in an enzymatic reaction to show the evolution of the ‘beliefs’ implicitly encoded by the expected value of the substrate about the product, and vice versa. The upper-right plot shows these beliefs, defined as q˜11=α2−q12 and q˜12=α2−q11, which converge towards q11 and q12, respectively as the steady state is attained. The implicit generative models are shown in the free energy plot, with the enzyme playing the role of the data being predicted. The free energy of each decreases as the beliefs converge upon the posterior probabilities of substrate and product given enzyme.

Figure 6
Figure 6

Metabolic networks and their pathologies. This figure shows the conditional dependencies in a generative model in the upper left, highlighting the directional influences at the lowest level of the model with pink arrows. These ensure S3 is a sensory state, while S5 and S7 are active states. In the upper right is the chemical message passing that solves this model. The two plots in the lower part of the figure illustrate the relative probability of the marginal probabilities (or concentrations) of each chemical species. The spatial configuration matches that of the network in the upper right. The sizes of the circles indicate the relative concentrations once steady state has been attained. The plots on the left and right show the steady states before and after introduction of a lesion that disconnects the reaction from S1 to S4. Here, we see a redistribution of the probability mass, resulting in an alternative (possibly pathological) steady state.

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