Detailed Balanced Chemical Reaction Networks as Generalized Boltzmann Machines Assignment

Assignment Task

Learning is a process during which a mathematical function is automatically optimized based on data. If we consider observed data X , we determine θ^∗ to minimize L, a loss function: θ^∗ = argminθ L(P(x |θ), Ψ)

Probabilistic inference gathers techniques for inferring unknown characteristics of a group from a sample of that group. It associates a degree of probability with each proposition.

Relative entropy measures the dissimilarity between two probability distributions. The relative entropy of ψ with respect to P is defined by

, where ψ is the true data distribution and P is the approximation of the former.

Boltzmann Machines 

Boltzmann Machines are probabilistic graphical models capable of probabilistic inference and learning. A Boltzmann machine is a stochastic neural network with the Gibbs-Boltzmann distribution as its equilibrium distribution, of the form:

P(x ) = 1/ Z e^−E(x)

energy function is given by E (x ) = Σi>j w_ij x_i x_j − Σi θ_i x_i , with w_ij where Z = Σx ∈{0,1}N e^−E(x) is the partition function and where the modeling the correlation between units i and j, θ is the bias, and the x_i take values of 0 or 1 according to their activation state.

A Boltzmann machine is composed of two types of units:

Visible units, which can be directly observed, represent the input variables of the system.

Hidden units, which are introduced to capture more abstract features or relationships in the data.

• They act as learned intermediate representations of the visible data.
•  Their activation allows the model to capture correlations between different input variables.

All units (visible and hidden) are connected to each other, allowing information sharing and generation of further data.

Properties of Boltzmann Machines

Boltzmann Machines have three particularly important characteristics:

• Boltzmann Machines are capable of representing a wide class of distributions.
• Boltzmann Machines can condition these distributions to perform probabilistic inference through the clamping process.
  •  π(u|v ) can be exactly sampled by clamping v to a constant value while simulating the Boltzmann Machine.
• There exists a simple algorithm to train the parameters of Boltzmann Machines from data.

Chemical Reaction Networks

Chemical Reaction Networks (CRNs) are composed of:

• a set of species S_i ∈ S
• reactions R, which convert a multi-set of species into another: is the number of species S_i in th input (resp. output) of reaction r . 
• rate constants k

Detailed balanced chemical Networks (dbCRNs) are a subclass of stochastic CRNs such that:

• Each species S_i ∈ S has an energy G_i .
•  All reactions are reversible, i.e., if I ^r k+ -> ∈ R then Or -> lr ∈ R.
• Reaction rates are such that K/K =e^−∆G wheres ∆G = Σi Gi (O_i − I_i ).

Energy clamping relies on modulating or adjusting the energies of chemical species.

Let D be a dbCRN, then the clamped dbCRN, D_Gc , retains the same species and reactions as D, but with different reaction rates defined as follows:

G^C = G if S ∈ S^F   

G_i + ∆_i if S_i ∈ S^C   

The clamped dbCRN thus possesses a new equilibrium distributio

π (s) = e, where Z^C = Σ e^−GC (s) and G^C (s) = Σ G^C s +lo

This type of clamping doesn’t fix the value of a species, but it fixes the mean of the clamped species, while allowing for fluctuations.

This method, therefore, does not change the accessibility class of the underlying dbCRN and can be applied to any species in any dbCRN.

Method: Introduction of a potentialized dbCRN in a small volume v , coupled with a large external bath volume V >> v .

The potentialized dbCRN includes additional potential species P_i , coupled with the clamped species S_i ∈ S^C and maintained at a constant concentration.

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