Binomial Models in Molecular Capture

• Binomial models of molecular capture are defined as frameworks where discrete molecular detection events follow a binomial distribution with probability p.
• They provide analytical tools for molecular communication channels, enabling the study of identification capacities and rate-renormalization in stochastic reaction networks.
• Practical applications include enhanced kinetic inference in single-cell measurements and scalable simulation methods using binomial moment equations.

Binomial models of molecular capture quantitatively describe systems in which discrete molecular entities are counted after undergoing probabilistic detection, arrival, or reaction. These models are foundational in analyzing molecular communication (MC) channels, interpreting single-cell gene-expression measurements, and stochastic transport phenomena involving random capture or adsorption. Molecular capture by its nature induces binomially distributed detection statistics, parameterized by the number of available molecules and the capture probability. Binomial frameworks support the development of channel and identification capacities, rate renormalizations, scalable moment-equation methods, and rigorous connections to classical deterministic kinetics.

1. Core Binomial Capture Law in Molecular Systems

In canonical molecular communication settings, each molecule released by a transmitter is captured at the receiver with independent probability $p$, influenced by the medium's diffusion, source-receiver separation, and geometric factors. For a transmitter releasing $N$ molecules in an interval, the register of observed molecules $Y$ follows

$$Y \sim \mathrm{Binomial}(N, p), \qquad P(Y = y \mid N) = \binom{N}{y}\,p^y (1-p)^{N-y}.$$

This model generalizes to vector-valued observations in networks, experimental platforms, or communication symbols. In single-cell technologies, per-molecule probabilities can vary by cell and molecular species, leading to heterogeneous binomial down-sampling. The law forms the basis for analyzing technical noise, information capacity, and inference in both discrete and continuous time.

2. Molecular Communication Channels and Deterministic Identification Capacity

Binomial models underpin discrete-time molecular communication channels (DTBC), where the received molecule count at each time slot is binomially distributed conditional on the transmitter's release rate. The channel law, for a vector input $\mathbf x = (x_1,\dots,x_n)$ and output $\mathbf y = (y_1,\dots,y_n)$, factorizes as

$$W^n(\mathbf y \mid \mathbf x) = \prod_{t=1}^n \binom{\lfloor T_s x_t \rfloor}{y_t}\,p^{y_t}(1-p)^{\lfloor T_s x_t \rfloor - y_t}.$$

Design constraints enforce peak and average molecule-release rates: $0 \leq x_t \leq P_{\max}$ and $n^{-1}\sum_t x_t \leq P_{\rm ave}$, typically combined via $A = \min(P_{\max}, P_{\rm ave})$.

For identification tasks, the deterministic-identification (DI) regime considers mapping $M$ messages to $M$ codewords under the binomial law, requiring vanishing probabilities for both type I and II identification errors. The identification codebook size scales as

$M(n,R) \approx 2^{(n \log n)R}$

where $R$ is the identification rate. BINOMIAL channels admit DI-capacity bounds under constraints:

$$\frac{1}{4}\leq \mathbb{C}_{\rm DI}(\mathcal{B}) \leq \frac{3}{2}$$

Every $R < 1/4$ is achievable; no code of rate $R > 3/2$ maintains minimal error (Salariseddigh et al., 2023).

3. Rate-Renormalization in Stochastic Reaction Networks

Single-cell gene-expression and regulatory networks, when subject to imperfect detection, experience binomial down-sampling in their observed statistics. Each molecule of species $i$ is detected with probability $p_i$, yielding

$$P(Y_i = k \mid X_i = n) = \binom{n}{k}\,p_i^k\,(1-p_i)^{n-k}$$

for true count $X_i$ and observed $Y_i$.

Kinetic rates in the Chemical Master Equation (CME) are renormalized under binomial capture:

• First-order synthesis: $k_{\rm synth}^{\rm app} = p\,k_{\rm synth}$
• Unimolecular decay: unaffected, $k_{\rm deg}^{\rm app} = k_{\rm deg}$
• Bimolecular binding: $k^{\rm app} = k/(p_A p_B)$
• General $n_i$-order: $k^{\rm app} = k \prod_i p_i^{-n_i}$

These modifications are exact under the generating-function mapping $G_{\rm obs}(z) = G_{\rm true}(1 - p(1 - z))$, holding when abundance is large or when promoter switching is separated in timescale (Zabaikina et al., 2 Dec 2025).

In bursty telegraph models (two-state promoter), observed burst size is $b_{\rm obs} = p b$, burst frequency $a_{\rm obs} = a$. Thus, imperfect detection causes apparent reductions in observable synthesis rates and apparent acceleration of binding reactions.

4. Binomial Moment Equations and Hierarchical Closure

The binomial-moment formalism generates a scalable hierarchy of moment equations suited for stochastic reaction networks. For $J$ species, binomial moments indexed by vector $v$ are

$$B_v \equiv \langle W(N, v) \rangle = \sum_N W(N, v)\,P(N)$$

with

$W(N, v) = \prod_{i=1}^J \binom{N_i}{v_i}$

and $|v| = \sum_i v_i$.

Moment evolution is governed by

$$\frac{dB_v}{dt} = \sum_{n,m} k_{n \rightarrow m} \left[C(v + n - m, n)\,B_{v + n - m} - C(v+n, n)\,B_{v+n}\right]$$

where $C$ denotes multi-index binomial coefficients. This hierarchy is truncated by a cutoff order $C$, yielding a polynomial-sized linear ODE system. In practical small-system regimes, cutoff $C=2$ or $3$ achieves accurate means, variances, and covariances with dramatically diminished computational cost compared to the master equation. This approach is extensible to complex biochemical networks and surface chemistry (Barzel et al., 2010).

5. Markov–Binomial Distribution in Reactive Transport

Stochastic reactive transport models combining molecular capture, adsorption, and desorption employ Markov–binomial distributions (MBD) for occupancy statistics. The state alternates between "free" (F) and "adsorbed" (A), governed by a two-state Markov chain with transition probabilities parameterized by adsorption rate $\lambda$ and desorption rate $\mu$. The number of time steps in the free state ($K_n$) admits a closed-form MBD via the probability generating function (pgf):

$$G_n(s) = A(s)\,\alpha(s)^{n-1} + B(s)\,\beta(s)^{n-1}$$

with eigenvalue-based expressions for $\alpha(s), \beta(s)$ and explicit formulas for moments. In the continuous-time limit, occupation times converge to deterministic kinetic transport PDEs; specifically, the limit densities for free and adsorbed states solve the coupled system

$$\begin{cases} \partial_t\left(C_F + C_A\right) = D\,\partial_{xx}C_F - v\,\partial_xC_F\ \partial_t C_A = -\mu\,C_A + \lambda\,C_F \end{cases}$$

The "double-peak" phenomenon in intermediate-time solute distributions is rigorously traced to the bimodality of the MBD. Such stochastic models thus clarify, in probabilistic terms, the physical origins of classical deterministic solutions (Dekking et al., 2011).

6. Practical Consequences and Design Considerations

Binomial models of molecular capture enable:

• Quantitative analysis of identification capacity in MC systems, with super-exponential growth in distinguishable event codebooks. This facilitates massive tagging, rare-event marking, and alarm signaling architectures, without sensitivity to reasonable rate constraints (Salariseddigh et al., 2023).
• Rigorous corrections to kinetic inference in single-cell measurements, via explicit rate-renormalization rules, crucial for the interpretation of noisy gene-regulatory dynamics (Zabaikina et al., 2 Dec 2025).
• Polynomially scalable simulation frameworks for stochastic network evolution using binomial moments, greatly expanding the tractable domain for physicochemical and biochemical modeling (Barzel et al., 2010).
• Mathematical insight into reactive transport, linking discrete Markov-binomial occupancy processes to classical PDE formulations and elucidating experimentally observed features (Dekking et al., 2011).

Designers and analysts must recognize that binomial down-sampling mixtures do not simply reduce mean counts, but systematically alter higher-order reaction rates, variances, and system dynamics in ways that depend sensitively on abundance, event coupling, and time-scale separation.

7. Limitations and Extensions

Binomial molecular capture models presuppose independent detection or arrival events, homogeneous or cell-/species-specific probabilities, and reaction schemes compatible with stochastic or combinatorial counting. Rate-renormalization holds exactly in certain regimes—high abundance, timescale separation, unimolecular processes—but can fail for higher-order coupled events. Similarly, identification capacity bounds rely on asymptotic sphere-packing arguments rather than constructive algorithms, and binomial-moment closures approximate stochastic hierarchies that may, in some chemical regimes, demand higher-order truncations. These caveats delineate promising avenues for constructing more expressive models, efficient codebooks, and experimentally informed parameterizations.