Markov State--Space Modeling and Channel Characterization for DNA-Based Molecular Communication

Abstract

In this paper, we study DNA-based molecular communication with microarray-style reception under reversible hybridization, where the bound-state observation exhibits both inter-symbol interference and colored counting noise. To capture these effects in a communication-oriented form, we develop a Markov state-space framework based on a voxelized reaction--diffusion model, in which a block-structured transition matrix describes molecular transport and binding/unbinding dynamics. For the microarray specialization, this representation yields the channel impulse response, the equilibrium gain, and a settling-time-based characterization of the effective channel memory. Building on the resulting symbol-rate observation model for on--off keying, we derive a grouped-binomial counting model and obtain a closed-form expression for the covariance of the counting noise. Based on these statistics, we further develop a differential-threshold detector and a finite-memory decision-feedback equalizer. Numerical results validate the theoretical correlation behavior and show that the relative performance of the proposed receivers depends strongly on the channel-memory regime.

Figures (8)

• Figure 1: Physical model of the considered DNA microarray channel. cDNA molecules released from the TX propagate in a bounded domain $\Omega$ toward a reactive receiver boundary $\Gamma_{\mathrm{RX}}$ with reversible hybridization, while the remaining boundary $\Gamma_{\mathrm{ref}}$ is reflecting. The RX observation is the number of bound cDNA molecules.
• Figure 2: Neighborhood set $\mathcal{N}(s_j)$ used to construct the diffusion block $\mathbf{Q}$ (2D example). (a) Interior voxel: $|\mathcal{N}(s_j)|=4$. (b) Voxel adjacent to a reflecting boundary: one diffusion direction is blocked, yielding $|\mathcal{N}(s_j)|=3$. The construction extends to 1D and 3D by replacing the interior neighborhood size with $2$ and $6$, respectively.
• Figure 3: Overview of the proposed Markov-based microarray channel model. (a) A 1D Markov chain with nearest-neighbor diffusion among the free states and reversible binding/unbinding between the receiver-adjacent free state $s_{N_{\mathrm f}}$ and the bound state $s_N$. (b) State occupancy probabilities $x_i(t)$ under a point release at the transmitter, illustrating transient propagation, binding, and convergence toward equilibrium; the highlighted rows correspond to the trajectories shown in (c) and (d). (c) Observable channel response $h(t)=x_N(t)$, with equilibrium gain $h^{\mathrm{eq}}$ and settling time $t^{\mathrm{eq}}$. (d) Example hidden-state trajectory $x_1(t)$, illustrating the evolution of an internal, non-observable state. Illustrative parameters are used here for visual clarity and may differ from those adopted in the numerical results.
• Figure 4: Bound-state response $h(t)$ under the point-release initial condition $\bm x[0]=\boldsymbol e_1$. (a) Varying the dissociation constant $K_{\mathrm D}$ changes the equilibrium gain $h^{\mathrm{eq}}$ predicted by \ref{['eq:heq_microarray_closedform']}, while the settling time $t^{\mathrm{eq}}$ remains relatively similar when $k_{\mathrm{off}}$ is fixed. (b) Varying the unbinding rate $k_{\mathrm{off}}$ mainly changes the settling time $t^{\mathrm{eq}}$, and hence the effective channel memory, while all curves share the same equilibrium gain $h^{\mathrm{eq}}$ when $K_{\mathrm D}$ is fixed.
• Figure 5: Time-averaged lag-dependent correlation coefficient $\rho[\ell]$ of the counting noise for independent equiprobable OOK signaling. Different subfigures correspond to different symbol intervals $T_b$, and different curves correspond to different settling times $t^{\mathrm{eq}}$ with fixed equilibrium gain $h^{\mathrm{eq}}$. Solid lines denote theory, and markers denote Monte Carlo estimates.

Theorems & Definitions (9)

• Proposition 1: Exponential convergence with characteristic time
• proof
• Corollary 1: Equilibrium Distribution and Gain
• proof
• Remark 1: Design implication
• Proposition 2: Covariance of the counting noise
• proof
• Corollary 2: Lag-dependent covariance decay
• proof