Ratio Shift Keying (RSK) Overview

• Ratio Shift Keying (RSK) is a molecular communication modulation scheme that encodes information in the relative concentration ratio of two ligand species.
• It leverages the invariance of the concentration ratio under common multiplicative channel variations to improve signal robustness in power-limited and time-varying environments.
• RSK offers practical advantages over traditional schemes like CSK and MoSK, demonstrating enhanced performance in terms of capacity and error probability under challenging mobile and noisy conditions.

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Ratio Shift Keying (RSK) is a molecular communication modulation scheme in which information is encoded in the relative abundance of two molecular species rather than in an absolute concentration or a molecule type alone. In its common formulation, the transmitted symbol is the concentration ratio
[
\alpha=\frac{c_1}{c_1+c_2}\in[0,1],
]
while alternative count-based formulations use a ratio such as (\eta=Q_A/Q_B). The defining premise is that the total molecule count may vary, but the information-bearing variable is the ratio itself. In diffusion-based channels that are linear and type-invariant, this ratio can be preserved at the receiver even when the overall signal amplitude fluctuates, which motivates RSK for power-limited and time-varying molecular communication scenarios [2205.13317][2302.10353].

1. Definition, notation, and conceptual scope

In RSK, the transmitter uses two distinct ligand species and maps each symbol to their relative concentration. One widely used notation defines the symbol as
[
\alpha=\frac{c_1}{c_1+c_2},
]
with (c_1) and (c_2) denoting the concentrations of type-1 and type-2 ligands. Under an impulse-release model at symbol time (t_0),
[
x_1(t)=N_{\mathrm{tx},1}\delta(t-t_0), \qquad x_2(t)=N_{\mathrm{tx},2}\delta(t-t_0),
]
and the encoded ratio is
[
\alpha=\frac{N_{\mathrm{tx},1}}{N_{\mathrm{tx},1}+N_{\mathrm{tx},2}}.
]
A count-based presentation, used in the absorbing-receiver literature, instead defines
[
\eta=\frac{Q_A}{Q_B},
]
with (Q_A) and (Q_B) the released numbers of molecules of types (A) and (B) [2205.13317][2302.10353][2412.20161].

The central distinction between RSK and concentration-based modulation is that absolute molecule counts (N_1,N_2) or concentrations (c_1,c_2) may vary without altering the conveyed symbol, provided the ratio is maintained. This makes RSK analytically and operationally different from concentration shift keying (CSK), where the input is the absolute received concentration (c), and from molecule shift keying (MoSK), where different species themselves carry the symbol identity [2205.13317][2412.20161].

A concise comparison is useful.

Scheme	Information-bearing quantity	Receiver statistic emphasized in the cited works
RSK	(\alpha=c_1/(c_1+c_2)) or (\eta=Q_A/Q_B)	Binding-time statistics or absorbed-count ratio
CSK	Absolute concentration (c)	Number of bound receptors (n_B)
MRSK	Successive ratios (\eta_i=Q_{i+1}/Q_i)	Multiple ratio estimates across molecule types

This formulation also bounds the conditions under which RSK retains its intended invariance. The cited analyses assume that both ligand types experience the same channel action; if the channel is not type-invariant, the received ratio need not equal the transmitted one. A common misconception is therefore that RSK is intrinsically immune to propagation variability. The more precise statement is that RSK is robust to channel variations that affect both molecule types equally [2205.13317][2302.10353].

2. End-to-end channel and receptor model

The standard RSK channel model assumes free diffusion in (3)D and, in several analyses, no intersymbol interference (ISI), either because the symbol interval (T_s) is sufficiently large or because enzyme-assisted degradation is present. In mobile molecular communication, both transmitter and receiver undergo Brownian motion with diffusion coefficient (D_{\mathrm{tx,rx}}\ll D), where (D) is the ligand diffusion coefficient [2302.10353].

For time-varying diffusion channels, the instantaneous channel impulse response at distance (r(t)) is
[
h(t,\tau)=[4\pi D\tau]^{-3/2}\exp!\left(-\frac{r(t)^2}{4D\tau}\right).
]
Sampling at the peak time obtained from (\partial h/\partial \tau=0) yields
[
\tau_{\mathrm{peak}}=\frac{r(t)^2}{6D},
]
and the peak concentration becomes
[
c=N_{\mathrm{tx}}\,h(\tau_{\mathrm{peak}})=N_{\mathrm{tx}}\,(2\pi r^2/3)^{-3/2}e^{-3/2}.
]
In the mobile case, (r) is random and is approximated as noncentral-(\chi), with mean (\mu_r) and variance (\sigma_r^2) known via standard formulas [2302.10353].

At the receiver, the canonical model uses a single receptor type with (N_R) independent receptors. Each receptor follows a monovalent two-state continuous-time Markov process,
[
U \rightleftarrows B,
]
with binding rate (k^+) and ligand-dependent unbinding rates (k_1^-) and (k_2^-). The dissociation constants are
[
K_{D,i}=\frac{k_i^-}{k^+}.
]
In the presence of a mixture ((c_1,c_2)), the equilibrium bound probability is
[
p_B=\frac{c_1/K_{D,1}+c_2/K_{D,2}}{1+c_1/K_{D,1}+c_2/K_{D,2}}.
]
A critical point is that individual concentrations cannot be recovered from (p_B) alone; bound-time statistics are needed [2302.10353].

Under equilibrium, the bound-time distribution of one receptor is a mixture of exponentials,
[
p(\tau_b)=\alpha k_1^-e^{-k_1^-\tau_b}+(1-\alpha)k_2^-e^{-k_2^-\tau_b},
]
where (\alpha=c_1/(c_1+c_2)). For (N_R) receptors, the maximum-likelihood objective may be written as
[
\mathcal{L}(\alpha)=\sum_{i=1}^{N_R}\ln p(\tau_{b,i}\mid \alpha).
]
This receptor model explains why RSK detection is commonly posed as a ratio-estimation problem from stochastic binding durations rather than as a direct concentration readout problem [2205.13317][2302.10353].

3. Information-theoretic analysis and capacity

The input-output relation for RSK is studied by taking (X=\alpha\in[0,1]) as the channel input and the set of observed bound times as the output. The capacity is
[
C=\max_{P(x)}I(X;Y),
]
or equivalently
[
C=\max_{p_X(\alpha)}I(X;R)
]
when the receptor output is denoted by (R). Closed-form maximization is intractable, so the cited works adopt the Jeffreys-prior approximation in the large-(N_R) regime:
[
P^*(x)\propto \sqrt{I(x)}, \qquad
C \simeq \log_2!\left((2\pi e)^{-1/2}\int \sqrt{I(x)}\,dx\right).
]
For RSK with optimal estimation, the Fisher information is
[
I_{\mathrm{RSK}}(\alpha)
= N_R k_2^- \int_0^\infty
\frac{\left[-1+\gamma e^{(1-\gamma)k_2^-\tau}\right]^2}
{1-\alpha+\alpha\gamma e^{(1-\gamma)k_2^-\tau}}
e^{-k_2^-\tau}\,d\tau,
]
where (\gamma=k_1^-/k_2^-) is the ligand similarity parameter. The capacity-achieving input then satisfies
[
P^*_{\mathrm{RSK}}(\alpha)\propto \sqrt{I_{\mathrm{RSK}}(\alpha)},
]
and
[
C_{\mathrm{RSK}}=\log_2!\left((2\pi e)^{-1/2}\int_0^1\sqrt{I_{\mathrm{RSK}}(\alpha)}\,d\alpha\right).
]
A suboptimal single-threshold estimator chooses a threshold (T), counts (n_T\sim \mathrm{Bin}(p_T,N_R)) with
[
p_T(\alpha)=\alpha e^{-k_1^-T}+(1-\alpha)e^{-k_2^-T},
]
and replaces (I_{\mathrm{RSK}}) by (I_{\mathrm{RSK,sub}}) in the same capacity formula [2205.13317][2302.10353].

Several concrete trends are established. When (\gamma=1), the two ligands are indistinguishable and (C_{\mathrm{RSK}}=0). As (\gamma) increases, (C_{\mathrm{RSK}}) rises rapidly and saturates around (4) bits/use for sufficiently distinguishable ligands; the suboptimal estimator tracks the optimal curve very closely. Capacity also increases with (N_R) [2205.13317].

The benchmark comparison is with CSK, whose input is (x=c\in[0,c_{\mathrm{Rx,max}}]), with receptor statistic (n_B\sim\mathrm{Bin}(p_B,N_R)) and
[
p_B(c)=\frac{c}{c+K_D}.
]
The corresponding Fisher information is
[
I_{\mathrm{CSK}}(c)=N_R\,\frac{K_D}{c(c+K_D)^2},
]
and
[
C_{\mathrm{CSK}}=\log_2!\left((2\pi e)^{-1/2}\int_0^{c_{\mathrm{Rx,max}}}\sqrt{I_{\mathrm{CSK}}(c)}\,dc\right).
]
The cited results report that RSK and CSK have similar asymptotic capacities, approximately (4) to (5) bits/use, but under power-limited conditions RSK outperforms CSK because (\alpha) is invariant to total molecule count. One specific comparison states that for a low power budget such as (c_{\mathrm{Rx,max}}=0.1K_D), CSK remains well below (2)–(3) bits/use even for large (N_R), whereas RSK can achieve (3)–(4) bits/use under the same receptor count [2205.13317][2302.10353].

These results delimit another common misconception: RSK is not presented as uniformly superior to CSK in all regimes. Rather, the advantage is strongest when transmitter power is constrained or when channel fluctuations preserve the inter-species ratio [2205.13317].

4. Time-varying channels, mobility, and detection performance

A principal motivation for RSK is robustness in dynamic channels. If both ligand types experience the same time-varying channel impulse response, then the received ratio is preserved by diffusion, and analogous invariance holds for equal diffusion constants, identical enzymatic degradation rates, and fluctuations in total transmitter output power that leave the ratio intact [2205.13317].

The mobile molecular communication analysis makes this premise explicit. In that setting, transmitter and receiver both move diffusively, so the received concentration is random through the random separation (r). For RSK, after sampling bound times, the suboptimal estimator (\hat{\alpha}) is approximately Gaussian with mean (\alpha), and its variance is obtained by a Method-of-Moments expression. For CSK, the sampled occupancy (n_B) is random through both (c) and (r), with
[
\mu_{n_B}=E[E[n_B\mid c]], \qquad
\sigma_{n_B}^2=E[\mathrm{Var}(n_B\mid c)]+\mathrm{Var}(E[n_B\mid c]).
]
Detection for a (4)-symbol constellation is optimized by choosing (\alpha_m) for RSK or (c_m) for CSK to minimize the Chernoff-bound surrogate
[
\epsilon'{12}+\epsilon'{23}+\epsilon'{34},
]
with pairwise terms
[
\epsilon'{ij}=\exp[-g(\lambda)],
]
and maximum-likelihood thresholds determined by solving
[
\mathcal{N}(\mu_{m-1},\sigma_{m-1}^2)=\mathcal{N}(\mu_m,\sigma_m^2).
]
For equally likely symbols, the symbol error probability is
[
P_e=\frac14\sum_{m=0}^3 \int_{z\notin D_m}P(z\mid H_m)\,dz
]
and reduces to complementary-error-function sums [2302.10353].

The reported performance trends are specific. In mobile channels, CSK symbol error probability degrades strongly as mobility (D_{\mathrm{tx,rx}}) increases, whereas RSK symbol error probability remains nearly constant. As a function of receptor number, CSK is nearly flat while RSK decreases proportionally to (1/N_R). For moderate (N_R), CSK may slightly outperform RSK, but for large (N_R\ge 100), RSK yields much lower symbol error probability in mobile scenarios. Dependence on ligand distinguishability is also strong: (\mathrm{SEP}_{\mathrm{RSK}}(\gamma)) is close to (0.75) at (\gamma\approx 1) and drops to approximately (10^{-8}) by (\gamma\approx 4.5) [2302.10353].

These findings clarify the operational meaning of “robustness” in RSK. The claim is not that ratio detection removes all estimation noise, but that it suppresses the effect of channel variations that act as common multiplicative distortions on both ligand types [2302.10353].

5. Multi Ratio Shift Keying (MRSK) and ratio distributions

Multi Ratio Shift Keying (MRSK) generalizes binary or two-species RSK by using (N) distinct molecule types and encoding information in multiple successive ratios. The transmitted ratios are
[
\eta_i=\frac{Q_{i+1}}{Q_i}, \qquad i=1,\ldots,N-1.
]
Each (\eta_i) is selected from a finite (2^M)-ary alphabet obtained by equally spacing a ratio-indicator
[
x'\in\left{-1,-1+\frac{2}{2^M-1},\ldots,+1\right}
]
and exponentiating,
[
x^{(i)}=\Omega^{x'}, \qquad \Omega>1.
]
If molecule type (1) has fixed count (Q), the remaining counts are generated by cumulative products,
[
Q_i=Q\cdot \prod_{k=1}^{i-1}\eta_k.
]
Each symbol interval therefore carries (M(N-1)) bits, since the symbol alphabet size is ((2^M)^{N-1}) [2412.20161].

The channel model for MRSK uses unbounded (3)D diffusion without drift, with a spherical absorbing receiver of radius (r) at distance (d). The cumulative hitting fraction for a release at (t=0) is
[
F_{\mathrm{hit}}(t)=\frac{r}{d}\,\mathrm{erfc}!\left(\frac{d-r}{\sqrt{4Dt}}\right),
]
and the hitting probability in the (k)-th symbol interval is
[
P_{\mathrm{hit}}(k)=F_{\mathrm{hit}}(kT_s)-F_{\mathrm{hit}}((k-1)T_s).
]
For large release counts, the total absorbed molecules in interval (k) are approximated as Gaussian:
[
N_{\mathrm{Rx}}[k]\sim \mathcal{N}(\mu[k],\sigma^2[k]),
]
with finite-memory ISI of length (L),
[
\mu[k]=\sum_{i=1}^L P_{\mathrm{hit}}(i)s[k-i+1], \qquad
\sigma^2[k]=\sum_{i=1}^L P_{\mathrm{hit}}(i)\bigl(1-P_{\mathrm{hit}}(i)\bigr)s[k-i+1].
]
Each molecule type is assumed independent, and the ISI is truncated to (L) taps [2412.20161].

A central analytical step is the ratio of two independent Gaussians. If (X\sim\mathcal{N}(\mu_X,\sigma_X^2)) and (Y\sim\mathcal{N}(\mu_Y,\sigma_Y^2)) are independent and (\eta=X/Y), the exact density is
[
\psi(\eta)=\int_{-\infty}^{+\infty}|y|\,\phi_X(y\eta)\phi_Y(y)\,dy,
]
with a closed-form evaluation based on Hinkley’s formula. For implementation, the paper uses a “solid approximation” and, when relative variances are small, an additional Gaussian approximation
[
\eta\approx \mathcal{N}(\beta,\lambda^2),
]
where
[
\beta=\frac{\mu_X}{\mu_Y}, \qquad
\lambda^2=\beta^2\left(\frac{\sigma_X^2}{\mu_X^2}+\frac{\sigma_Y^2}{\mu_Y^2}\right).
]
Under this Gaussian-ratio model, the maximum-likelihood threshold between adjacent levels (\beta_i) and (\beta_{i+1}) is
[
E_i=\sqrt{\beta_i\beta_{i+1}}.
]
For the predefined ratios
[
x^{(i)}=\Omega^{-1+2(i-1)/(2^M-1)},
]
the thresholds become
[
E_i=\exp!\left[-1+\frac{2i-1}{2^M-1}\right], \qquad i=1,\ldots,2^M-1.
]
Detection is then threshold comparison on the estimated ratio (\hat{\eta}) [2412.20161].

The reported performance comparison states that RSK, viewed as the (N=2), (M=1), (\Omega=e) case of MRSK, achieves much lower BER over a wide range of molecule counts (Q) and bit-times than On-Off Keying (OOK), CSK, MoSK, and Release-Time Shift Keying (RTSK). The paper attributes this to several factors: (E[\eta]=E[N_A]/E[N_B]=Q_A/Q_B) is independent of (D), (d), (r), and (T_s); dividing two noisy Gaussians yields a lighter-tailed distribution; and higher (M) or (N) can enlarge symbol-time structure in very high-rate regimes [2412.20161].

The trade-offs are equally explicit. MRSK consumes more total molecules, especially for high (M) or (N). Optimal detection with memory grows exponentially in (L\cdot M\cdot (N-1)), so practical implementations use fixed-threshold detection or adaptive decision-feedback memory cancellation:
[
Q_i^{\mathrm{Rx,adapt}}[k]=Q_i^{\mathrm{Rx}}[k]-P_{\mathrm{hit}}(2)\,\hat{Q}_i^{\mathrm{Tx}}[k-1].
]
The parameter (\Omega) balances mean separation against variance expansion, with empirical minima near (\Omega\approx 2). The paper further states that (M=1), (N=2) is optimal in moderate-rate scenarios, whereas larger (M) or (N) is beneficial chiefly in ultra-high-rate regimes [2412.20161].

6. Relation to Reaction Shift Keying and general molecular demodulation

The acronym “RSK” is not uniform across the molecular communication literature. In the 2016 paper “Generalized Solution for the Demodulation of Reaction Shift Keying Signals in Molecular Communication Networks,” RSK denotes Reaction Shift Keying rather than Ratio Shift Keying. There, symbols are mapped to different transmitter reaction circuits whose stochastic dynamics generate distinct concentration-versus-time waveforms of a signalling molecule (S), for example
[
\mathrm{RNA}_s \xrightarrow{\kappa_s} \mathrm{RNA}_s + S.
]
The same summary also notes a ratio-shift principle in which circuits produce two downstream signalling species (S_1,S_2) whose ratio
[
R(t)=\frac{S_1(t)}{S_2(t)}
]
follows a characteristic trajectory [1610.09785].

The receiver in that framework consists of a front-end molecular circuit and a back-end demodulator. A generic front-end is a multisite receptor, such as
[
S+E \rightleftarrows C_1, \qquad S+C_1 \rightleftarrows C_2.
]
The end-to-end model is a continuous-time Markov process with state
[
X(t)=\bigl(N(t),b_1(t),b_2(t),\dots\bigr),
]
where (N(t)) contains unobserved molecular counts and (b_i(t)) the observed complexes. The observation history is
[
\mathcal{B}(t)={b_i(\tau):0\le \tau\le t},
]
and demodulation is posed as posterior inference of
[
P[s\mid \mathcal{B}(t)],
]
with decision rule
[
\hat{s}=\arg\max_s P[s\mid \mathcal{B}(t)].
]
Using (L_s(t)=\ln P[s\mid \mathcal{B}(t)]), the continuous-time demodulator is represented by
[
\frac{dZ_s(t)}{dt}
=\sum_{i=1}^{m_R}\Bigl[\delta_{R_i}(t)\ln Q_i(t)-Q_i(t)\Bigr],
]
where (Q_i(t)=E[\rho_i(t)\mid s,\mathcal{B}(t)]) [1610.09785].

The main contribution of that work is a general graphical solution for the required Bayesian filtering kernel. A bipartite reaction graph is constructed with species nodes and reaction nodes; for each reaction,
[
a_{i1}O_1+\cdots+a_{i,m_O}O_{m_O}
+b_{i1}U_1+\cdots+b_{i,m_U}U_{m_U}
\xrightarrow{\kappa_i}
c_{i1}O_1+\cdots+c_{i,m_O}O_{m_O}
+d_{i1}U_1+\cdots+d_{i,m_U}U_{m_U},
]
the change in observed species is read off directly, and the mass-action rate is
[
\rho_i(n_u,n_o)=\kappa_i
\prod_{j=1}^{m_O} n_{O_j}^{\,a_{i,j}}
\prod_{k=1}^{m_U} n_{U_k}^{\,b_{i,k}}.
]
The transition coefficient is then
[
Q_i=E[\rho_i(N_u(t),b(t))\mid s,\mathcal{B}(t)]\,\Delta t.
]
Exact filtering is generally intractable because it requires the full posterior over the unobserved state, but a matched-filter approximation replaces posterior-conditioned means by priors (E[\cdot\mid s]) and reduces implementation to (K) parallel linear or bilinear filters [1610.09785].

This broader demodulation framework is not a formulation of Ratio Shift Keying in the later 2022–2024 sense, but it is relevant because it places ratio-based signalling within a more general program of Bayesian inference for biochemical front-ends. A plausible implication is that ratio-encoded schemes and reaction-circuit-based encoders can be analyzed within a common CTMP and filtering language when the receiver is itself a molecular reaction network [1610.09785].