Algae colonies are nature's own molecular network. This page turns the end-to-end channel models of Akyildiz et al. into live simulations: press Start to watch molecules, voltage pulses and photons travel from a transmitting cell to a receiving cell, and explore how distance, diffusion, wind, and receiver biology shape attenuation, delay — and, in a teaching extension, bit error rate.
A dense algae population releases signaling molecules (CSSF) into the water. They spread by pure diffusion and degrade on the way; a receiving cell responds by swimming faster once the local concentration crosses a threshold. Range: millimeters. Latency: hours — this is the slowest channel of the four.
Cell density N(t) grows logistically; molecules diffuse (coefficient \(D\)) and are lost at rate \(\delta = 1/\tau_{\mathrm{loss}}\). The receiver maps concentration to swimming speed through a thresholded Michaelis–Menten nonlinearity. Channel-only attenuation \(A_C(d)\) compares peak concentration at the receiver vs. the transmitter; end-to-end metrics are defined on the decoded swimming speed. Table I supplies the physical parameters; the unreported initial population and boundary-source convention are calibrated to reproduce Figs. 5–8 and documented under About & Citation.
On–off keying over the peak-concentration amplitude with additive Gaussian noise; \(\mathrm{BER}=Q\!\left(A/2\sigma\right)\). The paper's models are deliberately deterministic — this noise layer is added for teaching.
Each dot is a signaling molecule performing the random walk behind the diffusion equation; dots fade out as molecules degrade \((\tau_{\mathrm{loss}})\). The RX cell's tail beats faster once enough molecules accumulate. Particle view is a stochastic visualization of the same physics; the curves below are the paper's deterministic model.
Stressed algae release volatile compounds that escape into the air above the water and ride the wind — an advection–diffusion channel with chemical decay. Range jumps to meters or kilometers; delay drops to seconds–minutes. The receiver decodes by ligand–receptor binding.
An impulse of \(M_0\) molecules released at the surface propagates by wind advection (speed \(v\)) plus diffusion in\n air, decaying at rate \(\lambda\). Receptor binding at the receiver adds its own latency: the end-to-end peak arrives later than the airborne concentration peak. Parameters follow Table II of the paper. Because the paper does not report (M_0) in concentration units, the figure-calibrated reference concentration is documented under About & Citation.
OOK over the normalized airborne channel amplitude, \(\mathrm{BER}=Q\!\left(A/2\sigma\right)\). Teaching extension; the reference models are deterministic.
Molecules leave the emitting patch (left), drift with the wind and diffuse in air, and decay. The receiving patch (right) counts arrivals and its receptor-occupancy bar fills according to the binding kinetics. Curves below are computed from the closed-form Green's function of the paper.
The stimulus vector \(\sigma(t)=(\sigma_s,\sigma_\ell,\sigma_n)\) drives an \(N\)-species emission vector through \(s_i=a_{i,s}\sigma_s+a_{i,\ell}\sigma_\ell+a_{i,n}\sigma_n\); each compound crosses the air channel with its own \(D_i,\lambda_i\) and binds with its own \(k_{\mathrm{on},i}\), giving the bound-receptor vector \(R_b(t)\). The bars are that codeword — this is the “massive molecular shift keying” the paper points to. The paper tabulates no values for this block and plots no figure for it; the coupling matrix and the per-species spread around Table II are documented reconstructions, not reproductions.
Macroalgae branches conduct action-potential-like voltage pulses, modeled with the same cable equation that describes nerve fibers. Centimeter-to-meter range with millisecond-to-second latency — the colony's rapid response line. The receiver is a voltage-gated channel with a sigmoidal open probability.
The reduced cable equation with space constant \(\lambda\) and membrane time constant \(\tau_m\); a Gaussian voltage pulse is\n injected at \(x = 0\) and a sealed end is imposed at the branch tip. The receiver converts the arriving voltage into a gate open probability. Parameters follow Table III of the paper. Solved here live by finite differences. The transmitter selector switches between Eq. (40)'s Gaussian pulse — the approximation the paper adopts for its own numerics — and the explicit ion-channel encoding of Eqs. (28)–(29), \(g_i(t)=g_0+\Delta g\,f_{\mathrm{stim}}(t)\), \(C_m\dot V_m=-\sum_i g_i (V_m-E_i)\). Setting \(M>1\) adds the branch junction of §VI-A-3: voltage continuity plus axial-current conservation, \(V_J=(V_{J-1}+M\,V_{J+1})/(1+M)\) for equal-diameter daughters.
\(\Delta g\) acts only in the ion-channel mode, and \(x_J\) only when \(M>1\). The paper tabulates no conductances, reversal potentials, \(C_m\) or branch geometry, so these two blocks are normalized reconstructions of Eqs. (28)–(29) and of the junction conditions in §VI-A-3 — see About & Citation.
OOK over the peak received voltage, \(\mathrm{BER}=Q\!\left(A/2\sigma\right)\). Teaching extension.
Top: the branch, colored by instantaneous membrane voltage \(V(x,t)\) from a live finite-difference solution of the cable equation. Bottom: the voltage waveform arriving at the receiver (marker) and the gate open probability \(P_o\) it produces.
Bioluminescent flashes at 470 nm propagate through water with geometric spreading plus Beer–Lambert extinction. The numerical model gives roughly 0.15–0.25 s latency, set by the flash waveform and the photoreceptor, not by the water; propagation itself is effectively instantaneous at these ranges. The colony's urgent-alert channel.
The channel gain is closed-form: inverse-square spreading times exponential extinction (\(a\): absorption, \(b\):\n scattering — Petzold water types). The emitted flash has rise/decay constants \(\tau_r\), \(\tau_d\); the photoreceptor integrates with a saturating Hill nonlinearity, which buffers tens of dB of channel loss down to a few percent of end-to-end loss. Table IV supplies the physical parameters. The emitted signal is the flash train of Eq. (42), \(I_{\mathrm{tx}}(t)=\sum_i A(\sigma_i) f(t-t_i)\), with amplitudes spanning Table IV's \([1.0,1.1]\); the train's count and spacing are controls because the paper tabulates neither. \(\tau_I\) and \(\tau_S\) both use the paper's stated \(\rho=0.5\). Figure 22 is still evaluated on the quasi-steady Hill response rather than the finite-time relaxation peak — see About & Citation.
OOK over received optical power with additive Gaussian noise of std σ (same units as the normalized channel gain), \(\mathrm{BER}=Q\!\left(A/2\sigma\right)\). Teaching extension; a physically rigorous treatment would use a Poisson photon-counting channel — see §X-A of the paper.
The transmitter emits flash trains; each photon travels in a straight line and may be absorbed or scattered out of the path (probability set by the water's extinction coefficient). The eyespot counts surviving photons; the response bar \(S(t)\) shows the decoded photobiological output.
No single modality wins everywhere. The four channels occupy complementary niches in the range–latency plane — a natural multi-layer signaling architecture (paper §VIII, Table V).
Log–log operating regions from the comparative analysis. The optical box uses the 153–254 ms numerical delays in Figs. 21 and 23; Table V gives the broader qualitative label µs–ms. Energy cost per signaling event rises toward the optical channel.
| Modality | Range | Latency | Energy cost | Primary function |
|---|---|---|---|---|
| Quorum sensing | mm – cm | min – hours | Low | Local coordination |
| VOC | m – km | sec – min | Medium | Ecosystem broadcast |
| Electrical | cm – m | ms – s | Medium | Rapid response |
| Optical | cm – m | 153–254 ms in Figs. 21/23 µs–ms in Table V | High | Urgent alert |
A companion model from a second paper: Akyildiz, Sayin, Saygili and Unluturk, NANOCOM ’26, doi 10.1145/3818305.3830272. Every one of \(N\) VOC species is released at once and carries its own \(L\)-level concentration symbol, so a symbol vector holds \(N\log_2 L\) bits. This is a separate, micrometre-scale link — free 3-D diffusion to a fully absorbing receiver — and shares no parameter with the airborne channel on the VOC tab.
Each receptor samples its own species at \(t_{i,\mathrm{peak}}\), where \(dh_i/dt=0\). MRSK encodes in the ratio \(h_i/h_j\), which is time-invariant only when \(D_i=D_j\). Diffusion coefficients follow Stokes–Einstein, \(D\propto M_w^{-1/3}\), from DMS (62 Da, \(D_{\max}=10^{-9}\) m²/s) to the heaviest species; \(a=0.1\) µm. The paper does not say how intermediate masses are distributed, so they are spaced linearly.
Drives only the animation's decisions: each sub-channel count is Poisson with mean \(\bar n_{\max}\,s/(L-1)\), drawn for all \(N\) species every symbol.
Each dot is a VOC molecule taking a random walk with its own Stokes–Einstein \(D_i\) at true scale, so the heavy species visibly lags. Each lane is a 1-D projection toward the receiver: the first-arrival time then has exactly the time dependence of Eq. (1), and its histogram (right) peaks at \(t_{i,\mathrm{peak}}=(r-a)^2/6D_i\). The absorbed fraction \(a/r\) of the 3-D sphere is not drawn. Dashed lines are the sampling instants: each species' own peak for MTDM, the reference (heaviest) species' peak for MRSK-style ratio keying. Decisions come from the Poisson counts set by the SNR control, not from the dots; the counters are a stochastic illustration, not a validation of the Fig. 3 curves.
| Scheme | bits/symbol | b/sym | bps | needs equal \(D\)? |
|---|
Rates use the unrounded Stokes–Einstein \(D_{\min}\) (\(6.72\times10^{-10}\) m²/s at the defaults); the paper rounds it to 6.7 and \(T_s\) to 0.49 s and rounds its bps entries, so at the defaults the bps column differs from Table 1 by −2.3 % to +0.4 % (412 vs 410 for MTDM). The linear \(N\log_2 L\) growth assumes \(N\) orthogonal VOC–receptor pairs and a fixed budget per species; under a fixed total budget each species' SNR falls as \(1/N\) (paper §2). The paper's “below 5 % of peak” is the single-symbol tail; because that tail decays as \(t^{-3/2}\), the worst-case sum over all earlier symbols is larger, as the second readout shows.
\(t_s\) marks the heaviest species' peak, the common sampling instant assumed for MRSK below.
SNR \(=\bar n_{\max}\), the expected absorbed count of the top level, treated as an abstract normalized count (Eq. 1 is a rate and the paper names no counting window); every MTDM species is calibrated to the same \(\bar n_{\max}\) at its own peak; ISI is ignored. The paper does not state its detector. Its two MTDM curves, digitized, agree with the nearest-neighbour approximation for Gray-mapped \(L\)-level PAM under Gaussian noise of constant variance \(\bar n_{\max}\), \(\mathrm{BER}=\tfrac{2(L-1)}{L\log_2 L}\,Q\!\bigl(\sqrt{\bar n_{\max}}/2(L-1)\bigr)\), which is what is plotted; this is inferred from the figure, not a detector the paper states. Its two MRSK curves could not be matched by any stated decoder — including Kilic & Akan's canonical geometric-level MRSK — so the solid MRSK curves here are this page's own MRSK-style ratio keying under Poisson counts: the heaviest species is the reference, all species are sampled at its peak, and \(n_j/n_{\mathrm{ref}}\) is rounded to the nearest of the levels \(s/(L-1)\). On algae VOCs at the defaults its fixed decoder floors at a few %, whereas the paper's algae-VOC curve rises toward 0.5; the uniform-\(D\) curves also differ. The hollow markers are the paper's two MRSK curves, digitized from the published Fig. 3 (CC BY 4.0) at 1 dB steps. They are fixed reference data for the paper's configuration (\(N=100\), \(L=4\), 62–204 Da) and do not follow the controls.
This is an educational, interactive companion to the reference paper below. All attenuation and delay curves are computed in your browser from the paper's equations, parameter tables (Tables I–IV), and the explicitly documented figure-reproduction conventions below: the diffusion–reaction equation for quorum sensing, the advection–diffusion Green's function for VOCs, the cable equation for electrical signaling, and the Beer–Lambert / Hill-receptor chain for optical signaling. No figures are reproduced from the publication.
What is faithful to the paper: the deterministic channel and receiver equations, the tabulated parameters, and the channel-only vs. end-to-end distinction. For optical attenuation, the primary curve now evaluates Eq. (53) literally using the dynamic response of Eq. (47); a separately labelled quasi-steady curve preserves comparison with Fig. 22. The paper does not tabulate every numerical convention needed to regenerate its figures. To reproduce Figs. 5–8, the QS solver uses \(N(0)=0.003224K\), a 12 mm computational domain, and a 1.47 boundary-cell source weight. For Figs. 12–13, the unreported VOC impulse magnitude is represented by a \(5\times10^{-8}\) M reference concentration. For Figs. 20–23, the absolute emitted irradiance — also unreported — is represented by a single scale of 0.2445, together with a 0.15 s flash start. Everything else in the optical chain follows the paper directly: \(\tau_I\) and \(\tau_S\) use the stated \(\rho=0.5\) of Eqs. (52) and (54), and \(\tau_S(d)\) is integrated from Eq. (47) at every distance, so its flatness is a result of photoreceptor saturation rather than an assumption. These are figure-reproduction conventions inferred from the published plots, not independently measured biological parameters. What is added for teaching: (1) the particle animations are stochastic visualizations of the same transport physics — individual molecules and photons are drawn so the channel becomes visible, and the counters count those simulated particles; (2) the bit-error-rate-vs-distance panels of the four channel tabs assume on–off keying with additive Gaussian receiver noise, \(\mathrm{BER}=Q\!\left(A/2\sigma\right)\). The paper deliberately leaves noise modeling, SNR and capacity analysis as open problems (§X); the BER layer here is a standard textbook assumption, clearly separated so students can tell model from extension.