Appearance and suppression of Turing patterns under a periodically forced feed

The study investigates how periodic environmental forcing affects Turing pattern formation and robustness in reaction-diffusion systems. While most prior work assumes constant parameters, biological systems experience periodic changes (e.g., diurnal/seasonal). The authors ask whether sinusoidal modulation of a feed concentration can suppress existing Turing patterns, induce pattern formation from a homogeneous state, or alter morphology. Using the CIMA/CDIMA chemistry as a model, they explore whether time-periodic forcing can destabilize or stabilize stationary patterns and provide a means to control pattern selection and robustness, bridging laboratory conditions with biologically relevant, time-varying environments.

Early experimental realization of chemical Turing patterns used the CIMA reaction in two-side-fed gel reactors, where starch complexation slows the activator diffusion, enabling patterns. Extensive work has linked Turing mechanisms to biological patterning (e.g., animal skin patterns) and explored control strategies. Prior studies on global periodic forcing used light-sensitive CDIMA reactions, showing resonant suppression, entrainment, oscillating patterns, and superlattice formation via periodic illumination masks. Theoretical suggestions include time-dependent boundary temperature control. However, periodic modulation of reagent inflow (feed concentration) as a control parameter had not been systematically explored. The present work addresses this gap, focusing on sinusoidal forcing of one reagent’s feed concentration and its impact on pattern suppression and induction.

Experimental: A disc-shaped two-side-fed gel reactor (TSFR) was used. The agarose gel disc (4 m/V% agarose; thickness 2.5 mm; diameter 20 mm) was clamped between two continuously stirred tanks (A and B; each 26 mL) and immersed in a 6.0 ± 0.1 °C water bath. Reagents were fed from four stock solutions held at 6 °C: (1) NaClO2 + PVA; (2) KIO3 + NaOH + PVA to tank A (transparent); (3) KI + H2SO4 + PVA; (4) KI + H2SO4 + PVA + malonic acid (MA) to tank B (reddish). Fixed tank concentrations: [NaClO2] = 10 mM, [KIO3] = 2 mM, [NaOH] = 12 mM, [KI] = 2 mM, [H2SO4] = 10 mM, [PVA] = 1.5 g/L; [MA] varied 0–14 mM. Solutions (1–3) were delivered by constant HPLC pumps; solution (4) by a programmable peristaltic pump. The control parameter was the MA feed to tank B: either constant inflow rate (baseline, yielding constant [MA]0) or sinusoidal inflow q1(t) = q0 + A sin(2πt/T), producing a periodic [MA]1(t) in tank B computed via a CSTR mass balance. Typical parameters: tank volume V = 26 mL; q2 = 499 mL/h; q0 = 90–170 mL/h; A = 50–170 mL/h; forcing periods T = 5 or 10 min (chosen to exceed tank residence time Tres ≈ 2–3 min but shorter than pattern pre-stabilization time). Symmetric concentration modulation around [MA]0 was enforced. The average [MA] increased slightly under forcing (2.6–9.3%), deemed insufficient by itself to create new pattern states. Imaging used LED backlight through 600 nm band-pass; images captured from tank A side and processed with ImageJ. Experimental screens used forcing amplitudes of 50% and 100% of initial [MA]0 and varied [MA]0.

Numerical simulations: A reduced two-dimensional reaction-diffusion model based on the Lengyel–Epstein activator–inhibitor system was used, applying periodic forcing to the feed parameter a: ∂u/∂t = a + A sin(2πt/T) − u + 4uv/(1 + u^2) + ∇^2 u; ∂v/∂t = α(b − u − uv/(1 + u^2) + c∇^2 v). Periodic boundary conditions were used. Parameter estimation from experiments set b ≈ 1.9; typical values used: σ (α) = 8, b = 1.9, c = 1.5. Pattern regimes at constant a were mapped: starting from homogeneous initial conditions, Turing patterns formed for a between about 23 and 27 (spots to stripes), below 23 homogeneous steady state, and above 27 homogeneous oscillations. Simulations demonstrated bistability once patterns settled. Forcing studies explored amplitude A and period T to reproduce experimental scenarios. Spatial discretization used a uniform 2D square grid (e.g., 50 × 50 domain, grid spacing Δx = Δy = 1.25 × 10^-1), method of lines with finite differences (second-order five-point stencil), alternating direction implicit (ADI) for diffusion with LU decomposition, and Heun's method for reaction terms. Typical time step Δt = 3.124 × 10^-4; simulation time up to 600. Initial conditions: u = v = 0 plus small Gaussian noise (σ = 0.01).

Linear stability analysis: Stability of uniform periodic (time-oscillatory) solutions under periodic forcing was analyzed via a semi-analytic Floquet approach. First, uniform periodic solutions u_p(t), v_p(t) of the spatially uniform ODEs (without diffusion) were computed numerically for given a, A, T (e.g., a = 22, b = 1.9, c = 1.5, σ = 8, A varied, T = 1). Then small spatial perturbations with wavenumber k were introduced, yielding a time-periodic linear system dδ/dt = L(t;k)δ, integrated over one forcing period to construct the linearized Poincaré map P(k). The magnitudes of eigenvalues λ1(k), λ2(k) determined stability: values exceeding one indicate instability of the uniform periodic state. Phase diagrams in the T–A plane were obtained by scanning parameters.

Experimental:

• Periodic forcing can suppress pre-existing Turing patterns. Starting from a stable hexagonal spot pattern at constant [MA]0 = 7 mM, applying a large-amplitude sinusoidal modulation where [MA] in tank B varied between 0 and 14 mM rapidly erased the pattern. Removing forcing restored the original pattern, demonstrating robustness and reversibility.
• Periodic forcing can induce patterns below the Turing threshold. At [MA]0 = 3.9 mM (slightly below the spontaneous pattern onset), a moderate forcing with [MA] varying between 1.95 and 5.85 mM induced a hexagonal spot pattern that persisted during forcing but vanished upon returning to constant feed.
• Forcing can create patterns that remain after forcing ceases (locking). At lower [MA]0 = 3.5 mM with no initial pattern, forcing between 1.75 and 5.25 mM generated a hexagonal spot pattern that remained stable even after forcing was turned off.
• Forcing periods T = 5 or 10 min were effective (consistent with reactor residence time ~2–3 min and pattern response times of minutes). Forcing amplitudes of 50% and 100% of [MA]0 were explored based on simulations.

Numerical simulations:

• At constant parameters, pattern formation from homogeneous initial conditions occurred for a ≈ 23–27 (spots to stripes); below a < 23 homogeneous steady state; above a > 27 homogeneous oscillations. Once formed, patterns persisted over wider parameter ranges, indicating bistability (homogeneous vs. spots; spots vs. stripes).
• Large-amplitude forcing erased pre-existing spot patterns, which reappeared after forcing stopped (qualitatively matching experiments).
• Moderate forcing from a no-pattern state induced Turing spots that remained (locked) after forcing ceased (matching experimental locking). Generated patterns under forcing pulsated in intensity but maintained wavenumber.
• Some experimental transitions (no-pattern → pattern → no-pattern) could not be reproduced, suggesting model limitations; other transitions (stripes → no-pattern → stripes; mixed stripes-spots → spots) were observed.

Linear stability analysis:

• For a representative case (a = 22, b = 1.9, c = 1.5, σ = 8, T = 1), the uniform periodic solution destabilized over a range of wavenumbers; the dominant growth occurred near k ≈ 1.2 when A ≥ 2.
• The stability of the uniform oscillatory state exhibited resonance-like dependence on forcing amplitude and period; varying T at fixed A traversed stable/unstable regions. The T–A phase diagram showed sizable unstable regions where spatial structure can arise.
• Bifurcation points at constant parameters: Turing bifurcation near a ≈ 22.07; Hopf bifurcation near a ≈ 26.88.

Overall, periodic modulation of a single feed concentration can either suppress or promote Turing patterns, enabling controlled switching and pattern locking.

The findings demonstrate that time-periodic forcing of a single feed concentration can qualitatively change the stationary spatiotemporal behavior of a reaction-diffusion system. Patterns that are stable under constant feed can be erased by strong forcing and restored when forcing ceases, while moderate forcing near the Turing threshold can induce patterns from a homogeneous state and even lock them in after forcing is removed. This behavior leverages the subcriticality and bistability inherent to Turing transitions. Numerical simulations with a periodically forced Lengyel–Epstein model qualitatively reproduce key experimental scenarios (pattern suppression and induced locking) and highlight bistability between homogeneous and patterned states. The semi-analytic linear stability analysis indicates that uniform periodic states can become spatially unstable within specific amplitude–period regions, consistent with resonance-induced instabilities and offering a mechanism for forcing-enabled pattern onset. These results address the central question of robustness and control in pattern formation under time-varying conditions and suggest practical routes to manipulate patterns in biomimetic and materials contexts.

The study shows that periodic modulation of a reagent’s inflow (feed concentration) can serve as an effective control knob for Turing patterns in a boundary-fed gel reactor: large amplitudes suppress existing patterns reversibly; moderate amplitudes near threshold induce and can lock in patterns. Simulations and linear stability analysis support these observations, revealing parameter regimes where uniform periodic states lose stability to spatial modes. This work extends pattern control from global illumination to feed modulation, aligning better with biological scenarios of cyclic environments. Future research directions include: developing more detailed chemical models to capture experimentally observed transitions not reproduced by the two-variable model; analytical methods to predict pattern selection (spots vs. stripes) under forcing; stability analysis of pattern disappearance; exploring multi-parameter or multi-frequency forcing; and investigating three-dimensional effects and complex geometries.

• Model simplification: the two-variable Lengyel–Epstein model may not capture all kinetics relevant under periodic forcing; it failed to reproduce some experimental transitions (e.g., no-pattern → pattern → no-pattern).
• Pattern selection under forcing (spots vs. stripes) was not determined analytically; new techniques are required.
• Numerical modeling used a 2D reduction of an inherently 3D system; while reasonable qualitatively, certain 3D effects or spatial gradients along the feed direction could be missed.
• Slight shifts in average [MA] and residence times during forcing (2.6–9.3% increase in average [MA]; <10% changes in residence time) could introduce small biases, though deemed negligible for dynamics.
• Forcing choices (T = 5 or 10 min; amplitudes 50–100% of [MA]0) were guided by practical and prior considerations; a comprehensive parameter sweep in experiments is limited.