Model

Bacterial siRNA induction

We estimated the induction time of siRNA production using a heat-inducible dnaK promoter. In E. coli, dnaK promoter activity rises steeply within 5–20 minutes after a heat shock of 42–45°C (PMC107371). In our model, the promoter activation is captured by a sigmoidal function (T50 = 37°C, k_T = 1.5) to approximate partial activation at 40°C, reflecting realistic induction dynamics.

siRNA transcription & processing

Bacterial RNA polymerase elongates at 17–23 nucleotides per second (PMC10733919). For a short siRNA (~20–30 nt), transcription completes in under 10 seconds. Mature siRNA is generated through endoribonuclease processing (e.g., RNase III) within minutes of transcription (PMC10733919). These timescales inform the effective maximum siRNA production rate (alpha) in our model.

siRNA release and transfer to plants

siRNA produced in bacteria is released into the rhizosphere, potentially via extracellular vesicles, similar to mechanisms observed in fungal-plant interactions (PMC8369896). Uptake by plant roots is modeled using a first-order rate constant (k_up), and release from bacteria is captured by k_rel. Encapsulation and transport efficiencies are estimated from analogous studies using synthetic and lipid nanoparticles (PMC7314522).

siRNA stability in bacteria, rhizosphere, and plant cells

siRNA decay is modeled separately in bacteria (gamma_sb), the rhizosphere (gamma_sr), and plant cells (gamma_sp). Intracellular siRNA in plant cells can be stable from hours to days, and long-term RNAi silencing may persist for weeks (PMC7108305).

Plant target mRNA knockdown

Once siRNA is taken up by the plant, knockdown of the target mRNA (bHLH61) is typically observed within 24 hours, with maximal silencing occurring 1–2 days post-treatment (Sci Adv 2020).

Model Summary

In our first figure, we can look at siRNA pools over time in response to heat treatment (red). The siRNA inside the bacteria (S_b; blue), siRNA in the soil around the roots (S_r; green), and the siRNA inside the plant (S_p; magenta). We observe that when temperature rises, such as in a heat wave, the heat-inducible promoter (dnaK) turns on. This makes the bacteria start producing siRNA. First, the siRNA builds up inside the bacteria. Then, it gets released into the soil and is taken up by the plant roots. Once the heat wave is over and temperatures return to normal, siRNA levels rapidly decrease.

Figure 2 shows the level of target mRNA in the plant decreasing once siRNA enters the plant. As the plant’s RNAi system is triggered by the siRNA, the target mRNA is degraded but rises once the siRNA levels decrease as temperatures return to normal. Figure 3 shows the percent knockdown of the mRNA. While the model shows 100% gene knockdown, we wouldn’t expect this in real life situations, as mRNA production may be greater than mRNA degradation.

This modeling framework helps guide our project by connecting molecular processes to potential real life outcomes. By simulating heat-inducible promoter activity, we can estimate how long and at what temperature bacteria need to be exposed to reliably produce siRNA, allowing us to estimate effective induction conditions. These models also inform us on how much siRNA accumulates and how stable it is in bacteria, the rhizosphere, and plant cells, which can help us decide if the heat wave length is sufficient or whether longer times are required for siRNA production. Analyzing the transfer rates between bacteria, soil, and plant roots show us whether natural uptake is likely to be effective or whether engineered secretion or vesicle-based delivery strategies may be needed. Finally, modeling target mRNA knockdown allows us to predict how quickly and to what extent gene silencing will occur in the plant, allowing us to set realistic knockdown expectations.

Our model is really useful for seeing how siRNA might move from bacteria into plants, but it does have some limits.The model turns a really complicated process into just a few numbers and since a lot of those numbers come from studies in other systems, the predictions are more about general trends than exact values. Additionally, the model ignores spatial differences and assumes everything is mixed evenly, but in real soil siRNA spreads by diffusion, roots are only in certain spots, and conditions like pH or other microbes could make a big difference. The promoter response is oversimplified too. We treat it as a smooth curve that turns on with heat, but in reality heat-shock genes can pulse and adapt over time. We also combined siRNA transcription, processing, and packaging into one step, even though each part could slow things down (or be quite difficult for the bacteria to achieve). On the plant side, our model predicts siRNA knockdown of the mRNA in a saturating way, but in real plants, there can be amplification or long-term silencing that keeps going beyond what we modeled. So overall, the model is better for asking “what if” questions and comparing scenarios than for giving exact predictions.

MATLAB Code

  
  function promoter_siRNA_knockdown
      % Parameters
      p.alpha    = 10.0;   % max siRNA production (a.u./min)
      p.T50      = 37.0;   % midpoint temperature (°C)
      p.k_T      = 1.5;    % steepness for activation
      p.gamma_sb = 0.01;   % bacterial siRNA degradation
      p.k_rel    = 0.05;   % release rate
      p.gamma_sr = 0.02;   % rhizosphere decay
      p.k_up     = 0.02;   % uptake into plant
      p.gamma_sp = 0.05;   % plant siRNA decay
      p.r_m      = 1.0;    % target mRNA transcription
      p.gamma_m  = 0.01;   % target mRNA degradation
      p.k_sil    = 0.8;    % max silencing rate
      p.K        = 5.0;    % siRNA saturation constant
  
      % Time span (minutes)
      tspan = [0 2000];
  
      % Initial conditions
      M0 = p.r_m / p.gamma_m;
      y0 = [0 0 0 M0]; % [S_b, S_r, S_p, M]
  
      % Solve ODEs
      [t, y] = ode45(@(t,y) odesys(t,y,p), tspan, y0);
  
      S_b = y(:,1);
      S_r = y(:,2);
      S_p = y(:,3);
      M   = y(:,4);
      kd  = 100*(1 - M/M0);
  
      % Temperature profile
      T = temperature_profile(t);
  
      % --- Plot results ---
      figure;
      yyaxis left
      plot(t/60, T, 'r', 'LineWidth', 2);
      ylabel('Temperature (°C)');
      yyaxis right
      plot(t/60, S_b, 'b', t/60, S_r, 'g', t/60, S_p, 'm', 'LineWidth', 1.5);
      ylabel('siRNA (a.u.)');
      xlabel('Time (hours)');
      legend('Temp','S_b','S_r','S_p');
      title('siRNA pools over time');
  
      figure;
      plot(t/60, M, 'k', 'LineWidth', 1.5); hold on;
      yline(M0, '--');
      xlabel('Time (hours)');
      ylabel('Target mRNA (a.u.)');
      title('Plant target mRNA dynamics');
  
      figure;
      plot(t/60, kd, 'c', 'LineWidth', 1.5);
      xlabel('Time (hours)');
      ylabel('Knockdown (%)');
      title('Percent knockdown of target mRNA');
  end
  
  % --- ODE system ---
  function dydt = odesys(t, y, p)
      S_b = y(1); S_r = y(2); S_p = y(3); M = y(4);
  
      T = temperature_profile(t);
      A = activation(T, p.T50, p.k_T);
  
      dSb = p.alpha*A - p.gamma_sb*S_b - p.k_rel*S_b;
      dSr = p.k_rel*S_b - p.gamma_sr*S_r - p.k_up*S_r;
      dSp = p.k_up*S_r - p.gamma_sp*S_p;
      silencing = p.k_sil * (S_p/(p.K+S_p)) * M;
      dM = p.r_m - p.gamma_m*M - silencing;
  
      dydt = [dSb; dSr; dSp; dM];
  end
  
  % --- Activation function ---
  function A = activation(T, T50, k_T)
      A = 1.0 ./ (1.0 + exp(-(T - T50)/k_T));
  end
  
  % --- Temperature profile ---
  function T = temperature_profile(t)
      T = 25*ones(size(t));   % baseline 25°C
      T(t >= 300 & t <= 900) = 40.0;  % heat wave
  end