Fig 1.
Workflow of our proposed END-nSDE prediction on parameters altering stochastic dynamics.
A. Workflow for training and testing of the extrinsic-noise-driven neural SDE (END-nSDE). Predicted trajectories are simulated (see B) using a range of model parameters (see Sect 2.2) before splitting into training and testing sets (see Fig E in S1 Text for details on the splitting strategy). Model parameters and state variables serve as inputs to a neural network that reconstructs drift and diffusion terms (see C). Network weights are optimized by minimizing the Wasserstein distance (Eq 8) between the training set and predicted trajectories. B. Predicted trajectories are generated by the reconstructed SDE . C. The drift and diffusion functions,
and
, are approximated using parameterized neural networks. The parameterized neural-network-based drift function
and diffusion function
take the system state
and biological parameters ω as inputs. D. Table of three examples illustrating the nSDE input, along with training and testing datasets. For the last, NF
B example, a more detailed workflow for validation on experimental datasets is illustrated in Fig 8.
Fig 2.
A continuous-time discrete Markov chain model for multiple RPA molecules binding to long ssDNA.
The possible steps in the biomolecular kinetics of multiple RPA molecules binding to ssDNA. The RPA in the free solution can bind to ssDNA with rate k1 provided there are at least 20 nucleotides (nt) of consecutive unoccupied sites. This bound “20nt mode” RPA unbinds with rate k−1. When space permits, the 20nt-mode RPA can extend and bind an additional 10nt of DNA at a rate of k2, converting it to a 30nt-mode bound protein. The 30nt-mode RPA transforms back to 20nt-mode spontaneously with the rate k−2. However, when the gap is not large enough to accommodate the RPA, the binding or conversion is prohibited ( and
).
Fig 3.
Simplified schematic of the NFB Signaling Network.
TNF binds its receptor, activating IKK, which degrades IBα and releases NF
B. The free NF
B translocates to the nucleus and promotes I
Bα transcription. Newly synthesized I
Bα then binds NF
B and exports it back to the cytoplasm. Red arrows indicate noise that we consider in the corresponding SDE system.
Fig 4.
Reconstructing the circadian model using END-nSDE.
Temporally decoupled squared W2 losses Eq (3) and errors in the reconstructed drift and diffusion functions for different types of the diffusion function and different values of . A-C. The temporally decoupled squared W2 loss between the ground truth trajectories and the trajectories generated by the reconstructed nSDEs for the constant-type diffusion function Eq (11), Langevin-type diffusion function Eq (12), and the linear-type diffusion function Eq (13). D-F. Errors in the reconstructed drift function for the three different types of ground truth diffusion functions and the linear-type diffusion function Eq (13). G-I. Errors in the reconstructed diffusion function for the three different types of ground truth diffusion functions.
Fig 5.
Average temporally decoupled squared W2 losses Eq (3) and errors in the reconstructed drift and diffusion functions for different choices of intrinsic noise strength and extrinsic noise strength in Eq (16).
Table 1.
The extrinsic-noise-driven time-decoupled squared W2 distance Eq (8) between the ground truth and predicted trajectories generated by different models on the testing set.
Fig 6.
Reconstructed trajectories of the RPA-DNA binding model.
A. Sample ground truth and reconstructed trajectories evaluated at , where we use the convention that
. B. Sample ground truth and reconstructed parameters evaluated at
. C. Temporally decoupled squared W2 distances (see Eq (8)) between the ground truth and reconstructed trajectories evaluated at different
values. In A and B, blue and red trajectories represent the filling fractions of DNA by 20nt-mode and 30nt-mode RPA, respectively. The dashed lines represent the predicted trajectories, and the solid lines represent the ground truth. Throughout the figure, the data are generated by a single neural SDE model that accepts the conversion rate k2 as a parameter and outputs the trajectories.
Fig 7.
Reconstruction of NFB signaling dynamics.
A. Sample trajectories of nuclear NFB concentration as a function of time with
,
. B. Sample trajectories of nuclear NF
B concentration as a function of time with
,
. C. Reconstructed nuclear NF
B trajectories generated by the trained neural SDE versus the ground truth nuclear NF
B trajectories under noise intensities
,
in Eq (10). D. Reconstructed nuclear NF
B trajectories generated by the trained neural SDE versus the ground truth nuclear NF
B trajectories under noise intensities
,
. E. The squared W2 distance between the distributions of the predicted trajectories and ground truth trajectories on the training set under different noise strengths
. For training, we randomly selected 50% sample trajectories in 80 combinations of noise strengths
as the training dataset. Blank cells indicate that the corresponding parameter set is not included in the training set. F. Validation of the trained model by evaluating the squared W2 distance between the distributions of predicted trajectories and ground truth trajectories on the validation set.
Fig 8.
Workflow of reconstructing experimental data via END-nSDE.
Workflow for reconstructing experimental data using the trained parameterized nSDE and the parameter-inference neural network (NN). The boxes on the left outline the steps of the experimental data reconstruction process, while the boxes on the right illustrate the corresponding results at each step.
Fig 9.
Inferring intrinsic noise intensities and reconstructing experimental data via END-nSDE.
A. Plots showing the mean (solid circles) and variance (error bars) of the relative error in the reconstructed noise intensities predicted by the parameter-inference NN for the testing dataset, as a function of the group size of input trajectories. B. Heatmaps showing the relative error in the reconstructed noise intensities for the training dataset. Colored cells represent results from the parameter-inference NN for the training dataset, while blank cells indicate noise strength values not included in the training set. C. Heatmaps showing the relative error in the diffusion function for the testing dataset. D. The inferred intensity of I
Bα transcription noise (
) and NF
B translocation noise (
) in different groups of experimental trajectories, plotted against the group’s ranking in decreasing similarity with the representative ODE trajectory. E-H. Groups of experimental and nSDE-reconstructed trajectories ranked by decreasing cosine similarity: #1 (E), #4 (F), #16 (G), #29 (H). The squared W2-distance between experimental and SDE-generated trajectories are 0.157 (E), 0.143 (F), 0.212 (G), 0.236 (H). The inferred noises are (10−0.49,10−0.81) (E), (10−0.47,10−0.78) (F), (10−0.46,10−0.74) (G), (10−0.44,10−0.71) (H). I. The temporally decoupled squared W2 distance between reconstructed trajectories generated by the trained END-nSDE and groups of experimental trajectories, ordered according to decreasing similarity with the representative ODE trajectory.