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Physics-constrained neural ordinary differential equation models to discover and predict microbial community dynamics.

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The authors' code

Jupyter notebook · 290 lines · 7.9 KB · no license

  1. # %%
  2. import numpy as np
  3. import itertools
  4. import pandas as pd
  5. import matplotlib.pyplot as plt
  6. import seaborn as sns
  7. import scipy as sp
  8. from scipy.stats import linregress
  9. import pickle
  10. import time
  12. from nsm.nsm import *
  14. # set plot parameters
  15. params = {'legend.fontsize': 18,
  16. 'figure.figsize': (8, 7),
  17. 'axes.labelsize': 24,
  18. 'axes.titlesize':24,
  19. 'axes.linewidth':3,
  20. 'xtick.labelsize':20,
  21. 'ytick.labelsize':20,
  22. 'xtick.labelsize':20,
  23. 'ytick.labelsize':20}
  24. plt.rcParams.update(params)
  25. # plt.style.use('seaborn-colorblind')
  26. plt.rcParams['pdf.fonttype'] = 42
  28. np.random.seed(123456)
  30. # %% [markdown]
  31. # # Import data
  33. # %%
  34. # import Ryan's 25 mem community data
  35. df = pd.read_csv("data/clark.csv")
  37. # %%
  38. # df_4t = []
  39. # for t, df_t in df.groupby('Treatments'):
  40. # if df_t.shape[0]>2:
  41. # df_4t.append(df_t)
  42. # df = pd.concat(df_4t)
  44. # %%
  45. # define species and metabolites
  46. species = ['ER_OD','FP_OD','AC_OD','CC_OD','RI_OD','DP_OD','BH_OD','CA_OD','PC_OD','DL_OD','CG_OD','BF_OD','EL_OD','CH_OD','BO_OD','BT_OD','BU_OD','BV_OD','BC_OD','BY_OD','PJ_OD','DF_OD','BL_OD','BP_OD','BA_OD']
  47. mediators = ['Acetate', 'Butyrate', 'Lactate', 'Succinate']
  48. inputs = []
  50. # %%
  51. # normalize metabolites to [0, 1]
  52. df[mediators] /= df[mediators].max()
  53. df[mediators]
  55. # %%
  56. # init model
  57. model = NSM(dataframe=df,
  58. species=species,
  59. mediators=mediators,
  60. inputs=inputs,
  61. n_hidden=32, n_layers=1,
  62. nu2=.001, sigma2=.01)
  64. # init params
  65. model.init_params(sample=False)
  66. print(f"Number of parameters: {model.n_params}")
  68. # %%
  69. print(f"Total parameters: {model.n_params}")
  70. t0 = time.time()
  71. model.fit_posterior_EM(lr=.001,
  72. n_sample=1,
  73. n_sample_hypers=100,
  74. max_epochs=10000,
  75. patience=1, max_iterations=10)
  76. print("Elapsed time: {:.3f}s".format(time.time()-t0))
  78. # %%
  79. model.nu2
  81. # %%
  82. model.sigma2
  84. # %%
  85. # Define function to make predictions on test data
  86. def plot_sample(model, df_test):
  88. # save true values and predictions
  89. true = []
  90. pred = []
  91. spcs = []
  93. # pull a random community trajectory
  94. for treatment, comm_data in df_test.groupby("Treatments"):
  96. # make sure comm_data is sorted in chronological order
  97. comm_data.sort_values(by='Time', ascending=True, inplace=True)
  98. tspan = comm_data.Time.values
  100. # pull just the community data
  101. species_true = comm_data[species].values
  102. mediators_true = comm_data[mediators].values
  103. input_vals = comm_data[inputs].values[0]
  105. output_true = np.concatenate((species_true, mediators_true), -1)
  107. # determine initial condition
  108. s0 = species_true[0]
  109. r0 = mediators_true[0]
  110. x0 = np.concatenate((s0, r0))
  112. # increase teval
  113. t_eval = np.linspace(0, tspan[-1] + 5., 100)
  114. steps = len(t_eval)
  116. # generate posterior samples
  117. species_pred, mediators_pred = model.predict_sample(x0,
  118. t_eval,
  119. n_sample=50,
  120. inputs=input_vals)
  122. # compute summary statistics (mean and variance) of posterior predictive distribution
  123. E_species_pred, V_species_pred, E_mediators_pred, V_mediators_pred = model.predict(x0,
  124. t_eval,
  125. n_sample=100,
  126. inputs=input_vals)
  128. # concatenate species and mediator predictions
  129. output_pred = np.concatenate((species_pred, mediators_pred), -1)
  130. E_output_pred = np.concatenate((E_species_pred, E_mediators_pred), -1)
  131. V_output_pred = np.concatenate((V_species_pred, V_mediators_pred), -1)
  133. # predict end-point measured values
  134. true.append(output_true[-1])
  135. pred.append(E_output_pred[-1])
  136. spcs.append(species+mediators)
  138. # plot the results
  139. '''plt.figure(figsize=(8, 6))
  140. k = 0
  141. for i, sp in enumerate(species+mediators):
  143. if sp in mediators:
  145. out_true = output_true[:,i]
  146. plt.scatter(tspan, out_true, color='C{}'.format(k), edgecolor='k', s=100, zorder=2, label=sp)
  148. # plot posterior samples
  149. for out in output_pred:
  150. plt.plot(t_eval, out[:, i], color='C{}'.format(k), alpha=.15, zorder=0)
  152. # plot posterior predictive as a Gaussian
  153. plt.fill_between(t_eval,
  154. E_output_pred[:,i] - V_output_pred[:, i] ** .5,
  155. E_output_pred[:,i] + V_output_pred[:, i] ** .5,
  156. color='C{}'.format(k), alpha=0.2)
  158. # plot mean of posterior
  159. plt.plot(t_eval, E_output_pred[:,i], color='C{}'.format(k), linewidth=3, zorder=1)
  160. k += 1
  162. plt.axhline(y=0, linestyle='-', color='k', alpha=.5)
  163. plt.xlabel("Time (hr)")
  164. plt.ylabel("Species abundance")
  165. plt.legend(fontsize=14, loc='upper left')
  166. # plt.title(input_vals)
  168. # plt.ylim([0, .5])
  169. plt.tight_layout()
  170. plt.savefig("figures/trajectories/nsm.pdf")
  171. plt.title("NSM")
  172. plt.show()
  173. # break'''
  175. return true, pred, spcs
  177. # %%
  178. true, pred, spcs = plot_sample(model, df)
  180. # %%
  181. true = np.array(true).ravel()
  182. pred = np.array(pred).ravel()
  184. # %%
  185. mediator = 'AC_OD'
  186. inds = np.array(spcs).ravel() == mediator
  188. plt.scatter(true[inds], pred[inds])
  189. print(linregress(true[inds], pred[inds]))
  190. plt.title(mediator)
  191. plt.show()
  193. # %%
  194. for mediator in mediators:
  196. inds = np.array(spcs).ravel() == mediator
  198. plt.scatter(true[inds], pred[inds])
  199. print(linregress(true[inds], pred[inds]))
  200. plt.title(mediator)
  201. plt.show()
  203. # %%
  204. def get_params(self, n_sample=100):
  206. # parameter names
  207. param_names = []
  209. # consumption rates
  210. for md in self.mediators:
  211. for sp in self.species:
  212. param_names += [md+"->"+sp]
  214. # production rates
  215. for md in self.mediators:
  216. for sp in self.species:
  217. param_names += [sp+"->"+md]
  219. # sample parameters
  220. y = np.random.randn(n_sample, self.d)
  221. z = batch_T(self.shapes, y, self.lmbda)
  223. # init vector to store parameter samples
  224. params = np.zeros([n_sample, 2 * self.n_m * self.n_s])
  226. # transform and reshape parameters
  227. for i, z_i in enumerate(z):
  229. # transform / reshape parameters
  230. C_i, P_i, *rest = transform(reshape(self.shapes, z_i))
  231. params[i] = np.append(C_i.ravel(), P_i.ravel())
  233. # compute mean / stdv
  234. # C_mean = np.mean(C, 0).ravel()
  235. # C_stdv = np.std(C, 0).ravel()
  236. # P_mean = np.mean(P, 0).ravel()
  237. # P_stdv = np.std(P, 0).ravel()
  239. # save to dataframe
  240. param_df = pd.DataFrame()
  241. # param_df['Parameter'] = param_names
  242. # param_df['$\mu$'] = np.append(C_mean, P_mean)
  243. # param_df['$\sigma$'] = np.append(C_stdv, P_stdv)
  244. param_df[param_names] = np.array(params, float)
  246. return param_df
  248. # %%
  249. param_df = get_params(model)
  251. # %%
  252. plt.hist(param_df['Butyrate->AC_OD'].values)
  253. plt.show()
  255. # %%
  256. plt.hist(param_df['AC_OD->Butyrate'].values)
  257. plt.show()
  259. # %%
  260. C, P, d_m = params[0], params[1], params[2]
  262. # %%
  263. mediators
  265. # %%
  266. np.round(P[1][np.argsort(P[1])[::-1]], 3)
  268. # %%
  269. np.array(species)[np.argsort(P[1])[::-1]]
  271. # %%
  272. np.round(C[1][np.argsort(C[1])[::-1]], 3)
  274. # %%
  275. np.array(species)[np.argsort(C[1])[::-1]]
  277. # %%
  278. np.round(P[0][np.argsort(P[0])[::-1]], 2)
  280. # %%
  281. np.array(species)[np.argsort(P[0])[::-1]]
  283. # %%
  284. np.array(species)[np.argsort(P2[1])]
  286. # %%
  287. np.array(species)[np.argmax(C2[1])]
  289. # %%
  290. d_m

CR_clark_fit (P ds_dt + C s)-checkpoint.ipynb at commit 28caf3b, no license · at the source

Overview

Authors: Jaron Thompson1,2, Bryce M Connors1,2, Victor M Zavala1, Ophelia S Venturelli1,2,3,4
ORCID iDs: Victor M Zavala, Ophelia S Venturelli
  1. Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison, WI 53706
  2. Department of Biochemistry, University of Wisconsin-Madison, Madison, WI 53706
  3. Department of Biological Engineering, Duke University, Durham, NC 27708
  4. Department of Molecular Genetics and Microbiology, Duke University, Durham, NC 27708
Institutions: University of Wisconsin–Madison (United States); Duke University (United States)
Journal: Proceedings of the National Academy of Sciences of the United States of America, volume 123, issue 13, article e2517661123
Dates: received 9 July 2025; accepted 19 February 2026; published online 27 March 2026; in print 31 March 2026
Type: Research article · Language: English
License: CC BY-NC-ND
Identifiers: DOI 10.1073/pnas.2517661123 · PMID 41894324 · PMCID PMC13037942 · OpenAlex W7141338503
Open access: hybrid, a free copy (OpenAlex)
Status: code verified (how the code is found and verified)
Categories: computational modeling (no new data) (modality), none (in silico) (organism)
Keywords: microbial communities, microbial interaction networks, physics-informed machine learning, neural ordinary differential equations, nonlinear dynamical systems
MeSH: Microbiota*, Models, Biological*, Neural Networks, Computer*, Machine Learning, Predictive Learning Models (* major topic)
Topic: Microbial Metabolic Engineering and Bioproduction (Molecular Biology, Biochemistry, Genetics and Molecular Biology), according to OpenAlex
Funding: National Science Foundation (NSF) (CBET-2315963); DOD | ARPA | Defense Sciences Office, DARPA (DSO) (HR00112320001, HR00112530072); HHS | National Institutes of Health (R01EB030340); NIBIB NIH HHS (R01 EB030340)
Citations: cited by 5 papers (Europe PMC); 59 references in the paper

Abstract

The abstract is not reproduced here: the paper's license (CC BY-NC-ND) does not allow it. Read it in the paper, at the publisher or on Europe PMC.

Repository

Its files are read in the Code ↔ Paper reader above.

VenturelliLab/Thompson_et_al_2025

License: none: the authors keep all their rights
State: the link answers, verified on 30 September 2026
Evidence: files inventoried
Commit: 28caf3ba3ead37b2a796896cde9e960ba4c06ee6, 9 March 2026
Languages: Jupyter (172), Python (72), Shell (31)
Size: 11,758 files, 275 scripts
Software Heritage: not archived
Found in: “Data, Materials, and Software Availability”
Holds: 12 notebooks
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: Matplotlib (14 files), NumPy (14 files), pandas (14 files), SciPy (13 files), seaborn (10 files), SHAP (2 files)
Availability: 1 check, the latest on 30 September 2026: the link answers
  • 30 September 2026: the link answers
14 files
At the source: github.com/VenturelliLab/Thompson_et_al_2025

The paper's code and data availability statement is in the Data section.

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Read it in the paper: doi.org/10.1073/pnas.2517661123.

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Version 1, 30 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 4 authors, 5 keywords, 5 MeSH terms, 4 funders, 35 references.

Cite

This paper

Thompson, J., Connors, B. M., Zavala, V. M., & Venturelli, O. S. (2026). Physics-constrained neural ordinary differential equation models to discover and predict microbial community dynamics. Proceedings of the National Academy of Sciences of the United States of America, 123(13), e2517661123. https://doi.org/10.1073/pnas.2517661123

BibTeX

@article{thompson2026physics,
author = {Thompson, Jaron and Connors, Bryce M and Zavala, Victor M and Venturelli, Ophelia S},
title = {{Physics-constrained neural ordinary differential equation models to discover and predict microbial community dynamics}},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
year = {2026},
month = mar,
volume = {123},
number = {13},
pages = {e2517661123},
publisher = {National Academy of Sciences},
issn = {0027-8424},
doi = {10.1073/pnas.2517661123},
url = {https://doi.org/10.1073/pnas.2517661123},
pmid = {41894324},
pmcid = {PMC13037942}
}

RIS

TY - JOUR
AU - Thompson, Jaron
AU - Connors, Bryce M
AU - Zavala, Victor M
AU - Venturelli, Ophelia S
TI - Physics-constrained neural ordinary differential equation models to discover and predict microbial community dynamics
T2 - Proceedings of the National Academy of Sciences of the United States of America
J2 - Proc Natl Acad Sci U S A
PY - 2026
DA - 2026/03/27
VL - 123
IS - 13
SP - e2517661123
SN - 0027-8424
PB - National Academy of Sciences
DO - 10.1073/pnas.2517661123
UR - https://doi.org/10.1073/pnas.2517661123
LA - en
ER -

CSL-JSON

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"id": "10.1073/pnas.2517661123",
"type": "article-journal",
"title": "Physics-constrained neural ordinary differential equation models to discover and predict microbial community dynamics",
"container-title": "Proceedings of the National Academy of Sciences of the United States of America",
"author": [
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"family": "Thompson",
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},
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"volume": "123",
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"DOI": "10.1073/pnas.2517661123",
"PMID": "41894324",
"PMCID": "PMC13037942",
"ISSN": "0027-8424",
"publisher": "National Academy of Sciences",
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"date-parts": [
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2026,
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