Inferring Force Fields from Cross-Sectional Biological Data Using Probability Flow Inference (PFI) with a Focus on Molecular Noise

핵심 개념

Accurately inferring the dynamics of biological processes from cross-sectional data requires accounting for intrinsic noise, especially molecular noise, and the Probability Flow Inference (PFI) method offers a computationally efficient way to achieve this by leveraging score-based generative models and incorporating realistic noise priors.

초록

• Bibliographic Information: Maddu, S., Chardès, V., & Shelley, M. J. (2024). Inferring biological processes with intrinsic noise from cross-sectional data. arXiv preprint arXiv:2410.07501v1.
• Research Objective: This paper introduces Probability Flow Inference (PFI), a novel method for inferring the force field driving a stochastic dynamical system from cross-sectional data, focusing on the challenge of disentangling deterministic forces from intrinsic noise, particularly in the context of molecular noise in biological systems.
• Methodology: PFI leverages score-based generative models to estimate the gradient log-probability of time-evolving distributions and then fits a probability flow ODE to the data, incorporating prior knowledge about the noise model, such as the Chemical Langevin Equation (CLE) for molecular noise. The authors provide theoretical analysis for Ornstein-Uhlenbeck processes, demonstrating the role of regularization in ensuring identifiability and studying the bias-variance trade-off arising from finite sampling. They validate PFI on simulated data from stochastic reaction networks and a realistic model of hematopoietic stem cell differentiation, comparing its performance to existing methods like TrajectoryNet and PRESCIENT.
• Key Findings:
  • PFI accurately infers force fields and parameters of stochastic reaction networks, outperforming methods that ignore or simplify intrinsic noise.
  • The choice of noise model significantly impacts the accuracy of the inferred dynamics, with CLE-based models generally providing the best performance for systems with molecular noise.
  • PFI demonstrates superior generalization capabilities compared to other methods, accurately predicting the system's behavior under different initial conditions and perturbations.
  • In cell differentiation modeling, PFI captures the impact of molecular noise on cell fate decisions and correctly predicts the effects of gene knockdowns, highlighting the importance of incorporating realistic noise models in such applications.
• Main Conclusions: PFI offers a powerful and computationally efficient approach for inferring stochastic dynamics from cross-sectional data, effectively disentangling deterministic forces from intrinsic noise. The method's ability to incorporate realistic noise priors, particularly for molecular noise, makes it particularly well-suited for modeling complex biological processes like gene regulation and cell differentiation.
• Significance: This work significantly advances the field of stochastic dynamics inference by providing a flexible and accurate method for analyzing cross-sectional biological data, which is particularly relevant for single-cell omics studies where obtaining time-resolved trajectories is often infeasible.
• Limitations and Future Research: While PFI demonstrates strong performance, future work could explore its applicability to non-stationary noise models and develop strategies for inferring the noise model directly from data when prior knowledge is limited. Additionally, extending PFI to handle partially observed systems and incorporate spatial information could further broaden its applicability in biological research.

통계

The relative strength of the non-conservative and conservative forces in the isotropic Ornstein-Uhlenbeck process example is 3 to 1.
The time scales of diffusion and non-conservative forces in the isotropic Ornstein-Uhlenbeck process example are comparable (τforce/τdiff = D/(Σ0Ωmax)).
The score estimation using sliced score-matching achieved a Root Mean Square Error (RMSE) of ~2.5%.
The inferred force field in the isotropic Ornstein-Uhlenbeck process example achieved an RMSE of ~9%.
The mCAD gene regulatory network used in the study consists of d = 5 genes.
The Gillespie algorithm was used to generate K = 10 snapshots, each containing n = 6,000 samples for the mCAD network.
The PFI approach used a feed-forward neural network with four fully connected layers, each with 50 nodes and smooth ELU activation to parameterize the force.
The regularization parameter in the mCAD network analysis was set to λ = 10−4.
The Sinkhorn divergence with ϵ = 0.1 was used to compare the predicted and measured distributions in the mCAD network analysis.
The linear cyclic network used to evaluate parameter estimation consisted of d = 30 species and R = 30 reactions.
The HSC differentiation model included 11 transcription factors and captured differentiation into four cell types: Erythrocytes, Megakaryocytes, Monocytes, and Granulocytes.
The HSC regulatory network simulation generated K = 8 snapshots with n = 5000 samples.

인용구

"However, force and state-dependent noise models not only better capture biological variability, but they also have the capacity to shift, create, or eliminate fixed points in the energy landscape, which is of paramount importance to model processes like cell differentiation [1, 2, 6]."
"Therefore, accounting for intrinsic noise is crucial to accurately infer cellular processes from single-cell omics data."

핵심 통찰 요약

Inferring biological processes with intrinsic noise from cross-sectional data

by Sury... 게시일 arxiv.org 10-11-2024

https://arxiv.org/pdf/2410.07501.pdf

더 깊은 질문

How can the PFI approach be adapted to handle scenarios where the intrinsic noise model is not known a priori, and what are the potential challenges in inferring both the force field and the noise model simultaneously from cross-sectional data?

Adapting the PFI approach to scenarios where the intrinsic noise model is not known a priori presents a significant challenge, as it transforms the problem into a simultaneous inference of both the force field and the noise model. This is inherently difficult with cross-sectional data due to the lack of time-correlated information. Here are potential strategies and challenges:
Potential Strategies:

Parametric Noise Models: Instead of assuming a fixed noise model, one could parameterize a family of noise models (e.g., state-dependent with unknown parameters). The PFI loss function (Eq. 6) would then need to be jointly minimized over both the force field parameters and the noise model parameters.

Alternating Minimization: An iterative approach could be employed, where one alternates between optimizing the force field (given a noise model) and optimizing the noise model parameters (given a force field). This could be achieved by:

Step 1: Using the current noise model estimate, infer the force field using the standard PFI approach.
Step 2: Given the inferred force field, update the noise model parameters by minimizing a distance between the observed marginals and marginals generated by simulating the SDE (Eq. 1) with the current force field and parameterized noise.

Non-Parametric Noise Estimation: Explore non-parametric methods to estimate the diffusion tensor directly from the data. This could involve techniques like kernel density estimation or Gaussian process regression to approximate the diffusion tensor locally in state-space.

Challenges:

Identifiability: As highlighted in the paper, disentangling the force field from the noise model is fundamentally challenging with cross-sectional data. The lack of time correlations introduces ambiguities, potentially leading to multiple force field and noise model combinations that can explain the observed marginals.

Computational Complexity: Jointly inferring the force field and noise model significantly increases the dimensionality of the optimization problem, potentially making it computationally expensive, especially for high-dimensional systems.

Overfitting: With increased flexibility in the model (both force field and noise), there's a higher risk of overfitting the data, especially with limited sample sizes. Careful regularization and model selection strategies would be crucial.

While the PFI method shows promise in inferring stochastic dynamics, could its reliance on a pre-defined noise model potentially bias the inference towards that model even if the underlying biological system deviates from the assumed noise structure?

Yes, the PFI method's reliance on a pre-defined noise model could introduce bias if the true underlying biological system deviates significantly from the assumed noise structure. This is a general challenge in any inference problem where assumptions are made about the data-generating process.
Here's how the bias could manifest:

Inaccurate Force Field Estimation: If the true noise model is substantially different from the assumed one, the PFI approach might compensate for this mismatch by learning a biased force field that attempts to account for the discrepancies in the observed marginals. This could lead to incorrect interpretations of the underlying biological mechanisms.

Limited Sensitivity to Alternative Dynamics: A fixed noise model might restrict the PFI method's ability to capture alternative dynamical behaviors that could be better explained by a different noise structure. This could lead to overlooking potentially important biological phenomena.
Mitigating the Bias:

Sensitivity Analysis: Performing sensitivity analysis by varying the noise model parameters and observing the impact on the inferred force field can provide insights into the robustness of the results to the choice of the noise model.

Model Comparison: If feasible, comparing the PFI results obtained with different plausible noise models could help identify potential biases and assess the sensitivity of the inferred dynamics to the choice of the noise model.

Data-Driven Noise Model Selection: As discussed in the previous answer, exploring methods to estimate the noise model directly from the data, even if partially, could help reduce the reliance on pre-defined models and potentially lead to more accurate and unbiased inference.

If we consider the process of biological evolution as a form of "inference" where natural selection acts as the "regularization" mechanism, what insights can the PFI framework offer in understanding the interplay between deterministic and stochastic forces in shaping the evolution of complex biological systems?

The PFI framework, with its focus on disentangling deterministic forces from intrinsic stochasticity, offers intriguing parallels to biological evolution and provides a lens for understanding the interplay of these forces in shaping biological complexity.
Analogies and Insights:

Evolution as Inference: Evolution can be viewed as a process of "inferring" successful phenotypes that maximize fitness in a given environment. The "data" in this case are the survival and reproductive success of individuals with different traits.

Natural Selection as Regularization: Natural selection acts as a "regularization" mechanism by penalizing phenotypes with low fitness and favoring those with higher fitness. This selection pressure prevents the unconstrained exploration of the phenotypic space and guides the evolutionary trajectory towards regions of higher fitness.

Deterministic and Stochastic Forces: The PFI framework highlights the challenge of separating deterministic forces (e.g., selection pressure) from intrinsic stochasticity (e.g., genetic drift, mutations) in shaping evolutionary dynamics. Just as in PFI, accurately inferring the "force field" of natural selection requires accounting for the inherent stochasticity in the evolutionary process.

Noise-Driven Exploration: The PFI framework emphasizes that intrinsic noise can drive exploration of the phenotypic space, potentially leading to novel and potentially beneficial traits. This aligns with the understanding that genetic drift and mutations introduce randomness into the evolutionary process, allowing populations to explore a wider range of phenotypic possibilities.

Robustness and Evolvability: The PFI framework's focus on accurately modeling intrinsic noise could provide insights into the concept of "evolvability," which refers to a population's capacity to adapt to changing environments. Understanding how different noise structures influence a population's ability to explore the phenotypic space could shed light on the factors that promote or hinder evolvability.
Limitations and Future Directions:

Simplifying Assumptions: The PFI framework, like many evolutionary models, relies on simplifying assumptions about the underlying biological processes. For example, it assumes a fixed fitness landscape, which might not hold true in dynamically changing environments.

Complexity of Biological Systems: Biological systems are vastly more complex than the models typically used in PFI. Capturing the full intricacies of gene regulation, developmental processes, and ecological interactions remains a significant challenge.
Despite these limitations, the PFI framework provides a valuable conceptual framework for thinking about evolution as an inference process. By drawing parallels between the challenges of disentangling deterministic and stochastic forces in both PFI and evolution, we can gain a deeper appreciation for the complex interplay of these forces in shaping the diversity and complexity of life.