Path integral approach to generating functions for multistep post-transcription and post-translation processes and...

Area of Science:

• Systems Biology
• Computational Biology
• Biophysics

Background:

• Stochastic fluctuations in gene product numbers impact gene regulatory network (GRN) function.
• The master equation (ME) is a key theoretical tool for studying these stochastic effects in GRNs.
• Solving the ME is computationally challenging for complex systems.

Purpose of the Study:

• To develop a novel technique for solving the master equation for stochastic GRNs.
• To obtain the generating function (GF) for a single gene system with multi-step post-transcriptional and post-translational modifications.
• To provide a new method for analyzing gene product fluctuations.

Main Methods:

• Developed a novel approach to solve the generating function (GF) by considering paths in the time-copy number plane for mRNAs.
• Summed the GF over all possible mRNA processing paths.
• Proved a theorem showing the path summation is equivalent to an ME-like equation for mRNAs.

Main Results:

• Presented a new method for obtaining the GF for a one-gene system with complex post-transcriptional and post-translational processes.
• Demonstrated the equivalence of path summation to an ME-like equation for mRNAs.
• Validated the approach by comparing results with Gillespie simulations on a six-gene product system.

Conclusions:

• The novel GF approach offers a viable alternative for solving the ME in stochastic GRNs.
• This method simplifies the analysis of gene product fluctuations and network dynamics.
• The findings contribute to a better understanding of stochasticity in biological systems.