SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems
Abstract
:1. Introduction
- It is not necessary to define the state transition graph of the entire system. This can be a real advantage, since the state transition graph can become quite large. This holds in particular for catalytic systems in which the substrate exists in many conformations. A prominent example is post-translational modification of a protein, e.g., phosphorylation at different sites. The nodes of the transition graph correspond here to all possible configurations of substrate phosphorylation states and catalyst binding states. Due to the combinatorial complexity, this number grows rapidly even for small molecule numbers. Moreover, it is for most of those systems not possible to exploit the underlying structure of the graph.
- Intractable state transition graphs are replaced by a concatenation of much simpler graphs, resulting in lower dimensional differential equations for the CME approach. Instead of solving the conventional CME defined on the full state transition graph, one can solve the CME on these simpler graphs and concatenate the obtained solutions. This effectively enables the solution of the CME for complex systems, where it was formerly necessary to resort to simulation methods.
- The implementation of the SSA is considerably simplified.
2. Materials and Methods
Algorithm 1 SSA general scheme () |
1: Initialize and set 2: while do 3: Calculate and for 4: Draw waiting time until next reaction event from Exp 5: Draw reaction index j from discrete distribution 6: Update 7: if then 8: Update 9: end if 10: end while 11: return |
3. Results
3.1. An Intuitive Example: Linear Conversion Process
3.2. Applying SiCaSMA to a Model for DNMT1-Mediated DNA Methylation
3.3. Applying SiCaSMA to Larger Networks via the SSA
Algorithm 2 SSA DNA methylation system () |
Initialize and set 2: while do Calculate and 4: Draw independent random numbers and uniformly from Set i to be the smallest integer satisfying 6: Update if then 8: Update end if 10: end while return |
Algorithm 3 Single Catalyst SSA DNA methylation system () |
Initialize for do 3: Initialize Update end for 6: return |
4. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
CME | Chemical Master Equation |
SSA | Stochastic Simulation Algorithm |
SiCaSMA | Single Catalyst Stochastic Modeling Approach |
DNMT1 | DNA methyltransferase 1 |
Appendix A. Proof of Equivalence of X and Y for the DNA Methylation Model
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Wagner, V.; Radde, N.E. SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems. Mathematics 2021, 9, 1074. https://doi.org/10.3390/math9101074
Wagner V, Radde NE. SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems. Mathematics. 2021; 9(10):1074. https://doi.org/10.3390/math9101074
Chicago/Turabian StyleWagner, Vincent, and Nicole Erika Radde. 2021. "SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems" Mathematics 9, no. 10: 1074. https://doi.org/10.3390/math9101074
APA StyleWagner, V., & Radde, N. E. (2021). SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems. Mathematics, 9(10), 1074. https://doi.org/10.3390/math9101074