Transient Probability Analysis in CRNs

• Transient probability analysis is a method that estimates the evolution of stochastic behaviors in chemical reaction networks using continuous-time Markov chains.
• Semi-quantitative abstraction reduces complex state spaces by grouping populations into order-of-magnitude intervals while retaining key reaction dynamics.
• Efficient algorithms accelerate self-loop transitions and prune unlikely paths, enabling tractable and robust transient analysis even in multiscale biochemical systems.

A chemical reaction network (CRN) is a formal model describing the interactions of molecular species through stochastic reactions. Transient probability analysis of CRNs seeks to efficiently estimate and explain the time-evolving probabilistic behavior of such systems, particularly when modeled as continuous-time Markov chains (CTMCs). The challenge arises from the large or infinite state spaces and multiscale dynamics common in biochemical CRNs, which make direct numerical analysis intractable. Semi-quantitative abstraction, as introduced by Česka and Křetínský, enables a principled reduction of the underlying CTMC to a compact, analyzable system that preserves the key orders-of-magnitude and qualitative behaviors of the network, providing succinct yet informative transient analyses (Češka et al., 2019).

1. CRN Formalism and Stochastic CTMC Semantics

A CRN is defined by a finite species set $\Lambda = \{\lambda_1, \ldots, \lambda_m\}$ and a reaction set $\mathcal{R}$, where each reaction $\tau$ is a triple $(r_\tau, p_\tau, k_\tau)$: $r_\tau \in \mathbb{N}^m$ specifies reactant stoichiometry, $p_\tau \in \mathbb{N}^m$ product stoichiometry, and $k_\tau \in \mathbb{R}_{>0}$ the rate constant. For any reaction, the state-change vector is $\nu_\tau = p_\tau - r_\tau$.

Given initial population vector $X_0 \in \mathbb{N}^m$, the system evolves over $S \subseteq \mathbb{N}^m$, the set of all reachable population vectors. The CTMC $\mathcal{C} = (S, X_0, R)$ is constructed with rate matrix entries: $$R(X_i, X_j) = \sum_{\tau: X_i + \nu_\tau = X_j} k_\tau \cdot C_\tau(X_i)$$ where $C_\tau(X_i)$ is the combinatorial factor for reaction $\tau$ at state $X_i$.

This CTMC captures the state-dependent, stochastic mass-action dynamics of the CRN. The intractability of exact computation in large or stiff CRNs motivates the need for abstraction.

2. Semi-Quantitative Abstraction: Interval Partitions, Enabledness, and Acceleration

The semi-quantitative abstraction replaces the original CTMC with a much smaller one $\mathcal{A}$, tracking only orders of magnitude of populations but still preserving reaction structure and essential timing:

• Interval Abstraction: For each species $\lambda \in \Lambda$, the count domain $\mathbb{N}$ is partitioned into user-chosen intervals $$A_\lambda = \{I_{\lambda,0}, I_{\lambda,1}, \ldots\}$$, with singleton intervals for $\{0,1,2,\ldots\}$ up to the maximal reactant order to maintain enabledness. The abstract state space $A$ is the Cartesian product $\prod_{\lambda} A_\lambda$, and the abstraction map $\alpha: \mathbb{N}^m \to A$ assigns each concrete state to its interval block.
• May-Abstraction and Interval Rates: From each abstract state $a \in A$, a may-abstraction would consider all reactions enabled in some concretization $\gamma(a)$, labeling outgoing transitions with the interval of attainable rates over $\gamma(a)$. This over-approximation is narrowed by subsequent steps.
• Acceleration of Self-Loops: Reactions that do not exit the current abstract block are accelerated. Instead of modeling the geometric sequence of internal self-loops, a single transition is introduced out of the block at a rate equal to $n_\tau \cdot r$, where $n_\tau$ is the minimal number of firings needed to exit and $r$ is the reaction rate at a chosen representative of the block.
• Concrete Representative: Non-determinism is eliminated by picking a representative population $\gamma(a) \in \mathbb{N}^m$ for each abstract state; typically a midpoint or lower bound of the interval(s). Rates are computed concretely at representatives, yielding a deterministic, reduced CTMC.

The combination of these steps yields a tractable, finite-state abstraction, amenable to direct analysis.

3. Algorithms for Abstraction Construction and Transient Path Analysis

The abstraction and analysis procedure involves two principal algorithms, each with complexity linear in the size of the reduced model:

• Algorithm 1 (Building Abstract CTMC $\mathcal{A}$):

1. Generate the abstract state space $A$ and map $a_0 = \alpha(X_0)$. 2. For each $a\in A$, compute representatives. 3. For each reaction $\tau$ enabled at the representative, determine whether the transition is a self-loop or crosses to a new block; if the former, compute the accelerated transition. 4. Output the CTMC $(A, a_0, R^A)$.

This process is $O(|A|\cdot|\mathcal{R}|)$ and dramatically reduces the state space size.

• Algorithm 2 (Semi-Quantitative Transient Analysis):
  • Compute steady-state distributions via reciprocal sojourn times.
  • Identify the most likely exit states (maximize exit rate/stay rate).
  • Estimate expected sojourn times and exit probabilities.
  • Sequentially follow the most probable path through SCCs.

This approach produces an ordered sequence of abstract SCCs, stay times, and dominant exit probabilities, forming a compressed “skeleton” of the network's typical transient behavior.

4. Quantitative Interpretation: Lifting Abstract Results to the CTMC

Semi-quantitative analysis enables bounding the true transient probabilities and mean times in the original CTMC by leveraging the correspondence between abstract transitions and representatives:

• For a sojourn time $T_C$ in abstract SCC $C$, the true sojourn in the concrete CTMC satisfies:

$$T_C(\mathrm{true}) \in \left[\frac{T_C}{\xi},\, \xi\,T_C\right]$$

where $\xi$ is the maximal ratio between representative and actual rates/populations within the block (often $<10$).

• Exit probabilities labeled as “nearly 1” in the abstract analysis are at least $0.1$ in the concrete CTMC, still within the dominant order of magnitude.

This provides explicit, controllable error bounds: as interval refinement increases and order-pruning is relaxed, the predictions converge to the exact analysis, but always remain within one or two orders of magnitude per the chosen abstraction.

5. Worked Example: Gene-Expression Motif

The gene-expression CRN with four species $\Lambda = \{$D_off, D_on, RNA, P$\}$ and seven reactions (including slow DNA switching, mRNA/protein production and degradation) illustrates the methodology:

• Intervals: For RNA, $A_{RNA} = \{0\} \cup [1..9] \cup [10..\infty]$; for P, $A_{P} = \{0\} \cup [1..99] \cup [100..\infty]$; both DNA variables are binary.
• Representatives: Chosen as midpoints or lower bounds; e.g., $[10..\infty] \to 10$.
• Abstraction: The state space collapses to $\sim10$ blocks (orders-of-magnitude classes).
• Accelerated Self-Loops: For the initial state, RNA and protein degradations are collapsed, yielding single transitions that efficiently approximate the depletion process. DNA switching reactions yield cross-block transitions of rate $0.05$ h$^{-1}$.
• Order-0 Pruning: Keeping only maximal-rate edges reveals two bottom SCCs, $C_{off}$ (DNA off) and $C_{on}$ (DNA on with high expression), with rare transitions between them.

The extracted skeleton reflects the known qualitative dynamics: $$C_{off}\xrightarrow{20h}C_{on}\xrightarrow{0.1h}C_{off}$$, rapidly cycling in high-expression states and slowly switching DNA states. This outcome matches the orders-of-magnitude and modal behavior obtained by direct numerical simulation and known analytical results.

6. Scalability, Accuracy, Limitations, and Applicability

• Scalability: Both abstraction and analysis are $O(|A|\cdot|\mathcal{R}|)$ or $O(|A|+|E|)$, with $|A|\ll|S|$ for realistic biomolecular models. Case studies include networks with over $10^4$ concrete states and $|A|\approx50$.
• Trade-offs: The method gives up precise probability values in favor of orders-of-magnitude accuracy; finer intervals and higher pruning parameters ($n$) improve precision but with increased cost.
• Formal Guarantees: As interval partitioning is refined and $n \to \infty$, abstract predictions converge to the true CTMC values; for any fixed abstraction, accuracy is guaranteed within one–two orders of magnitude.
• Limitations: There is no formal guarantee of small absolute error (e.g., $\pm 0.01$); the approach can underperform if population rapidly traverses multiple intervals in a single epoch, but this can be remedied by refining intervals.
• Scope: This methodology applies to any population-based CTMC with stiff or multi-scale behavior, including Petri nets and stochastic population protocols. It is particularly effective where time-scale separation exists and a qualitative or order-of-magnitude explanation is adequate (Češka et al., 2019).

• M. Česka & J. Křetínský, “Semi-Quantitative Abstraction and Analysis of Chemical Reaction Networks,” (Češka et al., 2019) (CAV 2020).
• D. Baier & J. Katoen, “Principles of Model Checking,” MIT Press, 2008.