Conference version

paper.tex

@@ -567,11 +567,11 @@ anchorcolor=black   % make links/references 'clickable' without ugly frames
 		\subfloat[Probabilistic update scheme\label{fig:repr_prob_stationary}]{
 			\includegraphics[width=0.48\textwidth]{repressilator_prob025.pdf}}
 		\caption{
-			Comparison between theoretical probability distribution and empirical data from a test run using asynchronous random (\cref{fig:repr_async_stationary}) and probabilistic (\cref{fig:repr_prob_stationary}) with one million simulated time steps, both starting from initial state $x=000$. 
+			Comparison between theoretical stationary probability distribution and empirical data from a test run using asynchronous random (\cref{fig:repr_async_stationary}) and probabilistic (\cref{fig:repr_prob_stationary}) with one million simulated time steps, both starting from initial state $x=000$. 
 		}
 	\end{figure*}
 	
-	Under the synchronous scheme, the network has two attractors, shown in the upper part of \cref{fig:bn_repr_state_graph}. The first is a period-6 oscillation cycle through the states $100 \rightarrow 101 \rightarrow 001 \rightarrow 011 \rightarrow 010 \rightarrow 110 \rightarrow 100$, which corresponds to the alternating gene expression observed in experimental implementations of the Repressilator. The second is an artificial period-2 cycle alternating between $000$ (all genes off) and $111$ (all genes on). This second attractor has no biological basis: it arises purely because updating all three repressors simultaneously maps $000$ to $111$ and back, a dynamic that would not occur if the genes were updated at slightly different times.
+	Under the synchronous scheme, the network has two attractors, shown in the upper part of \cref{fig:bn_repr_state_graph}. The first is a period-6 oscillation cycle through the states $100 \rightarrow 101 \rightarrow 001 \rightarrow 011 \rightarrow 010 \rightarrow 110 \rightarrow 100$, which corresponds to the alternating gene expression observed in experimental implementations of the repressilator. The second is an artificial period-2 cycle alternating between $000$ (all genes off) and $111$ (all genes on). This second attractor has no biological basis: it arises purely because updating all three repressors simultaneously maps $000$ to $111$ and back, a dynamic that would not occur if the genes were updated at slightly different times.
 	
 	Under random asynchronous update, the picture changes. From state $000$, any node selected for updating will have $f_i = \neg 0 = 1$, meaning the network is immediately pushed into one of the oscillatory states regardless of which node is chosen. The same applies to $111$. Neither state can sustain itself, so they become transient states and the artificial 2-cycle disappears entirely. Only the biologically meaningful oscillation remains as an attractor. This confirms that the period-2 cycle in the synchronous case was indeed an artifact of simultaneous updating \cite{bornholdt2008boolean}, and it illustrates why asynchronous update is often considered more biologically faithful for oscillatory GRN models.
 	
@@ -581,14 +581,22 @@ anchorcolor=black   % make links/references 'clickable' without ugly frames
 	
 	Under a probabilistic scheme, the clean periodic oscillation gives way to an irregular, noisy one. Introducing a small probability for an alternative update function on any of the three genes, modeling the stochastic fluctuations that occur in real gene expression, means the network occasionally deviates from the regular cycle. The long-term behavior is no longer a fixed attractor but a stationary distribution over states, reflecting how much time the system spends in each expression configuration on average \cite{schwab2020concepts}. This type of model is particularly useful for understanding phenotypic variability within a population of genetically identical cells.
 	
-	The Repressilator example illustrates a key point for GRN modeling in general: the choice of update scheme is not a neutral decision. It directly affects which attractors exist, which states are biologically accessible, and how realistically the model captures the noise and timing characteristics of the underlying system. For large-scale GRNs where the biological timing of regulatory events is not known, synchronous update remains the practical standard due to its computational simplicity and the ease of attractor analysis. Sequential or block-sequential update is appropriate when relative timing information is available. Asynchronous update is biologically motivated but becomes slow for large networks, and probabilistic update is the method of choice when transcriptional noise is a central modeling goal.
+	The repressilator example illustrates a key point for GRN modeling in general: the choice of update scheme is not a neutral decision. It directly affects which attractors exist, which states are biologically accessible, and how realistically the model captures the noise and timing characteristics of the underlying system. For large-scale GRNs where the biological timing of regulatory events is not known, synchronous update remains the practical standard due to its computational simplicity and the ease of attractor analysis. Sequential or block-sequential update is appropriate when relative timing information is available. Asynchronous update is biologically motivated but becomes slow for large networks, and probabilistic update is the method of choice when transcriptional noise is a central modeling goal.
 	
 	
+	The long-term behavior of stochastic update schemes can be analyzed formally through the theory of Markov chains. Both the random asynchronous and the probabilistic synchronous schemes define a stochastic process over the $2^n$ possible states, where the probability of moving to the next state depends only on the current one. This process is fully captured by a \textit{transition matrix} $P \in [0,1]^{2^n \times 2^n}$, where the entry $P_{ij}$ gives the probability of transitioning from state $i$ to state $j$ in one time step. Each row of $P$ sums to one, reflecting that the network must always move to some state.
 	
-	The long-term behavior of stochastic update schemes can be characterized precisely by their stationary distribution, and comparing this theoretical prediction against empirical simulation provides a practical validation of the Markov chain model. We write $\pi(x)$ for the stationary probability of state $x$. Under random asynchronous update of the Repressilator, the states $000$ and $111$ are transient: once the network leaves them it never returns, so $\pi(000) = \pi(111) = 0$. The remaining six oscillatory states $\{100, 101, 001, 011, 010, 110\}$ form a single irreducible recurrent class, and by the symmetry of the random walk each is visited equally often, giving $\pi(x) = \frac{1}{6}$ for each state. \Cref{fig:repr_async_stationary} shows that a long simulation run closely matches this theoretical distribution.
+	The stationary distribution $\pi$ is a probability vector over the $2^n$ states satisfying
+	\begin{equation}
+		\label{eq:stationary}
+		\pi P = \pi,
+	\end{equation}
+	meaning $\pi$ is a left eigenvector of $P$ with eigenvalue 1. Intuitively, $\pi(x)$ gives the long-run fraction of time the network spends in state $x$: transient states receive $\pi(x) = 0$, while recurrent states satisfy $\pi(x) > 0$ and are revisited indefinitely. For an irreducible Markov chain, a unique $\pi$ satisfying \cref{eq:stationary} always exists. Comparing $\pi$ against empirical visit frequencies from a long simulation then provides a practical validation of the model.
 	
-	Under the probabilistic synchronous scheme, the analysis differs. When a per-node bit-flip probability $p \neq 0$ is introduced alongside the standard repression function, the resulting transition matrix is doubly stochastic regardless of the value of $p$: every row and column sums to one. A doubly stochastic matrix over a finite irreducible Markov chain has the uniform distribution as its unique stationary distribution, so all eight states are visited with equal probability $\pi(x) = \frac{1}{8}$ in the long run, including $000$ and $111$, which are never reached under the asynchronous scheme. \Cref{fig:repr_prob_stationary} confirms this result empirically, illustrating how the two stochastic schemes produce qualitatively different long-term behavior despite both introducing randomness into the system.
+	Under random asynchronous update of the repressilator, the states $000$ and $111$ are transient: from either state, every possible node selection immediately moves the network away, so once left they are never revisited. No stationary distribution can assign positive probability to a transient state, hence $\pi(000) = \pi(111) = 0$. The remaining six oscillatory states $\{100, 101, 001, 011, 010, 110\}$ form a single irreducible recurrent class. Since the update rule selects each of the three nodes with equal probability $\frac{1}{3}$, the transition matrix restricted to this class is symmetric, and the unique solution to \cref{eq:stationary} is the uniform distribution $\pi(x) = \frac{1}{6}$ for each of the six states. \Cref{fig:repr_async_stationary} shows that a long simulation run closely matches this theoretical result.
 	
+	Under the probabilistic synchronous scheme, the analysis differs. When a per-node bit-flip probability $p \neq 0$ is introduced alongside the standard repression function, the resulting transition matrix $P$ is doubly stochastic: not only does every row sum up to one, but every column does as	well. For any doubly stochastic matrix over a finite irreducible Markov chain, the uniform distribution is the unique solution to \cref{eq:stationary}, giving $\pi(x) = \frac{1}{8}$ for all eight states in the long run, including $000$ and $111$, which carry zero probability under the asynchronous scheme. \Cref{fig:repr_prob_stationary} confirms this result empirically, illustrating how the two stochastic schemes produce qualitatively different long-term behavior despite both introducing randomness.
+		
 	\section{Conclusion}
 	This paper gave an overview of four update schemes for boolean networks and analyzed how they affect the dynamics of a model. The synchronous scheme, where all nodes update simultaneously, produces fully deterministic and easily enumerable dynamics, making it the most widely used approach in practice despite its biological simplifications. Sequential and block-sequential schemes allow for a more realistic encoding of temporal ordering between regulatory events while retaining determinism. The probabilistic scheme introduces stochastic variation at the level of the update functions, capturing noise and fluctuations in the system. The asynchronous scheme updates one node at a time and, in its random variant, introduces non-determinism at the level of node selection rather than function selection.