Gene networks
A gene network represents regulatory relationships among genes and gene products. The network shows who regulates whom; the dynamical model determines how expression changes through time.
Nodes and regulatory edges
Depending on the modelling scale, a node may represent a gene, mRNA species, transcription factor, protein or regulatory module.
A directed edge
\[i\to j\]means that component \(i\) regulates component \(j\).
Direction is essential because regulation need not be reciprocal.
Activation and inhibition
Gene-regulatory edges often carry a sign. A convenient signed adjacency representation is
\[A_{ij}>0\quad\text{activation},\qquad A_{ij}<0\quad\text{inhibition},\qquad A_{ij}=0\quad\text{no direct regulatory edge}.\]The sign records the qualitative effect, while the magnitude can represent regulatory strength in a weighted model.
Incoming and outgoing regulation
With the convention that \(A_{ij}\) represents regulation from \(i\) to \(j\), the out-degree of node \(i\) counts how many targets it regulates, while the in-degree of node \(j\) counts how many regulators act on it.
These two quantities can have different biological meanings.
Network structure alone is not enough
Knowing that gene \(i\) activates gene \(j\) does not determine how strongly or how quickly \(j\) responds.
A dynamical model must specify the functional relationship between regulator concentration and target production.
A generic ODE network model
Let \(x_i(t)\) denote the concentration or expression level associated with node \(i\). A general deterministic form is
\[\boxed{\frac{dx_i}{dt}=F_i(x_1,\ldots,x_n)-\delta_i x_i}.\]The function \(F_i\) describes regulatory production, while \(\delta_i x_i\) represents degradation or removal.
The graph indicates which variables are allowed to appear in \(F_i\).
Activation with a Hill function
If regulator \(x_j\) activates target \(x_i\), one common response term is
\[\boxed{H_+(x_j)=\frac{x_j^n}{K^n+x_j^n}}.\]This rises from near zero to near one as \(x_j\) increases.
The parameter \(K\) is the half-response scale and \(n\) controls steepness.
Repression with a Hill function
A simple inhibitory response is
\[\boxed{H_-(x_j)=\frac{K^n}{K^n+x_j^n}}.\]This decreases as regulator concentration increases.
These functions are phenomenological approximations unless derived from a more detailed biochemical mechanism.
Combining multiple regulators
If several regulators control one gene, their effects can be combined in different ways. For example, an AND-like requirement may multiply activation terms, while an additive model may sum contributions.
Different choices represent different regulatory logic and can generate different dynamics even on the same graph.
Boolean gene networks
When only ON/OFF expression is modelled, let
\[x_i(t)\in\{0,1\}.\]Each node updates according to a Boolean rule
\[\boxed{x_i(t+1)=F_i(x_1(t),\ldots,x_n(t))}.\]This sacrifices concentration detail but can efficiently represent qualitative regulatory logic in large networks.
Synchronous and asynchronous Boolean updates
In synchronous updating, every node is updated simultaneously. In asynchronous updating, one or a subset of nodes changes at a time.
The choice can change attractors and transition structure, so update timing is part of the model rather than a purely numerical detail.
Fixed points
A regulatory state \(x^*\) is a fixed point when
\[F_i(x^*)=x_i^*\]for every node in a Boolean model, or
\[\frac{dx_i}{dt}=0\]for every node in an ODE model.
Stable fixed points are often interpreted as persistent expression states or cell states.
Positive feedback
A positive feedback loop occurs when regulation reinforces a change. For example,
\[A\to B\to A\]with net positive sign can support switching or multiple stable states under appropriate nonlinear dynamics.
Negative feedback
Negative feedback tends to oppose deviations and can stabilise expression around a steady level.
With sufficient delay or additional dynamical structure, negative feedback can also produce oscillations.
Thus the sign of a feedback loop alone does not fully determine the time-dependent behaviour.
Feed-forward loops
A common three-node motif has one regulator acting on a target both directly and indirectly:
\[A\to C,\qquad A\to B\to C.\]Depending on signs and time scales, feed-forward loops can filter short signals, create delays or accelerate responses.
Network motifs
A motif is a small recurring connection pattern. Examples include feedback loops, feed-forward loops and mutual inhibition.
Motif frequency can be compared with suitable random-network null models, but overrepresentation alone does not prove a particular biological function.
Mutual inhibition
Two genes that repress one another form a toggle-like architecture:
\[A\dashv B,\qquad B\dashv A.\]With sufficiently nonlinear repression, this can support two alternative stable expression states.
The graph suggests the possibility, while the dynamical equations determine whether bistability actually occurs.
Oscillatory circuits
Regulatory cycles containing effective negative feedback can generate oscillations when production, degradation and delays interact appropriately.
Examples include transcriptional repressilator-type architectures and cell-cycle control networks.
Jacobian and local stability
For an ODE system
\[\frac{d\mathbf x}{dt}=\mathbf F(\mathbf x),\]local stability near an equilibrium \(\mathbf x^*\) is studied using the Jacobian
\[\boxed{J_{ij}=\left.\frac{\partial F_i}{\partial x_j}\right|_{\mathbf x=\mathbf x^*}}.\]The regulatory graph constrains which Jacobian entries can be non-zero, while the actual values determine local stability.
Linearised network dynamics
Near equilibrium, small perturbations \(\boldsymbol\eta\) satisfy approximately
\[\frac{d\boldsymbol\eta}{dt}=J\boldsymbol\eta.\]Eigenvalues of \(J\) determine whether perturbations decay, grow or oscillate locally.
This connects network architecture to dynamical-systems analysis.
Stochastic gene networks
Gene expression often involves small molecule numbers and random reaction events. A stochastic model may therefore treat transcription, translation and degradation as discrete reactions.
For reaction \(r\) with propensity \(a_r(\mathbf n)\), the probability of that reaction occurring in a short interval is approximately
\[a_r(\mathbf n)\Delta t.\]The network indicates which reactions couple regulatory components.
Noise and switching
In a bistable regulatory system, stochastic fluctuations can occasionally drive transitions between stable expression states.
This means a deterministic attractor structure can coexist with random switching in finite molecular systems.
Static versus context-dependent regulation
A static gene network assumes regulatory edges remain fixed. In reality, chromatin state, signalling, cell type and environment can change which interactions are active.
More detailed models may therefore use context-dependent or time-varying regulatory networks.
Inferring gene networks from data
Gene-network reconstruction attempts to infer regulatory relationships from data such as expression measurements, perturbation experiments or binding assays.
Correlation alone does not establish direct regulation, because two genes can correlate through a common upstream regulator or indirect pathway.
Network topology versus parameter values
Two models can share the same regulatory graph but behave very differently if production rates, degradation rates, thresholds or cooperativity differ.
Conversely, different network architectures can sometimes produce similar observed expression trajectories.
This creates parameter-identifiability and model-selection challenges.
Relation to earlier gene-regulatory modelling
The earlier cell-and-molecular modelling section focused on the equations and dynamical behaviour of specific regulatory systems. Here the emphasis is different: the regulatory interactions are treated explicitly as a network and connected to graph-theoretic ideas such as direction, degree, motifs and topology.
Transition to ecological networks
The next lesson changes the meaning of the nodes and edges again. Nodes become species or ecological groups, while links represent feeding, competition, mutualism or other ecological interactions.