Cell signalling
Cell signalling describes how a cell detects a signal and converts it into a biochemical response. The signal may come from outside the cell, from another cell, or from an internal process.
From ligand to response
A signalling model asks how strongly and how quickly the output responds to the input.
Receptor binding
A simple receptor-binding reaction is
\[L+R\;\underset{k_-}{\overset{k_+}{\rightleftharpoons}}\;LR.\]Here \(L\) is ligand, \(R\) free receptor and \(LR\) ligand-bound receptor.
Under mass-action kinetics,
\[\frac{d[LR]}{dt}=k_+[L][R]-k_-[LR].\]Receptor conservation
If receptors are either free or ligand-bound,
\[R_T=[R]+[LR].\]Therefore
\[[R]=R_T-[LR].\]Substituting gives
\[\frac{d[LR]}{dt}=k_+[L](R_T-[LR])-k_-[LR].\]Steady receptor occupancy
At equilibrium,
\[0=k_+[L](R_T-[LR]^*)-k_-[LR]^*.\]Define the dissociation constant
\[K_d=\frac{k_-}{k_+}.\]Then
\[\boxed{[LR]^*=R_T\frac{[L]}{K_d+[L]}.}\]The fraction of occupied receptors is
\[\boxed{\theta=\frac{[LR]^*}{R_T}=\frac{[L]}{K_d+[L]}.}\]What does \(K_d\) mean?
When \([L]=K_d\),
\[\theta=\frac12.\]So \(K_d\) is the ligand concentration giving half receptor occupancy in this simple binding model.
Binding is only the first step
Ligand binding often changes receptor activity, which then affects intracellular proteins. A simple activation model is
\[\frac{dX^*}{dt}=k_{on}[LR](X_T-X^*)-k_{off}X^*.\]Here \(X^*\) is an active signalling protein and \(X_T-X^*\) is the inactive fraction.
Why activation saturates
There is only a finite amount \(X_T\). Once nearly all of it is active, increasing upstream signal further cannot increase \(X^*\) much more. Saturation therefore appears naturally from conservation.
Signal amplification
One activated receptor can activate many downstream molecules. A pathway can therefore produce a larger intracellular response than the number of bound receptors alone might suggest.
A simple cascade may be written
Each stage can amplify or reshape the signal.
A model-generated signalling cascade
The next graph solves a three-stage activation cascade after a step input turns the receptor signal on at \(t=0\).
Why downstream signals can be delayed
Even if the input changes instantly, molecular concentrations cannot generally jump to their new steady values. Each reaction has its own timescale.
In a cascade, these finite response times accumulate, so the final output often responds later than the receptor.
Transient versus sustained signalling
Some pathways respond strongly at first and then decline even while the input remains present. Others stay active as long as the input persists.
This distinction is biologically important because cells may interpret a transient pulse differently from a sustained signal.
Adaptation
Adaptation occurs when a pathway responds to a change in input but later returns close to its original output despite the continued presence of that input.
A simple mechanism is negative feedback: the output activates an inhibitor that suppresses the pathway.
A simple adaptation model
One phenomenological system is
\[\frac{dX}{dt}=k_sS-k_iIX-k_xX,\]\[\frac{dI}{dt}=k_fX-k_dI.\]The input \(S\) raises \(X\), but \(X\) also builds up inhibitor \(I\), which pushes \(X\) back down.
Depending on parameters, this can create a strong transient response followed by partial or near-complete adaptation.
Positive feedback in signalling
Positive feedback occurs when an active component promotes more of its own activation directly or indirectly.
This can sharpen responses, create memory and, if the feedback is sufficiently nonlinear, produce bistability.
Negative feedback in signalling
Negative feedback can limit signal amplitude, speed recovery after stimulation and reduce sensitivity to fluctuations. With delays, it can also generate oscillations.
Ultrasensitivity
Some signalling pathways respond weakly below a threshold but strongly above it. A Hill function is often used phenomenologically:
\[\boxed{Y(S)=Y_{\max}\frac{S^n}{K^n+S^n}.}\]Larger \(n\) produces a steeper input–output relationship.
Signal threshold
In the Hill model, \(K\) is the input giving half-maximal output:
\[Y(K)=\frac{Y_{\max}}2.\]This makes \(K\) a useful response threshold parameter.
Phosphorylation cycles
A common signalling motif is reversible phosphorylation:
\[X\;\underset{\text{phosphatase}}{\overset{\text{kinase}}{\rightleftharpoons}}\;X^*.\]Kinases activate or modify proteins, while phosphatases reverse the modification.
The balance between these opposing enzyme activities determines the active fraction.
Zero-order ultrasensitivity
If both the activating and deactivating enzymes operate near saturation, a phosphorylation cycle can become very steep even without cooperative binding.
This is called zero-order ultrasensitivity and shows that switch-like signalling does not always require a Hill coefficient arising from cooperativity.
Signal integration
Cells often receive several signals simultaneously. A downstream component may respond only when both are present, when either is present, or according to a weighted combination.
Mathematical models make these logical rules explicit through the production or activation function.
Spatial signalling
Signals are not always well mixed. Receptors are located at membranes, proteins can move between cellular compartments, and second messengers can diffuse through the cytoplasm.
Spatial signalling may therefore require reaction–diffusion equations or compartmental models rather than ordinary ODEs.
Stochastic signalling
At low molecule numbers, receptor binding, phosphorylation and molecular production occur as discrete random events.
Stochastic models can explain cell-to-cell variability in response time, signal amplitude and switching behaviour.
Timescale separation
Some signalling reactions happen in milliseconds or seconds, while gene-expression responses may require minutes or hours. This separation of timescales is useful mathematically.
Fast binding steps can sometimes be approximated as being near equilibrium while slower downstream processes are modelled explicitly through time.
Steady state is not the whole story
Two pathways can have the same final output but very different transient responses. One may respond quickly, another slowly; one may overshoot, another approach smoothly.
For signalling biology, timing can be as important as the final steady-state value.
What can signalling models study?
They can investigate sensitivity, amplification, saturation, adaptation, thresholds, switching, oscillation, pathway cross-talk, drug inhibition and how information is transmitted through molecular networks.
A practical modelling workflow
Identify the input signal, receptor and key intracellular components. Write the molecular reactions. Decide which species are conserved. Derive rate equations from mass action or appropriate effective kinetics. Separate fast and slow timescales when justified. Analyse steady states and transient responses. Then test amplification, feedback, adaptation, thresholds and stochastic variability according to the biological question.