Signal transduction networks are responsible for transferring biochemical signals from the extracellular to the intracellular environment. Understanding the dynamics of these networks helps understand their biological processes. Signals are often delivered in pulses and oscillations. Therefore, understanding the dynamics of these networks under pulsatile and periodic stimuli is useful. One tool to do this is the transfer function. This tutorial outlines the basic theory behind the transfer function approach and walks through some examples of simple signal transduction networks.

The ERK pathway is an important biochemical cascade and acts as a master regulator of myriad cell processes including cell proliferation, differentiation, and survival. Early biochemical work established that the timing of ERK phosphorylation was an important determinant of PC12 cell fate, with extended phosphorylation (with nerve growth factor treatment) linked to differentiation but rapid on–off ERK phosphorylation kinetics (with epidermal growth factor treatment) linked to cell proliferation. Recent work from several laboratories has revealed that periodic forcing the phosphorylation of ERK with growth factors, light (optogenetics) or electronically can switch cell fate from proliferative to differentiated depending on type of stimulus (amplitude and frequency). Here, we take an ERK model and analyze it from the frequency domain perspective. The key is the transfer function, which provides a compact description of input (growth factor)–output (ERK activation) behavior over a range of input frequencies, allowing an understanding of system dynamics in terms of amplitude modulations, phase shifts, and signaling bandwidths. Our analysis of transfer functions indicates that, at normal receptor levels, the ERK pathway acts as a negative feedback amplifier to growth factor fluctuations, amplifying them at low receptor occupancy but suppressing them at high receptor occupancy. The frequency dependence is best described as a resonant low pass filter, which selectively filters out high frequency input oscillations. We use the transfer function to predict how different growth factor input dynamics shape ERK activation.

Understanding the function of network motifs in an attempt to gain insight into how their combinations create larger reaction networks that drive cellular functions, has been a longstanding pursuit of systems biology. One specific objective within this pursuit is understanding how individual motifs respond to pulsatile and oscillatory signals. This is especially relevant because biochemical networks are often activated by signals that, in nature, occur in the form of pulses and oscillations. A powerful analytical tool for studying such dynamics is the transfer function: a compact frequency-domain description of input-output dynamics. In this work, we derive transfer functions for a set of commonly studied network motifs and characterise their responses to pulsatile and oscillatory inputs. The novelty of this review does not lie in the introduction of new mathematical theorems or biological discoveries, but in bridging well-established frequency domain formalisms from control theory with the analysis of biochemical networks under periodic forcing. In doing so, our contributions are threefold: 1. A systematic derivation and compilation of transfer functions for common network motifs: consolidating results scattered across the literature and establishing a consistent formalism for motif-level transfer functions. 2. Contextualisation of these transfer functions within biological models: extending abstract transfer functions to concrete biological settings so that the results are readily applicable without extensive mathematical labour. 3. Resolution of ambiguity between biological and control-theoretic treatments of feedback: clarifying how feedback loops should be understood within the transfer function formalism and reconciling differences between biology literature and control-oriented literature. This is done by formalising the notion of an intrinsic transfer function.

Natural biological systems use a complex network of feedback regulation to effectively respond to their changing environment. Even though in engineered systems we understand how accurate feedback can be depending on the electronic or mechanical parts that it is implemented with, we largely lack a similar theoretical framework to study biological feedback. Specifically, it is not fully understood or quantified how accurate or robust the implementation of biological feedback actually is. In this paper, we study the sensitivity of biological feedback to variations in biochemical parameters using five example circuits: positive autoregulation, negative autoregulation, double-positive feedback, positive-negative feedback and double-negative feedback (the toggle switch). We find that of these examples of biological feedback are subjected to fundamental trade-offs, and we propose multi-objective optimisation as a framework to study them. The impact of this work is to improve robust circuit design for synthetic biology and to improve our understanding of systems biology.

Advances in microfluidics, optogenetics and electronics have enabled the study of dynamically controlled inputs on cellular fate. Here, we applied a microfluidic system to deliver periodic inputs of growth factors to pheochromocytoma cells and measured the extent of premature differentiation as a function of input frequency. Epidermal growth factor-triggered differentiation peaked at two cycles/hour, while nerve growth factor-triggered differentiation peaked at one cycle/hour. To interpret the results, we analyzed a published model that attributed pheochromocytoma cell differentiation to the linear combination of activated enzymes extracellular signal-regulated kinase (ERK), cAMP response element binding protein (CREB), protein kinase B (AKT) and c-Jun N-terminal kinase (JNK) at specific times after step input stimulation. Transfer functions for enzyme activation were derived from the published time-domain activation kinetics and these transfer functions were combined in a parallel architecture as a predictor of neurite outgrowth, as a function of input frequency. Qualitative agreement was observed between the model and the experiments.