Linear models of activation cascades: analytical solutions and coarse-graining of delayed signal transduction

Multiscale mobility patterns and the restriction of human movement

From the perspective of human mobility, the COVID-19 pandemic constituted a natural experiment of enormous reach in space and time. Here, we analyse the inherent multiple scales of human mobility using Facebook Movement maps collected before and during the first UK lockdown. Firstly, we obtain the pre-lockdown UK mobility graph and employ multiscale community detection to extract, in an unsupervised manner, a set of robust partitions into flow communities at different levels of coarseness. The partitions so obtained capture intrinsic mobility scales with better coverage than nomenclature of territorial units for statistics (NUTS) regions, which suffer from mismatches between human mobility and administrative divisions. Furthermore, the flow communities in the fine-scale partition not only match well the UK travel to work areas but also capture mobility patterns beyond commuting to work. We also examine the evolution of mobility under lockdown and show that mobility first reverted towards fine-scale flow communities already found in the pre-lockdown data, and then expanded back towards coarser flow communities as restrictions were lifted. The improved coverage induced by lockdown is well captured by a linear decay shock model, which allows us to quantify regional differences in both the strength of the effect and the recovery time from the lockdown shock.

The ups and downs of biological oscillators: a comparison of time-delayed negative feedback mechanisms

Many biochemical oscillators are driven by the periodic rise and fall of protein concentrations or activities. A negative feedback loop underlies such oscillations. The feedback can act on different parts of the biochemical network. Here, we mathematically compare time-delay models where the feedback affects production and degradation. We show a mathematical connection between the linear stability of the two models, and derive how both mechanisms impose different constraints on the production and degradation rates that allow oscillations. We show how oscillations are affected by the inclusion of a distributed delay, of double regulation (acting on production and degradation) and of enzymatic degradation.

Systematic inference identifies a major source of heterogeneity in cell signaling dynamics: The rate-limiting step number

Identifying the sources of cell-to-cell variability in signaling dynamics is essential to understand drug response variability and develop effective therapeutics. However, it is challenging because not all signaling intermediate reactions can be experimentally measured simultaneously. This can be overcome by replacing them with a single random time delay, but the resulting process is non-Markovian, making it difficult to infer cell-to-cell heterogeneity in reaction rates and time delays. To address this, we developed an efficient and scalable moment-based Bayesian inference method (MBI) with a user-friendly computational package that infers cell-to-cell heterogeneity in the non-Markovian signaling process. We applied MBI to single-cell expression profiles from promoters responding to antibiotics and discovered a major source of cell-to-cell variability in antibiotic stress response: the number of rate-limiting steps in signaling cascades. This knowledge can help identify effective therapies that destroy all pathogenic or cancer cells, and the approach can be applied to precision medicine.

Limited inhibition of multiple nodes in a driver network blocks metastasis

Metastasis suppression by high-dose, multi-drug targeting is unsuccessful due to network heterogeneity and compensatory network activation. Here, we show that targeting driver network signaling capacity by limited inhibition of core pathways is a more effective anti-metastatic strategy. This principle underlies the action of a physiological metastasis suppressor, Raf Kinase Inhibitory Protein (RKIP), that moderately decreases stress-regulated MAP kinase network activity, reducing output to transcription factors such as pro-metastastic BACH1 and motility-related target genes. We developed a low-dose four-drug mimic that blocks metastatic colonization in mouse breast cancer models and increases survival. Experiments and network flow modeling show limited inhibition of multiple pathways is required to overcome variation in MAPK network topology and suppress signaling output across heterogeneous tumor cells. Restricting inhibition of individual kinases dissipates surplus signal, preventing threshold activation of compensatory kinase networks. This low-dose multi-drug approach to decrease signaling capacity of driver networks represents a transformative, clinically relevant strategy for anti-metastatic treatment.

Memorizing environmental signals through feedback and feedforward loops

Cells in diverse organisms can store the information of previous environmental conditions for long periods of time. This form of cellular memory adjusts the cell's responses to future challenges, providing fitness advantages in fluctuating environments. Many biological functions, including cellular memory, are mediated by specific recurring patterns of interactions among proteins and genes, known as 'network motifs.' In this review, we focus on three well-characterized network motifs - negative feedback loops, positive feedback loops, and feedforward loops, which underlie different types of cellular memories. We describe the latest studies identifying these motifs in various molecular processes and discuss how the topologies and dynamics of these motifs can enable memory encoding and storage.

Molecular switch architecture determines response properties of signaling pathways

Many intracellular signaling pathways are composed of molecular switches, proteins that transition between two states-on and off Typically, signaling is initiated when an external stimulus activates its cognate receptor that, in turn, causes downstream switches to transition from off to on using one of the following mechanisms: activation, in which the transition rate from the off state to the on state increases; derepression, in which the transition rate from the on state to the off state decreases; and concerted, in which activation and derepression operate simultaneously. We use mathematical modeling to compare these signaling mechanisms in terms of their dose-response curves, response times, and abilities to process upstream fluctuations. Our analysis elucidates several operating principles for molecular switches. First, activation increases the sensitivity of the pathway, whereas derepression decreases sensitivity. Second, activation generates response times that decrease with signal strength, whereas derepression causes response times to increase with signal strength. These opposing features allow the concerted mechanism to not only show dose-response alignment, but also to decouple the response time from stimulus strength. However, these potentially beneficial properties come at the expense of increased susceptibility to upstream fluctuations. We demonstrate that these operating principles also hold when the models are extended to include additional features, such as receptor removal, kinetic proofreading, and cascades of switches. In total, we show how the architecture of molecular switches govern their response properties. We also discuss the biological implications of our findings.

Exact solutions and equi-dosing regimen regions for multi-dose pharmacokinetics models with transit compartments

Compartmental models which yield linear ordinary differential equations (ODEs) provide common tools for pharmacokinetics (PK) analysis, with exact solutions for drug levels or concentrations readily obtainable for low-dimensional compartment models. Exact solutions enable valuable insights and further analysis of these systems. Transit compartment models are a popular semi-mechanistic approach for generalising simple PK models to allow for delayed kinetics, but computing exact solutions for multi-dosing inputs to transit compartment systems leading to different final compartments is nontrivial. Here, we find exact solutions for drug levels as functions of time throughout a linear transit compartment cascade followed by an absorption compartment and a central blood compartment, for the general case of n transit compartments and M equi-bolus doses to the first compartment. We further show the utility of exact solutions to PK ODE models in finding constraints on equi-dosing regimen parameters imposed by a prescribed therapeutic range. This leads to the construction of equi-dosing regimen regions (EDRRs), providing new, novel visualisations which summarise the safe and effective dosing parameter space. EDRRs are computed for classical and transit compartment models with two- and three-dimensional parameter spaces, and are proposed as useful graphical tools for informing drug dosing regimen design.

It's about time: Analysing simplifying assumptions for modelling multi-step pathways in systems biology

Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics. Here, we perform a computational investigation of sequential multi-step pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a pathway leads to detectable signatures in model dynamics and its optimised parameters. An alternative assumption is suggested for simplifying such pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and, instead, letting the length of the pathway be a free parameter. We first focus on linear pathways that are sequential and have first-order kinetics, and we show how this assumption results in a three-parameter model that consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics. We then show how the proposed assumption allows for similarly terse and effective models of non-linear pathways. Our results provide a foundation for well-informed decision making during model simplifications.

A network-based analysis for mining the risk pathways in glioblastoma

The most malignant type of brain tumour is glioblastoma multiforme (GBM). Patients with GBM often have a poor prognosis, as a result of incomplete or inaccurate diagnoses. Regulatory pathways have been demonstrated to serve important roles in complex human diseases. Therefore, deciphering these risk pathways may shed light on the molecular mechanisms underlying GBM progression. In the present study, differentially expressed genes and microRNAs (miRNAs) in a publicly available database were identified between normal and tumour samples. To determine the pathophysiology and molecular mechanisms underlying GBM, integrated network analysis was performed to mine GBM-specific risk pathways. Specifically, a GBM-specific regulatory network was constructed that integrated manually curated GBM-associated transcription and post-transcriptional data resources, including transcription factors and miRNAs. A total of 1,827 differentially expressed genes and 30 miRNAs were identified. The differentially expressed genes were significantly enriched in a number of immune response-associated functions. Based on the GBM-specific regulatory network, 15 risk regulatory pathways containing not only known regulators, but also potential novel targets that might be involved in tumourigenesis were identified. Network analysis provides a strategy for leveraging genomic data to identify potential oncogenic pathways and molecular targets for GBM.

Tensor clustering with algebraic constraints gives interpretable groups of crosstalk mechanisms in breast cancer

We introduce a tensor-based clustering method to extract sparse, low-dimensional structure from high-dimensional, multi-indexed datasets. This framework is designed to enable detection of clusters of data in the presence of structural requirements which we encode as algebraic constraints in a linear program. Our clustering method is general and can be tailored to a variety of applications in science and industry. We illustrate our method on a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. Each experiment consists of a cell line–ligand combination, and contains time-course measurements of the early signalling kinases MAPK and AKT at two different ligand dose levels. By imposing appropriate structural constraints and respecting the multi-indexed structure of the data, the analysis of clusters can be optimized for biological interpretation and therapeutic understanding. We then perform a systematic, large-scale exploration of mechanistic models of MAPK–AKT crosstalk for each cluster. This analysis allows us to quantify the heterogeneity of breast cancer cell subtypes, and leads to hypotheses about the signalling mechanisms that mediate the response of the cell lines to ligands.

Spatial modeling of the membrane-cytosolic interface in protein kinase signal transduction

The spatial architecture of signaling pathways and the interaction with cell size and morphology are complex, but little understood. With the advances of single cell imaging and single cell biology, it becomes crucial to understand intracellular processes in time and space. Activation of cell surface receptors often triggers a signaling cascade including the activation of membrane-attached and cytosolic signaling components, which eventually transmit the signal to the cell nucleus. Signaling proteins can form steep gradients in the cytosol, which cause strong cell size dependence. We show that the kinetics at the membrane-cytosolic interface and the ratio of cell membrane area to the enclosed cytosolic volume change the behavior of signaling cascades significantly. We suggest an estimate of average concentration for arbitrary cell shapes depending on the cell volume and cell surface area. The normalized variance, known from image analysis, is suggested as an alternative measure to quantify the deviation from the average concentration. A mathematical analysis of signal transduction in time and space is presented, providing analytical solutions for different spatial arrangements of linear signaling cascades. Quantification of signaling time scales reveals that signal propagation is faster at the membrane than at the nucleus, while this time difference decreases with the number of signaling components in the cytosol. Our investigations are complemented by numerical simulations of non-linear cascades with feedback and asymmetric cell shapes. We conclude that intracellular signal propagation is highly dependent on cell geometry and, thereby, conveys information on cell size and shape to the nucleus.

Mathematical model of early Reelin-induced Src family kinase-mediated signaling

Reelin is a large glycoprotein with a dual role in the mammalian brain. It regulates the positioning and differentiation of postmitotic neurons during brain development and modulates neurotransmission and memory formation in the adult brain. Alterations in the Reelin signaling pathway have been described in different psychiatric disorders. Reelin mainly signals by binding to the lipoprotein receptors Vldlr and ApoER2, which induces tyrosine phosphorylation of the adaptor protein Dab1 mediated by Src family kinases (SFKs). In turn, phosphorylated Dab1 activates downstream signaling cascades, including PI3-kinase-dependent signaling. In this work, a mechanistic model based on ordinary differential equations was built to model early dynamics of the Reelin-mediated signaling cascade. Mechanistic models are frequently used to disentangle the highly complex mechanisms underlying cellular processes and obtain new biological insights. The model was calibrated on time-resolved data and a dose-response measurement of protein concentrations measured in cortical neurons treated with Reelin. It focusses on the interplay between Dab1 and SFKs with a special emphasis on the tyrosine phosphorylation of Dab1, and their role for the regulation of Reelin-induced signaling. Model selection was performed on different model structures and a comprehensive mechanistic model of the early Reelin signaling cascade is provided in this work. It emphasizes the importance of Reelin-induced lipoprotein receptor clustering for SFK-mediated Dab1 trans-phosphorylation and does not require co-receptors to describe the measured data. The model is freely available within the open-source framework Data2Dynamics (www.data2dynamics.org). It can be used to generate predictions that can be validated experimentally, and provides a platform for model extensions both to downstream targets such as transcription factors and interactions with other transmembrane proteins and neuronal signaling pathways.

Statistical physics approaches to subnetwork dynamics in biochemical systems

We apply a Gaussian variational approximation to model reduction in large biochemical networks of unary and binary reactions. We focus on a small subset of variables (subnetwork) of interest, e.g. because they are accessible experimentally, embedded in a larger network (bulk). The key goal is to write dynamical equations reduced to the subnetwork but still retaining the effects of the bulk. As a result, the subnetwork-reduced dynamics contains a memory term and an extrinsic noise term with non-trivial temporal correlations. We first derive expressions for this memory and noise in the linearized (Gaussian) dynamics and then use a perturbative power expansion to obtain first order nonlinear corrections. For the case of vanishing intrinsic noise, our description is explicitly shown to be equivalent to projection methods up to quadratic terms, but it is applicable also in the presence of stochastic fluctuations in the original dynamics. An example from the epidermal growth factor receptor signalling pathway is provided to probe the increased prediction accuracy and computational efficiency of our method.