Bulletin of Mathematical Biology

Cancer-associated fibroblasts (CAFs) are a central component of the tumor microenvironment that facilitate a supportive niche for cancer progression and metastasis. Experimental evidence suggests that CAFs can facilitate estrogen-independent tumor growth, thereby reducing the efficacy of anti-hormonal therapies. Understanding and quantifying the complex interactions between tumor cells, hormonal signalling, and the microenvironment are crucial for designing more effective and individualized treatment strategies. We propose a mathematical framework to explore the influence of CAFs on ER+ breast cancer progression and to evaluate strategies to mitigate their impact. We develop a system of nonlinear ordinary differential equations that substantiates the experimental observations by providing a mechanistic basis for the role of CAFs in regulating estrogen-independent growth dynamics. We then employ optimal control theory to evaluate distinct therapeutic approaches involving monotherapy or combinations of: (i) inhibition of tumor-to-CAF signaling, (ii) inhibition of CAF-to-tumor proliferative signaling, and (iii) endocrine therapy. Taken together, our results demonstrate that CAF-targeted strategies can enhance treatment efficacy across various estrogen dosing regimens. Our study provides new insights into the potential of CAF as a therapeutic target that could help to improve existing approaches for endocrine therapies.

Variability in gene expression among single cells and growing cell populations can arise from the stochastic nature of protein synthesis, which often occurs in random bursts. This study investigates the variability in the expression of a growth-sustaining protein, whose concentration is regulated by a negative feedback loop due to cell growth-induced dilution. We model the distribution of protein concentration using a Chapman-Kolmogorov equation for single cells and a population balance equation for growing cell populations. For single cells, we derive an explicit solution for the protein concentration distribution in state space and represent it as a Bessel function in Laplace space. For growing populations, we find that the distribution satisfies a Heun differential equation with singular boundary conditions. By addressing the central connection problem for the Heun equation, we quantify the population-level protein distribution and determine the Mathusian parameter, which characterizes population growth. This work provides a comprehensive analytical framework for understanding how stochastic protein synthesis impacts gene expression variability and population dynamics.

Cytokine-mediated communication is a central mechanism by which immune cells coordinate activation, differentiation and proliferation. While mechanistic reaction-diffusion models provide detailed descriptions of cytokine secretion and uptake at the cellular scale, their computational cost limits their applicability to large and densely packed cell populations. Previously employed approximations of cytokine diffusion fields rely on assumptions that neglect the influence of cellular geometry and volume exclusion. In this work, we study a macroscopic description of cytokine diffusion and reaction dynamics based on homogenization techniques, rigorously linking microscopic reaction-diffusion formulations to effective continuum models. The resulting homogenized equations replace discrete responder cells with a continuous density, while retaining essential features of cellular uptake and excluded-volume effects. Further, we show that in regimes with approximate radial symmetry, classical Yukawa-type solutions emerge as limiting cases of the homogenized model, provided appropriate correction factors are included. Overall, our approach allows efficient multiscale modeling of cytokine signaling in complex immune-cell environments.

Neuronal activity alters extracellular ion concentrations and electric potentials. Ephaptic effects refer to the feedback influence that these extracellular changes can have on neuronal activity. While electric ephaptic effects occur on a fast timescale due to extracellular potential perturbations, ionic ephaptic effects are driven by slower, accumulative changes in ion concentrations. Among the previous computational studies of ephaptic effects, the vast majority have focused exclusively on electric effects, while ionic ephaptic effects have largely been neglected. In this work, we present an electrodiffusive computational framework consisting of two-compartment neurons that interact via a shared extracellular space. By accounting for both electric potentials and ion-concentration dynamics in a self-consistent manner, our framework enables us to explore the relative roles of electric and ionic ephaptic effects. Through numerical experiments, we demonstrate that ionic and electric ephaptic interactions play very different roles. While ionic ephaptic interactions increase population firing rates, electric ephaptic interactions primarily drive subtle shifts in spike timing. Furthermore, we show that these spike shifts cause the phase difference (the distance in spike times between a small collection of neurons) to converge to a stable, unique phase difference, which we coin the ephaptic intrinsic phase preference. Author summaryNeurons predominantly communicate through synapses: specialized contact points where a brief electrical signal, known as a spike or action potential, in one neuron influences another. Neurons generate these spikes by exchanging ions with the surrounding extracellular space. This way, spiking neurons alter extracellular ion concentrations and electric potentials. Since neurons are sensitive to such changes in their environment, they can also influence one another indirectly through the shared extracellular medium. This form of non-synaptic interaction is known as ephaptic coupling. Most computational models of neuronal activity neglect ephaptic interactions, and those that include them typically consider only electric effects while ignoring ionic contributions. As a result, the relative roles of electric and ionic ephaptic effects remain poorly understood. Here, we introduce a computational framework that accounts for both mechanisms in a self-consistent way. Our results show a functional distinction: ionic ephaptic effects act slowly, regulating population firing rates, whereas electric ephaptic effects act on millisecond timescales and subtly shift spike timing. These shifts cause spike-time differences between neurons to converge to a stable value, a phenomenon we call ephaptic intrinsic phase preference.

Cerebrospinal fluid (CSF) is a Newtonian fluid that bathes the brain and spinal cord and oscillates in response to the physiological periodic changes in brain volume, of which the cardiac cycle is a major driver. Understanding this motion is essential for clarifying its contribution to solute transport, waste clearance, and drug delivery. In this work, we study oscillatory and steady streaming flow in the cranial subarachnoid space using a lubrication-based theoretical framework. The model represents the cranial CSF compartment as a thin fluid layer bounded internally by the brain surface and externally by the dura, driven by time-dependent brain surface displacements. We first derive simplified governing equations for flow over an arbitrary smooth sphere-like brain surface and obtain analytical solutions for an idealised spherical geometry with uniform displacements. We then incorporate realistic displacement fields reconstructed from MRI measurements in healthy subjects and solve the reduced equations numerically. The results show that oscillatory forcing produces a steady streaming component that may enhance solute transport compared with diffusion alone. This work provides a mechanistic description of the flow generated by physiological brain motion and highlights the potential presence of steady streaming in cranial subarachnoid fluid dynamics.

The Bayesian Ordered Lattice Design (BOLD) method for Phase I clinical trials is extended to address an important challenge. It is widely understood that conventional Phase I trial designs are not consistently effective in determining safe and active dose levels. The US FDA launched the Project Optimus, aimed at reforming the paradigms of dose optimization and selection. We propose a backfill BOLD design (BF-BOLD) that centers on BOLD for dose-finding but also adds an activity evaluation for each patient. Our method for determining the optimal biological dose (OBD) first involves identifying the maximum tolerated dose (MTD) and then assessing activity rates among dose levels below the identified MTD. This approach is straightforward and does not require complex statistical modeling. The results of the simulation indicate that performing dose-finding trials with backfilling can both enhance safety and activity assessment, thereby improving treatment sustainability while also preserving the potential for efficacy of the Recommended Phase II Dose (RP2D). We also demonstrate the applicability of the backfill design for reducing overdose rates, and as a more attractive alternative to small-scale dose expansion trials that follow dose escalation. Backfill designs are an important design approach for early phase trials.

Mass-conserved reaction-diffusion systems are used as mathematical models for various phenomena such as cell polarity. Numerical simulations of this system present transient dynamics in which multiple stripe patterns converge to spatially monotonic patterns. Previous studies indicated that the transient dynamics are driven by a mass conservation law and by variations in the amount of substance contained in each pattern, which we refer to as "pattern flux". However, it is challenging to mathematically investigate these pattern dynamics. In this study, we introduce a reaction-diffusion compartment model to investigate the pattern dynamics in view of the conservation law and the pattern flux. This model is defined on multiple intervals (compartments), and diffusive couplings are imposed on each boundary of the compartments. Corresponding to the transient dynamics in the original system, we consider the dynamics around stripe patterns in the compartment model. We derive ordinary differential equations describing the pattern dynamics of the compartment model and analyze the existence and stability of equilibria for the reduced ODE with respect to the boundary parameters. For a specific parameter setting, we obtained results consistent with previous studies. Moreover, we present that the stripe patterns in the compartment model are potentially stabilized by changing the parameter, which is not observed in the original system. We expect that the methodology developed in this paper is extendable to various directions, such as membrane-induced pattern control.

We extend previous models of cumulative cultural evolution by incorporating structured populations and social networks. We examine how connectivity and network topology shape the accumulation of cultural complexity under unbiased (copy randomly), indirectly biased (copy successful individuals), and directly biased (copy successful traits) transmission. We consider random, scalefree, and small-world networks, as well as the communication structures introduced by Mason and Watts, and derive analytical approximations for the homogeneous case. We find that the effects of social structure depend strongly on the form of transmission bias. Under unbiased transmission, network effects are weak except at very low connectivity. Under indirect bias, cultural complexity increases with connectivity, whereas direct bias shows optimal performance at intermediate connectivity, reflecting a trade-off between diffusion and diversity. Differences across topologies are generally modest once the average degree is fixed. Overall, our results show that no single social structure universally promotes cumulative cultural evolution; instead, its effects depend primarily on the dynamics of learning and innovation.

We introduce a spectral existence criterion for the evolution of cooperation in the form of the inequality{lambda} maxb > c, where{lambda} max is the leading eigenvalue of an interaction operator encoding population structure, and b and c represent benefit and cost tradeoffs, respectively. Nowaks five rules for the evolution of cooperation correspond to cases in which the cooperation condition reduces to a scalar assortment coefficient. These results follow from the Price equation, which sheds light on a long-standing debate on the role of inclusive fitness and evolutionary dynamics in explaining the evolution of cooperation.

BackgroundTo control leishmaniasis, chemotherapy drugs are currently under development. However, these drugs often exhibit poor efficacy and are associated with toxicity, adverse effects, and drug resistance. At present, no specific drug is available for the treatment of leishmaniasis. Meanwhile, vaccine research is ongoing. Recent studies have analysed some experimental vaccines using mathematical models. AimIn previous work, drug targeting was focused on the entire human body rather than specifically addressing infected macrophages and parasites. In our current approach, we aim to eliminate infected macrophages and parasites through nano-drug design. Specifically, we utilise two types of nanoparticles: iron oxide and citric acid-coated iron oxide. Moving forward, we plan to advance this strategy using mathematical modelling of macrophage-parasite interactions. MethodsWe design PDE-based models of macrophages and parasites, incorporating cytokine dynamics, to support nano-drug development. Drug efficacy is estimated using posterior distributions to analyse phenotypic fluctuations of macrophages and parasites during the design phase. We investigate implicit and semi-implicit treatment schemes, focusing on energy decay properties. To model drug flow during treatment, we introduce a three-phase moving boundary problem. Comparative analyses are conducted to evaluate macrophage and parasite behaviour with and without treatment. Finally, the entire framework is implemented within a virtual lab environment. ResultsThe results show that the nano-drug exhibits better efficacy compared to combined drug doses. We analysed and compared two types of nano-drug particles: iron oxide and citric acid-coated iron oxide. We discuss how the drug effectively targets and eliminates infected macrophages and parasites. ConclusionOur models results and simulations will support researchers conducting further studies in nano-drug design for leishmaniasis. These simulations are performed within a virtual lab environment.

Spatio-temporal interactions shape microbial community dynamics. Metabolism, through competition and cross-feeding, is a foundational mechanism of these interactions. Flux balance analysis enables efficient simulation of steady-state metabolism. Integrating these simulations through time, using dynamic flux balance analysis, provides temporal predictions of growth and metabolism. Incorporating spatial context, through partial differential equations, enables spatio-temporal simulation of microbial communities. In this chapter, we step through this sequential process, moving from steady-state, to temporal, to spatio-temporal simulation of microbial community metabolism. We provide an illustrative example using the modeling software COMETS (Computation of Microbial Ecosystems in Time and Space) to simulate interacting bacterial colonies of Bifidobacterium longum subsp. infantis and Anaerobutyricum hallii (previously Eubacterium hallii). Within this simulation, both competition and cross-feeding influenced the production of butyrate leading to an intermediate optimal interaction distance for metabolite production. We outline each step and provide open-source code such that this simulation can serve as a template for future spatio-temporal simulations of microbial community metabolism.

Mitochondria often form branching membrane networks distributed throughout the cell interior. In many, though not all, cell types, these networks are observed to consist of one large connected component together with many smaller fragments. Why does this pattern arise? Does it reflect a specific biological function, an external biophysical constraint, or something simpler? Using results from extremal graph theory, we prove a new theorem which suggests that, under a sufficiently broad sampling of the space of mitochondria-like graphs, the predominance of three-way junctions makes the appearance of a large component likely. This suggests that, in some settings, a large component may serve as a useful null model for mitochondrial network structure rather than requiring a dedicated explanation. More broadly, our result points towards testable predictions, since systematic deviations from this baseline may help reveal additional constraints or mechanisms shaping mitochondrial morphology.

Trees accumulate somatic mutations throughout their long lifespan, resulting in genetic mosaicism among branches. While recent genomic studies quantified these mutations, they were largely limited to describing static patterns of variation. In this study, we developed a mathematical model to infer the dynamic processes of somatic mutation accumulation from snapshot genomic data obtained from four tropical trees (Dipterocarpaceae), which dominate tropical rain forests in Southeast Asia. Our model focus on genetic differences between shoot apical meristems (SAMs) at branch tips and explicitly incorporate stem cell dynamics within SAMs during shoot elongation and branching, enabling us to quantify somatic genetic drift arising from stem cell lineage replacement. By comparing model predictions with empirical data from Dipterocarpaceae trees, we estimated key parameters governing stem cell dynamics and somatic mutation rates. Our results indicate that both shoot elongation and branching involve replacement of stem cell lineages, leading to a moderate degree of somatic genetic drift. Accounting for stem cell dynamics resulted in slightly lower mutation rate estimates than previous approaches that ignored these processes. Using the estimated parameters, we further performed stochastic simulations to predict patterns of somatic mutations, including features not directly observed in the sampled trees, such as occasional deviations of somatic mutation phylogenies from physical architecture. Together, our modeling framework provides insights into how genetic mosaicism is shaped within tropical trees and reveals the stem cell dynamics underlying their long-term growth and accumulation of somatic mutations. (236 words) Highlights- We built mathematical models to predict the genetic differences between branch tips by somatic mutations. - The model considers the varying dynamics of stem cells in shoot meristem during shoot elongation and branching. - We compared the model prediction with empirical data from tropical trees, Dipterocarpaceae, and estimated the dynamics of stem cells and mutation rate. - Somatic mutation dynamics were shaped by somatic genetic drift arising from stem cell lineage replacement during shoot elongation and branching. - Accounting for stem cell dynamics led to slightly smaller estimates of mutation rates compared with previous estimates that ignored the dynamics. - Our models offer insights into how genetic variability is shaped in the tropical trees and the stem cell dynamics underlying their long-term growth.

Alzheimers disease (AD) is characterized by the accumulation of amyloid-{beta} (A{beta}), yet the specific link between plaque burden and cognitive decline remains a subject of intense investigation. This paper presents a mathematical model that simulates the coupled dynamics of A{beta} monomers, soluble oligomers, and fibrillar species in the brain tissue. By modifying existing moment equations to include a dedicated conservation equation for A{beta} monomers, the model explores how various microscopic processes, such as primary nucleation, surface-catalyzed secondary nucleation, fibril elongation, and fragmentation, contribute to macroscopic disease progression. Central to this study is the concept of "accumulated neurotoxicity" as a surrogate marker of biological age, defined as the time-integrated concentration of soluble A{beta} oligomers. Unlike plaque burden, accumulated neurotoxicity cannot be reversed, and the harm it causes depends critically on the sequence of events that produced it. Numerical results demonstrate that while plaque burden and neurotoxicity both increase over time, their relationship is non-linear and highly sensitive to the efficiency of protein degradation machinery. Specifically, impaired degradation leads to a rapid advancement of biological age relative to calendar age. The model further identifies oligomer dissociation and fibril fragmentation as potential protective mechanisms that can counterintuitively reduce neurotoxic burden by diverting monomers away from the soluble oligomer pool. These findings provide a quantitative framework for understanding why individuals with similar plaque burdens may experience vastly different cognitive outcomes, underscoring the importance of targeting soluble oligomers early in therapeutic interventions.

Timeliness of therapy initiation is a fundamental determinant of outcomes for many medical conditions, most importantly, cancer. Yet, existing inefficiencies in healthcare systems mean that delays between diagnosis and treatment frequently adversely affect the clinical outcome for cancer patients. Although estimates of effects of lag time to therapy would be informative to policymakers considering resource allocation to minimize delays in oncology, causal methods are seldom explicitly discussed in epidemiologic analyses of these lag times. Here, we propose causal estimands for such studies, and outline the protocol of a target trial that could be emulated with observational data on lag times. To illustrate the application of this approach, we simulate studies of lag time to treatment under two scenarios: one in which indication bias (Waiting Time Paradox) is present and another in which it is absent. Although our discussion focuses on oncologic outcomes, components of the proposed target trial could be adapted to study delays for other medical conditions. We believe that the clarity with which causal questions are posed under the target trial emulation framework would lead to improved quantification of the effects of lag times in oncology, and hence to better informed policy decisions.

Cancer cells in hypoxic environments often proliferate less but exhibit enhanced migration relative to their normoxic counterparts. Recent in vitro and in silico studies have characterized the role of hypoxic memory - the ability of cancer cells to retain their hypoxic phenotype even when reoxygenated - in tumor invasion. However, the observations have been limited either to exposing cancer cells to hypoxia for a fixed duration or by assuming a fixed-time persistence of the hypoxic state upon reoxygenation independent of the duration of hypoxia exposure. Thus, time-dependent cell-state changes during hypoxia and their impact on hypoxic memory remains unclear. Here, we first analyze transcriptomic data from breast cancer samples to show that the genes upregulated at transcriptional level and hypomethylated at epigenetic level are enriched in cell invasion, indicating hypoxic memory-driven process of tumor invasion. Next, we used a computational model to investigate how the spatial-temporal dynamics of oxygen levels in a tumor drive time-dependent changes in hypoxic memory and influence tumor invasion dynamics. Our simulation results show that such dynamic hypoxic memory can drive enhanced tumor invasion over a fixed hypoxic memory by a) enriching hypoxic cell density at the tumor front, b) reducing sensitivity of hypoxic cell state to fluctuations in oxygen supply, and c) enhancing effective diffusion of hypoxic cells. Our results highlight the crucial role of dynamic hypoxic memory in shaping tumor invasion dynamics, underscoring the need to elucidate its underlying mechanisms in future studies.

Patient-derived tumour spheroids are increasingly used as engineered three-dimensional tissue models for studying tumour growth, nutrient limitation, and therapeutic response. However, extracting quantitative, mechanistically interpretable information from longitudinal imaging data remains challenging. Here, we present a three-dimensional phase-field framework for modelling patient-derived tumour spheroids as continuum, self-organising tissues. The model captures the coupled evolution of viable and necrotic cell fractions through nutrient-limited growth, death, and mechanically and thermodynamically mediated motion, using seven biologically interpretable effective parameters. Key experimental observables emerge naturally from nutrient-growth coupling, without imposing explicit species interfaces or quiescent layers. The framework was quantitatively calibrated against longitudinal imaging data from melanoma spheroids across two cell lines and three initial seeding densities. Across all conditions, simulations reproduced the temporal evolution of all measured observables with low relative error ({approx} 3{sigma} of experimental data), and direct comparison with an established Greenspan-type ODE model demonstrated comparable or improved predictive accuracy. Parameter identifiability analysis revealed weak individual parameter constraints, yet model predictions remained robust, a profile consistent with biological models. We demonstrate that a general PDE-based growth framework can match or outperform a dedicated spheroid model while remaining fully biologically interpretable. Beyond predictive accuracy, the phase-field formulation naturally resolves internal mechanical structure, providing access to quantities that are not directly experimentally observable. These results establish that mechanistically grounded continuum models can be quantitatively calibrated to routine spheroid imaging data, offering a foundation for integrating spatial and mechanical information into the interpretation of organoid-based assays. Graphical Abstract O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=77 SRC="FIGDIR/small/717345v1_ufig1.gif" ALT="Figure 1"> View larger version (21K): org.highwire.dtl.DTLVardef@1cb3b45org.highwire.dtl.DTLVardef@1a053d5org.highwire.dtl.DTLVardef@dffe34org.highwire.dtl.DTLVardef@1aa0b72_HPS_FORMAT_FIGEXP M_FIG C_FIG

Clavicle fractures often exhibit markedly different clinical outcomes: some patients recover acceptable function despite shortening or displacement, whereas others with apparently similar deformity develop persistent pain, functional loss, or poor healing. To explain this distinction, we propose a minimal nonlinear mechanical model for prognostic analysis of clavicle fractures. The model describes the interaction between fracture-related shortening and compensatory shoulder-girdle posture through a reduced equilibrium equation incorporating stiffness, geometric nonlinearity, and shortening-posture coupling. Within this framework, we analyze equilibrium branches, local stability, and the emergence of critical thresholds. We show that post-fracture destabilization can be interpreted as a fold bifurcation, while more complex parameter dependence gives rise to cusp-type structures and multistability. These bifurcation mechanisms provide a mathematical explanation for sudden deterioration after injury or treatment, as well as for strong inter-individual variability. We further introduce an optimization principle based on a utility functional to guide treatment planning. The analysis predicts that the optimal safe correction should lie strictly below the bifurcation threshold, thereby generating a natural safety margin. Although the model is simplified and has not yet been calibrated against patient data, it nevertheless provides a theoretical framework for understanding why fracture prognosis may deteriorate abruptly near critical mechanical conditions and offers a dynamical-systems interpretation of empirical treatment thresholds used in clinical practice.

How genetically identical cells spontaneously break symmetry to assume divergent fates is a fundamental problem in developmental biology. While modern genomics has mapped the vast molecular repertoire involved in gene regulation, understanding the mechanism of cell state transitions that drive differentiation remains a formidable challenge. To address this, we use a reaction-kinetic framework to analyze recurring motifs of two and three competing master regulators. While typically such circuits are studied numerically, we show that assuming symmetry in nodes and interactions provides exact analytical description of the bifurcations governing cell fate transitions. We find that the possible cell fates across all considered topologies are dictated by a single dimensionless quantity, {beta}--the ratio of protein degradation to production rates. In the binary Toggle Switch (TS), decreasing {beta} destabilizes the symmetric (stem cell) state, giving rise to two asymmetric (differentiated) fates via a supercritical pitchfork bifurcation. In the three-component Toggle Triad (TT), low values of {beta} yield three asymmetric fates through subcritical pitchfork bifurcation, creating an intermediate range of {beta} where both symmetric and asymmetric fates are simultaneously stable. For the Self-Activating Toggle Switch (SATS), we identify a new parameter for the self-activation threshold ({theta}) and show that decreasing{theta} progressively stabilizes the uncommitted state, leading to a regime of tristability. Building on these temporal bifurcations, we next address the feasibility of spatial structure formation: can these multistable fates stably coexist within a spatial domain? Through a minimal model of cell-cell communication via free diffusion, we extend these motifs into reaction-diffusion systems, which reveals a direct role of network topology on spatial organization. We prove that any heterogeneous pattern in two-node circuits is inherently transient and unstable. In contrast, the three-node repressive network supports the stable spatial coexistence of differentiated phenotypes through pure diffusion, a phenomenon we analyze by studying heteroclinic interface solutions as building blocks. By reducing complex regulatory dynamics to tractable models with physically meaningful parameters, we establish a minimal framework which relates topology to cell fate. Finally, the effects of temporal multistability on pattern formation provide an excellent studying ground for morphogenesis, synthetic biology, and the overarching problem of spatiotemporal self-organization.

Stratified epithelial tissues such as the skin epidermis maintain barrier integrity during development and homeostasis through the coordinated action of cell proliferation, differentiation, delamination, and tissue-scale mechanical forces. During development, the orientation of cell division within the basal layer plays a pivotal role in tissue stratification; however, the mechanical principles linking the orientation of the division plane to these processes across developmental stages remain poorly understood. Here, we expand a recently developed three-dimensional vertex model for stratified epithelia, composed of the basement membrane, basal, and suprabasal layers, to study the mechanical and structural impact of cell divisions with a wider range of orientations. The model integrates developmental stage via specific changes in heterotypic interfacial tensions (arising from actomyosin cortical contractility and adhesion molecules at the basal-suprabasal interface) and tissue stiffness that have been quantified previously in experiments. By systematically varying background mechanical parameters, we investigate how heterotypic tension, division orientation, and tissue fluidity collectively influence the outcome of cell division. Our goal is to uncover the strategies that the embryo may employ to generate stratified phenotypes at different developmental stages, recognizing that these strategies might evolve over time. Although our focus is on the embryonic developmental stages of the epidermis, this framework may also be extended to investigate transformed cells, such as in cancer, to explore how altered division orientation contributes to precancerous or transformed phenotypes.