Modelling spatially regulated beta-catenin dynamics and invasion in intestinal crypts

Mechanical Cell Interactions on Curved Interfaces

We propose a simple mathematical model to describe the mechanical relaxation of cells within a curved epithelial tissue layer represented by an arbitrary curve in two-dimensional space. This model generalises previous one-dimensional models of flat epithelia to investigate the influence of curvature for mechanical relaxation. We represent the mechanics of a cell body either by straight springs, or by curved springs that follow the curve's shape. To understand the collective dynamics of the cells, we devise an appropriate continuum limit in which the number of cells and the length of the substrate are constant but the number of springs tends to infinity. In this limit, cell density is governed by a diffusion equation in arc length coordinates, where diffusion may be linear or nonlinear depending on the choice of the spring restoring force law. Our results have important implications about modelling cells on curved geometries: (i) curved and straight springs can lead to different dynamics when there is a finite number of springs, but they both converge quadratically to the dynamics governed by the diffusion equation; (ii) in the continuum limit, the curvature of the tissue does not affect the mechanical relaxation of cells within the layer nor their tangential stress; (iii) a cell's normal stress depends on curvature due to surface tension induced by the tangential forces. Normal stress enables cells to sense substrate curvature at length scales much larger than their cell body, and could induce curvature dependences in experiments.

The force of cell-cell adhesion in determining the outcome in a nonlocal advection diffusion model of wound healing

A model of wound healing is presented to investigate the connection of the force of cell-cell adhesion to the sensing radius of cells in their spatial environment. The model consists of a partial differential equation with nonlocal advection and diffusion terms, describing the movement of cells in a spatial environment. The model is applied to biological wound healing experiments to understand incomplete wound closure. The analysis demonstrates that for each value of the force of adhesion parameter, there is a critical value of the sensing radius above which complete wound healing does not occur.

Relating simulation studies by provenance-Developing a family of Wnt signaling models

For many biological systems, a variety of simulation models exist. A new simulation model is rarely developed from scratch, but rather revises and extends an existing one. A key challenge, however, is to decide which model might be an appropriate starting point for a particular problem and why. To answer this question, we need to identify entities and activities that contributed to the development of a simulation model. Therefore, we exploit the provenance data model, PROV-DM, of the World Wide Web Consortium and, building on previous work, continue developing a PROV ontology for simulation studies. Based on a case study of 19 Wnt/β-catenin signaling models, we identify crucial entities and activities as well as useful metadata to both capture the provenance information from individual simulation studies and relate these forming a family of models. The approach is implemented in WebProv, a web application for inserting and querying provenance information. Our specialization of PROV-DM contains the entities Research Question, Assumption, Requirement, Qualitative Model, Simulation Model, Simulation Experiment, Simulation Data, and Wet-lab Data as well as activities referring to building, calibrating, validating, and analyzing a simulation model. We show that most Wnt simulation models are connected to other Wnt models by using (parts of) these models. However, the overlap, especially regarding the Wet-lab Data used for calibration or validation of the models is small. Making these aspects of developing a model explicit and queryable is an important step for assessing and reusing simulation models more effectively. Exposing this information helps to integrate a new simulation model within a family of existing ones and may lead to the development of more robust and valid simulation models. We hope that our approach becomes part of a standardization effort and that modelers adopt the benefits of provenance when considering or creating simulation models.

We revise a provenance ontology for simulation studies of cellular biochemical models. Provenance information is useful for understanding the creation of a simulation model because it not only contains information about the entities and activities that have led to a simulation model but also their relations, all of which can be visualized. It provides additional structure by explicitly recording research questions, assumptions, and requirements and relating them along with data, qualitative models, simulation models, and simulation experiments through a small set of predefined but extensible activities.

We have applied our concept to a family of 19 Wnt signaling models and implemented a web-based tool (WebProv) to store the provenance information from these studies. The resulting provenance graph visualizes the story line of simulation studies and demonstrates the creation and calibration of simulation models, the successive attempts of validation and extension, and shows, beyond an individual simulation study, how the Wnt models are related. Thereby, the steps and sources that contributed to a simulation model are made explicit.

Our approach complements other approaches aimed at facilitating the reuse and assessment of simulation products in systems biology such as model repositories as well as annotation and documentation guidelines.

A logical network-based drug-screening platform for Alzheimer's disease representing pathological features of human brain organoids

Developing effective drugs for Alzheimer's disease (AD), the most common cause of dementia, has been difficult because of complicated pathogenesis. Here, we report an efficient, network-based drug-screening platform developed by integrating mathematical modeling and the pathological features of AD with human iPSC-derived cerebral organoids (iCOs), including CRISPR-Cas9-edited isogenic lines. We use 1300 organoids from 11 participants to build a high-content screening (HCS) system and test blood-brain barrier-permeable FDA-approved drugs. Our study provides a strategy for precision medicine through the convergence of mathematical modeling and a miniature pathological brain model using iCOs.

A Systems Biology Approach to Identifying a Master Regulator That Can Transform the Fast Growing Cellular State to a Slowly Growing One in Early Colorectal Cancer Development Model

Colorectal cancer (CRC) has been most extensively studied for characterizing genetic mutations along its development. However, we still have a poor understanding of CRC initiation due to limited measures of its observation and analysis. If we can unveil CRC initiation events, we might identify novel prognostic markers and therapeutic targets for early cancer detection and prevention. To tackle this problem, we establish the early CRC development model and perform transcriptome analysis of its single cell RNA-sequencing data. Interestingly, we find two subtypes, fast growing vs. slowly growing populations of distinct growth rate and gene signatures, and identify CCDC85B as a master regulator that can transform the cellular state of fast growing subtype cells into that of slowly growing subtype cells. We further validate this by in vitro experiments and suggest CCDC85B as a novel potential therapeutic target that may prevent malignant CRC development by suppressing stemness and uncontrolled cell proliferation.

Mechanical Cell Competition in Heterogeneous Epithelial Tissues

Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than an intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.

Realizing Cancer Precision Medicine by Integrating Systems Biology and Nanomaterial Engineering

Many clinical trials for cancer precision medicine have yielded unsatisfactory results due to challenges such as drug resistance and low efficacy. Drug resistance is often caused by the complex compensatory regulation within the biomolecular network in a cancer cell. Recently, systems biological studies have modeled and simulated such complex networks to unravel the hidden mechanisms of drug resistance and identify promising new drug targets or combinatorial or sequential treatments for overcoming resistance to anticancer drugs. However, many of the identified targets or treatments present major difficulties for drug development and clinical application. Nanocarriers represent a path forward for developing therapies with these "undruggable" targets or those that require precise combinatorial or sequential application, for which conventional drug delivery mechanisms are unsuitable. Conversely, a challenge in nanomedicine has been low efficacy due to heterogeneity of cancers in patients. This problem can also be resolved through systems biological approaches by identifying personalized targets for individual patients or promoting the drug responses. Therefore, integration of systems biology and nanomaterial engineering will enable the clinical application of cancer precision medicine to overcome both drug resistance of conventional treatments and low efficacy of nanomedicine due to patient heterogeneity.

Feedback analysis identifies a combination target for overcoming adaptive resistance to targeted cancer therapy

Targeted drugs aim to treat cancer by directly inhibiting oncogene activity or oncogenic pathways, but drug resistance frequently emerges. Due to the intricate dynamics of cancer signaling networks, which contain complex feedback regulations, cancer cells can rewire these networks to adapt to and counter the cytotoxic effects of a drug, thereby limiting the efficacy of targeted therapies. To identify a combinatorial drug target that can overcome such a limitation, we developed a Boolean network simulation and analysis framework and applied this approach to a large-scale signaling network of colorectal cancer with integrated genomic information. We discovered Src as a critical combination drug target that can overcome the adaptive resistance to the targeted inhibition of mitogen-activated protein kinase pathway by blocking the essential feedback regulation responsible for resistance. The proposed framework is generic and can be widely used to identify drug targets that can overcome adaptive resistance to targeted therapies.

Combined Positive and Negative Feedback Allows Modulation of Neuronal Oscillation Frequency during Sensory Processing

A key step in sensory information processing involves modulation and integration of neuronal oscillations in disparate frequency bands, a poorly understood process. Here, we investigate how top-down input causes frequency changes in slow oscillations during sensory processing and, in turn, how the slow oscillations are combined with fast oscillations (which encode sensory input). Using experimental connectivity patterns and strengths of interneurons, we develop a system-level model of a neuronal circuit controlling these oscillatory behaviors, allowing us to understand the mechanisms responsible for the observed oscillatory behaviors. Our analysis discovers a circuit capable of producing the observed oscillatory behaviors and finds that a detailed balance in the strength of synaptic connections is the critical determinant to produce such oscillatory behaviors. We not only uncover how disparate frequency bands are modulated and combined but also give insights into the causes of abnormal neuronal activities present in brain disorders.

Signal flow control of complex signaling networks

Complex disease such as cancer is often caused by genetic mutations that eventually alter the signal flow in the intra-cellular signaling network and result in different cell fate. Therefore, it is crucial to identify control targets that can most effectively block such unwanted signal flow. For this purpose, systems biological analysis provides a useful framework, but mathematical modeling of complicated signaling networks requires massive time-series measurements of signaling protein activity levels for accurate estimation of kinetic parameter values or regulatory logics. Here, we present a novel method, called SFC (Signal Flow Control), for identifying control targets without the information of kinetic parameter values or regulatory logics. Our method requires only the structural information of a signaling network and is based on the topological estimation of signal flow through the network. SFC will be particularly useful for a large-scale signaling network to which parameter estimation or inference of regulatory logics is no longer applicable in practice. The identified control targets have significant implication in drug development as they can be putative drug targets.

Minimal intervening control of biomolecular networks leading to a desired cellular state

A cell phenotype can be represented by an attractor state of the underlying molecular regulatory network, to which other network states eventually converge. Here, the set of states converging to each attractor is called its basin of attraction. A central question is how to drive a particular cell state toward a desired attractor with minimal interventions on the network system. We develop a general control framework of complex Boolean networks to provide an answer to this question by identifying control targets on which one-time temporary perturbation can induce a state transition to the boundary of a desired attractor basin. Examples are shown to illustrate the proposed control framework which is also applicable to other types of complex Boolean networks.

A one-dimensional individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation

Mechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty, we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an individual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the individual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. We discuss applications of the model, such as determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.

Systems analysis identifies potential target genes to overcome cetuximab resistance in colorectal cancer cells

Cetuximab (CTX), a monoclonal antibody against epidermal growth factor receptor, is being widely used for colorectal cancer (CRC) with wild-type (WT) KRAS. However, its responsiveness is still very limited and WT KRAS is not enough to indicate such responsiveness. Here, by analyzing the gene expression data of CRC patients treated with CTX monotherapy, we have identified DUSP4, ETV5, GNB5, NT5E, and PHLDA1 as potential targets to overcome CTX resistance. We found that knockdown of any of these five genes can increase CTX sensitivity in KRAS WT cells. Interestingly, we further found that GNB5 knockdown can increase CTX sensitivity even for KRAS mutant cells. We unraveled that GNB5 overexpression contributes to CTX resistance by modulating the Akt signaling pathway from experiments and mathematical simulation. Overall, these results indicate that GNB5 might be a promising target for combination therapy with CTX irrespective of KRAS mutation.

The Hidden Control Architecture of Complex Brain Networks

The brain controls various cognitive functions in a robust and efficient way. What is the control architecture of brain networks that enables such robust and optimal control? Is this brain control architecture distinct from that of other complex networks? Here, we developed a framework to delineate a control architecture of a complex network that is compatible with the behavior of the network and applied the framework to structural brain networks and other complex networks. As a result, we revealed that the brain networks have a distributed and overlapping control architecture governed by a small number of control nodes, which may be responsible for the robust and efficient brain functions. Moreover, our artificial network evolution analysis showed that the distributed and overlapping control architecture of the brain network emerges when it evolves toward increasing both robustness and efficiency.

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We develop a framework to delineate the control architecture of brain networks

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The control architecture of brain networks is compared with other complex networks

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Brain networks have a distributed and overlapping control architecture

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Robust and efficient brain functions might be rooted in its control architecture

Global Stabilization of Boolean Networks to Control the Heterogeneity of Cellular Responses

Boolean networks (BNs) have been widely used as a useful model for molecular regulatory networks in systems biology. In the state space of BNs, attractors represent particular cell phenotypes. For targeted therapy of cancer, there is a pressing need to control the heterogeneity of cellular responses to the targeted drug by reducing the number of attractors associated with the ill phenotypes of cancer cells. Here, we present a novel control scheme for global stabilization of BNs to a unique fixed point. Using a sufficient condition of global stabilization with respect to the adjacency matrix, we can determine a set of constant controls so that the controlled BN is steered toward an unspecified fixed point which can then be further transformed to a desired attractor by subsequent control. Our method is efficient in that it has polynomial complexity with respect to the number of state variables, while having exponential complexity with respect to in-degree of BNs. To demonstrate the applicability of the proposed control scheme, we conduct simulation studies using a regulation influence network describing the metastatic process of cells and the Mitogen-activated protein kinase (MAPK) signaling network that is crucial in cancer cell fate determination.

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

Attractor landscape analysis of the cardiac signaling network reveals mechanism-based therapeutic strategies for heart failure

Apoptosis and hypertrophy of cardiomyocytes are the primary causes of heart failure (HF), a global leading cause of death, and are regulated through the complicated intracellular signaling network, limiting the development of effective treatments due to its complexity. To identify effective therapeutic strategies for HF at a system level, we develop a large-scale comprehensive mathematical model of the cardiac signaling network by integrating all available experimental evidence. Attractor landscape analysis of the network model identifies distinct sets of control nodes that effectively suppress apoptosis and hypertrophy of cardiomyocytes under ischemic or pressure overload-induced HF, the two major types of HF. Intriguingly, our system-level analysis suggests that intervention of these control nodes may increase the efficacy of clinical drugs for HF and, of most importance, different combinations of control nodes are suggested as potentially effective candidate drug targets depending on the types of HF. Our study provides a systematic way of developing mechanism-based therapeutic strategies for HF.

Topological estimation of signal flow in complex signaling networks

In a cell, any information about extra- or intra-cellular changes is transferred and processed through a signaling network and dysregulation of signal flow often leads to disease such as cancer. So, understanding of signal flow in the signaling network is critical to identify drug targets. Owing to the development of high-throughput measurement technologies, the structure of a signaling network is becoming more available, but detailed kinetic parameter information about molecular interactions is still very limited. A question then arises as to whether we can estimate the signal flow based only on the structure information of a signaling network. To answer this question, we develop a novel algorithm that can estimate the signal flow using only the topological information and apply it to predict the direction of activity change in various signaling networks. Interestingly, we find that the average accuracy of the estimation algorithm is about 60–80% even though we only use the topological information. We also find that this predictive power gets collapsed if we randomly alter the network topology, showing the importance of network topology. Our study provides a basis for utilizing the topological information of signaling networks in precision medicine or drug target discovery.

Multi-scale modeling of APC and [Formula: see text]-catenin regulation in the human colonic crypt

Stem cell renewal and differentiation in the human colonic crypt are linked to the [Formula: see text]-catenin pathway. The spatial balance of Wnt factors in proliferative cells within the crypt maintain an appropriate level of cellular reproduction needed for normal crypt homeostasis. Mutational events at the gene level are responsible for deregulating the balance of Wnt factors along the crypt, causing an overpopulation of proliferative cells, a loss of structure of the crypt domain, and the initiation of colorectal carcinomas. We formulate a PDE model describing cell movement and reproduction in a static crypt domain. We consider a single cell population whose proliferative capabilities are determined by stemness, a quantity defined by intracellular levels of adenomatous polyposis coli (APC) scaffold protein and [Formula: see text]-catenin. We fit APC regulation parameters to biological data that describe normal protein gradients in the crypt. We also fit cell movement and protein flux parameters to normal crypt characteristics such as renewal time, total cell count, and proportion of proliferating cells. The model is used to investigate abnormal crypt dynamics when subjected to a diminished APC gradient, a scenario synonymous to mutations in the APC gene. We find that a 25% decrease in APC synthesis leads to a fraction of 0.88 proliferative, which is reflective of normal-appearing FAP crypts. A 50% drop in APC activity yields a fully proliferative crypt showing a doubling of the level of stemness, which characterizes the initial stages of colorectal cancer development. A sensitivity analysis of APC regulation parameters shows the perturbation of factors that is required to restore crypt dynamics to normal in the case of APC mutations.

Percolation transition of cooperative mutational effects in colorectal tumorigenesis

Cancer is caused by the accumulation of multiple genetic mutations, but their cooperative effects are poorly understood. Using a genome-wide analysis of all the somatic mutations in colorectal cancer patients in a large-scale molecular interaction network, here we find that a giant cluster of mutation-propagating modules in the network undergoes a percolation transition, a sudden critical transition from scattered small modules to a large connected cluster, during colorectal tumorigenesis. Such a large cluster ultimately results in a giant percolated cluster, which is accompanied by phenotypic changes corresponding to cancer hallmarks. Moreover, we find that the most commonly observed sequence of driver mutations in colorectal cancer has been optimized to maximize the giant percolated cluster. Our network-level percolation study shows that the cooperative effect rather than any single dominance of multiple somatic mutations is crucial in colorectal tumorigenesis.

Cancer is caused by accumulating genetic mutations. Here, the authors investigate the cooperative effect of these mutations in colorectal cancer patients and identify a giant cluster of mutation-propagating modules that undergoes percolation transition during tumorigenesis.

Paracrine and autocrine regulation of gene expression by Wnt-inhibitor Dickkopf in wild-type and mutant hepatocytes

Background

Cells are able to communicate and coordinate their function within tissues via secreted factors. Aberrant secretion by cancer cells can modulate this intercellular communication, in particular in highly organised tissues such as the liver. Hepatocytes, the major cell type of the liver, secrete Dickkopf (Dkk), which inhibits Wnt/ β-catenin signalling in an autocrine and paracrine manner. Consequently, Dkk modulates the expression of Wnt/ β-catenin target genes. We present a mathematical model that describes the autocrine and paracrine regulation of hepatic gene expression by Dkk under wild-type conditions as well as in the presence of mutant cells.

Results

Our spatial model describes the competition of Dkk and Wnt at receptor level, intra-cellular Wnt/ β-catenin signalling, and the regulation of target gene expression for 21 individual hepatocytes. Autocrine and paracrine regulation is mediated through a feedback mechanism via Dkk and Dkk diffusion along the porto-central axis. Along this axis an APC concentration gradient is modelled as experimentally detected in liver. Simulations of mutant cells demonstrate that already a single mutant cell increases overall Dkk concentration. The influence of the mutant cell on gene expression of surrounding wild-type hepatocytes is limited in magnitude and restricted to hepatocytes in close proximity. To explore the underlying molecular mechanisms, we perform a comprehensive analysis of the model parameters such as diffusion coefficient, mutation strength and feedback strength.

Conclusions

Our simulations show that Dkk concentration is elevated in the presence of a mutant cell. However, the impact of these elevated Dkk levels on wild-type hepatocytes is confined in space and magnitude. The combination of inter- and intracellular processes, such as Dkk feedback, diffusion and Wnt/ β-catenin signal transduction, allow wild-type hepatocytes to largely maintain their gene expression.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0470-9) contains supplementary material, which is available to authorized users.

An agent-based model of anoikis in the colon crypt displays novel emergent behaviour consistent with biological observations

Colorectal cancer (CRC) is a major cause of cancer mortality. Colon crypts are multi-cellular flask-shaped invaginations of the colonic epithelium, with stem cells at their base which support the continual turnover of the epithelium with loss of cells by anoikis from the flat mucosa. Mutations in these stem cells can become embedded in the crypts, a process that is strongly implicated in CRC initiation. We describe a computational model which includes novel features, including an accurate representation of the geometry of the crypt mouth. Model simulations yield previously unseen emergent phenomena, such as localization of cell death to a small region of the crypt mouth which corresponds with that observed in vivo. A mechanism emerges in the model for regulation of crypt cellularity in response to changes in either cell proliferation rates or membrane adhesion strengths. We show that cell shape assumptions influence this behaviour, with cylinders recapitulating biology better than spheres. Potential applications of the model include determination of roles of mutations in neoplasia and exploring factors for altered crypt morphodynamics.

A kinetic model to study the regulation of β-catenin, APC, and Axin in the human colonic crypt

The Wnt/[Formula: see text]-catenin pathway plays a crucial role in stem cell renewal and differentiation in the normal human colonic crypt. The balance between [Formula: see text]-catenin and APC along the crypt axis determines its normal functionality. The mechanism that deregulates this balance may give insight into the initiation of colorectal cancer. This is significant because the spatial dysregulation of [Formula: see text]-catenin by the mutated tumor suppressor gene/protein APC in human colonic crypts is responsible for the initiation and growth of colorectal cancer. We consider a regulatory function that promotes APC synthesis within the cell and its effect on the accumulation of the Wnt target protein, [Formula: see text]-catenin. It is evident that an APC gradient exists along the crypt axis; however, the mechanism by which APC expression is regulated within the cell is not well known. We investigate the dynamics of an APC regulatory mechanism with an increased level of Axin at the subcellular level. Model output shows an increase of APC for a diminished Wnt signal, which explains the APC gradient along the crypt. We find that the dynamic interplay between [Formula: see text]-catenin, APC, and Axin produces oscillatory behavior, which is controlled by the Wnt stimulus. In the presence of reduced functional APC, the oscillations are amplified, which suggests that the cell remains in a more proliferative state for longer periods of time. Increased Axin levels (typical of mammalian cells) reduce oscillatory behavior and minimize the levels of [Formula: see text]-catenin within the cell while raising the levels of APC.

Mathematical and Statistical Techniques for Systems Medicine: The Wnt Signaling Pathway as a Case Study

The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non-exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.

The Interplay between Wnt Mediated Expansion and Negative Regulation of Growth Promotes Robust Intestinal Crypt Structure and Homeostasis

The epithelium of the small intestinal crypt, which has a vital role in protecting the underlying tissue from the harsh intestinal environment, is completely renewed every 4–5 days by a small pool of stem cells at the base of each crypt. How is this renewal controlled and homeostasis maintained, particularly given the rapid nature of this process? Here, based on the recent observations from in vitro “mini gut” studies, we use a hybrid stochastic model of the crypt to investigate how exogenous niche signaling (from Wnt and BMP) combines with auto-regulation to promote homeostasis. This model builds on the sub-cellular element method to account for the three-dimensional structure of the crypt, external regulation by Wnt and BMP, internal regulation by Notch signaling, as well as regulation by internally generated diffusible signals. Results show that Paneth cell derived Wnt signals, which have been observed experimentally to sustain crypts in cultured organs, have a dramatically different influence on niche dynamics than does mesenchyme derived Wnt. While this signaling can indeed act as a redundant backup to the exogenous gradient, it introduces a positive feedback that destabilizes the niche and causes its uncontrolled expansion. We find that in this setting, BMP has a critical role in constraining this expansion, consistent with observations that its removal leads to crypt fission. Further results also point to a new hypothesis for the role of Ephrin mediated motility of Paneth cells, specifically that it is required to constrain niche expansion and maintain the crypt’s spatial structure. Combined, these provide an alternative view of crypt homeostasis where the niche is in a constant state of expansion and the spatial structure of the crypt arises as a balance between this expansion and the action of various sources of negative regulation that hold it in check.

The small intestinal epithelium, like our skin, is constantly being renewed. In the intestine however, this epithelium is exposed to the harsh digestive environment, necessitating much more rapid renewal. Remarkably, the entire epithelium is renewed every 4–5 days. This raises the question, how can the size and structure of this tissue be maintained given this pace. Motivated by recent experimental observations, we construct a three-dimensional, hybrid stochastic model to investigate the mechanisms responsible for homeostasis of this tissue. We find that there are redundant signals created by both the epithelium itself and surrounding tissues that act in parallel to maintain epithelial structure. This redundancy comes at a price however: it introduces the possibility of explosive stem cell population growth. Additional results suggest that other signals along with choreographed motion of cells are responsible for repressing this expansion. Taken together, our results provide a novel hypothesis for how robust but fast renewal of the crypt is achieved: as a balance between expansion, which drives fast renewal and repression, which holds that expansion in check to maintain the crypt’s structure.

The Cdc42 Effector Kinase PAK4 Localizes to Cell-Cell Junctions and Contributes to Establishing Cell Polarity

The serine/threonine kinase PAK4 is a Cdc42 effector whose role is not well understood; overexpression of PAK4 has been associated with some cancers, and there are reports that correlate kinase level with increased cell migration in vitro. Here we report that PAK4 is primarily associated with cell-cell junctions in all the cell lines we tested, and fails to accumulate at focal adhesions or at the leading edge of migrating cells. In U2OS osteosarcoma and MCF-7 breast cancer cell lines, PAK4 depletion did not affect collective cell migration, but affected cell polarization. By contrast, Cdc42 depletion (as reported by many studies) caused a strong defect in junctional assembly in multiple cells lines. We also report that the depletion of PAK4 protein or treatment of cells with the PAK4 inhibitor PF-3758309 can lead to defects in centrosome reorientation (polarization) after cell monolayer wounding. These experiments are consistent with PAK4 forming part of a conserved cell-cell junctional polarity Cdc42 complex. We also confirm β-catenin as a target for PAK4 in these cells. Treatment of cells with PF-3758309 caused inhibition of β-catenin Ser-675 phosphorylation, which is located predominantly at cell-cell junctions.

The APC network regulates the removal of mutated cells from colonic crypts

Self-renewal is essential for multicellular organisms but carries the risk of somatic mutations that can lead to cancer, which is particularly critical for rapidly renewing tissues in a highly mutagenic environment such as the intestinal epithelium. Using computational modeling and in vivo experimentation, we have analyzed how adenomatous polyposis coli (APC) mutations and β-catenin aberrations affect the maintenance of mutant cells in colonic crypts. The increasing abundance of APC along the crypt axis forms a gradient of cellular adhesion that causes more proliferative cells to accelerate their movement toward the top of the crypt, where they are shed into the lumen. Thus, the normal crypt can efficiently eliminate β-catenin mutant cells, whereas APC mutations favor retention. Together, the molecular design of the APC/β-catenin signaling network integrates cell proliferation and migration dynamics to translate intracellular signal processing and protein gradients along the crypt into intercellular interactions and whole-crypt physiological or pathological behavior.

Investigation of inflammation and tissue patterning in the gut using a Spatially Explicit General-purpose Model of Enteric Tissue (SEGMEnT)

The mucosa of the intestinal tract represents a finely tuned system where tissue structure strongly influences, and is turn influenced by, its function as both an absorptive surface and a defensive barrier. Mucosal architecture and histology plays a key role in the diagnosis, characterization and pathophysiology of a host of gastrointestinal diseases. Inflammation is a significant factor in the pathogenesis in many gastrointestinal diseases, and is perhaps the most clinically significant control factor governing the maintenance of the mucosal architecture by morphogenic pathways. We propose that appropriate characterization of the role of inflammation as a controller of enteric mucosal tissue patterning requires understanding the underlying cellular and molecular dynamics that determine the epithelial crypt-villus architecture across a range of conditions from health to disease. Towards this end we have developed the Spatially Explicit General-purpose Model of Enteric Tissue (SEGMEnT) to dynamically represent existing knowledge of the behavior of enteric epithelial tissue as influenced by inflammation with the ability to generate a variety of pathophysiological processes within a common platform and from a common knowledge base. In addition to reproducing healthy ileal mucosal dynamics as well as a series of morphogen knock-out/inhibition experiments, SEGMEnT provides insight into a range of clinically relevant cellular-molecular mechanisms, such as a putative role for Phosphotase and tensin homolog/phosphoinositide 3-kinase (PTEN/PI3K) as a key point of crosstalk between inflammation and morphogenesis, the protective role of enterocyte sloughing in enteric ischemia-reperfusion and chronic low level inflammation as a driver for colonic metaplasia. These results suggest that SEGMEnT can serve as an integrating platform for the study of inflammation in gastrointestinal disease.

Risk factor-dependent dynamics of atopic dermatitis: modelling multi-scale regulation of epithelium homeostasis

Epithelial tissue provides the body with its first layer of protection against harmful environmental stimuli by enacting the regulatory interplay between a physical barrier preventing the influx of external stimuli and an inflammatory response to the infiltrating stimuli. Importantly, this interdependent regulation occurs on different time scales: the tissue-level barrier permeability is regulated over the course of hours, whereas the cellular-level enzymatic reactions leading to inflammation take place within minutes. This multi-scale regulation is key to the epithelium's function and its dysfunction leads to various diseases. This paper presents a mathematical model of regulatory mechanisms in the epidermal epithelium that includes processes on two different time scales at the cellular and tissue levels. We use this model to investigate the essential regulatory interactions between epidermal barrier integrity and skin inflammation and how their dysfunction leads to atopic dermatitis (AD). Our model exhibits a structure of dual (positive and negative) control at both cellular and tissue levels. We also determined how the variation induced by well-known risk factors for AD can break the balance of the dual control. Our model analysis based on time-scale separation suggests that each risk factor leads to qualitatively different dynamic behaviours of different severity for AD, and that the coincidence of multiple risk factors dramatically increases the fragility of the epithelium's function. The proposed mathematical framework should also be applicable to other inflammatory diseases that have similar time-scale separation and control architectures.

Toward a quantitative understanding of the Wnt/β-catenin pathway through simulation and experiment

Wnt signaling regulates cell survival, proliferation, and differentiation throughout development and is aberrantly regulated in cancer. The pathway is activated when Wnt ligands bind to specific receptors on the cell surface, resulting in the stabilization and nuclear accumulation of the transcriptional co-activator β-catenin. Mathematical and computational models have been used to study the spatial and temporal regulation of the Wnt/β-catenin pathway and to investigate the functional impact of mutations in key components. Such models range in complexity, from time-dependent, ordinary differential equations that describe the biochemical interactions between key pathway components within a single cell, to complex, multiscale models that incorporate the role of the Wnt/β-catenin pathway target genes in tissue homeostasis and carcinogenesis. This review aims to summarize recent progress in mathematical modeling of the Wnt pathway and to highlight new biological results that could form the basis for future theoretical investigations designed to increase the utility of theoretical models of Wnt signaling in the biomedical arena.

Modeling Wnt/β-Catenin Target Gene Expression in APC and Wnt Gradients Under Wild Type and Mutant Conditions

The Wnt/β-catenin pathway is involved in the regulation of a multitude of physiological processes by controlling the differential expression of target genes. In certain tissues such as the adult liver, the Wnt/β-catenin pathway can attain different levels of activity due to gradients of Wnt ligands and/or intracellular pathway components like APC. How graded pathway activity is converted into regionally distinct patterns of Wnt/β-catenin target gene expression is largely unknown. Here, we apply a mathematical modeling approach to investigate the impact of different regulatory mechanisms on target gene expression within Wnt or APC concentration gradients. We develop a minimal model of Wnt/β-catenin signal transduction and combine it with various mechanisms of target gene regulation. In particular, the effects of activation, inhibition, and an incoherent feedforward loop (iFFL) are compared. To specify activation kinetics, we analyze experimental data that quantify the response of β-catenin/TCF reporter constructs to different Wnt concentrations, and demonstrate that the induction of these constructs occurs in a cooperative manner with Hill coefficients between 2 and 5. In summary, our study shows that the combination of specific gene regulatory mechanisms with a time-independent gradient of Wnt or APC is sufficient to generate distinct target gene expression patterns as have been experimentally observed in liver. We find that cooperative gene activation in combination with a TCF feedback can establish sharp borders of target gene expression in Wnt or APC gradients. In contrast, the iFFL renders gene expression independent of gradients of the upstream signaling components. Our subsequent analysis of carcinogenic pathway mutations reveals that their impact on gene expression is determined by the gene regulatory mechanism and the APC concentration of the cell in which the mutation occurs.

Mathematical model of colitis-associated colon cancer

As a result of chronic inflammation of their colon, patients with ulcerative colitis or Crohn's disease are at risk of developing colon cancer. In this paper, we consider the progression of colitis-associated colon cancer. Unlike normal colon mucosa, the inflammed colon mucosa undergoes genetic mutations, affecting, in particular, tumor suppressors TP53 and adenomatous polyposis coli (APC) gene. We develop a mathematical model that involves these genes, under chronic inflammation, as well as NF-κB, β-catenin, MUC1 and MUC2. The model demonstrates that increased level of cells with TP53 mutations results in abnormal growth and proliferation of the epithelium; further increase in the epithelium proliferation results from additional APC mutations. The model may serve as a conceptual framework for further data-based study of the early stage of colon cancer.

Cytokine combination therapy prediction for bone remodeling in tissue engineering based on the intracellular signaling pathway

The long-term performance of tissue-engineered bone grafts is determined by a dynamic balance between bone regeneration and resorption. We proposed using embedded cytokine slow-releasing hydrogels to tune this balance toward a desirable final bone density. In this study we established a systems biology model, and quantitatively explored the combinatorial effects of delivered cytokines from hydrogels on final bone density. We hypothesized that: 1) bone regeneration was driven by transcription factors Runx2 and Osterix, which responded to released cytokines, such as Wnt, BMP2, and TGFβ, drove the development of osteoblast lineage, and contributed to bone mass generation; and 2) the osteoclast lineage, on the other hand, governed the bone resorption, and communications between these two lineages determined the dynamics of bone remodeling. In our model, Intracellular signaling pathways were represented by ordinary differential equations, while the intercellular communications and cellular population dynamics were modeled by stochastic differential equations. Effects of synergistic cytokine combinations were evaluated by Loewe index and Bliss index. Simulation results revealed that the Wnt/BMP2 combinations released from hydrogels showed best control of bone regeneration and synergistic effects, and suggested optimal dose ratios of given cytokine combinations released from hydrogels to most efficiently control the long-term bone remodeling. We revealed the characteristics of cytokine combinations of Wnt/BMP2 which could be used to guide the design of in vivo bone scaffolds and the clinical treatment of some diseases such as osteoporosis.

A theoretical investigation of the effect of proliferation and adhesion on monoclonal conversion in the colonic crypt

The surface epithelium lining the intestinal tract renews itself rapidly by a coordinated programme of cell proliferation, migration and differentiation events that is initiated in the crypts of Lieberkühn. It is generally believed that colorectal cancer arises due to mutations that disrupt the normal cellular dynamics of the crypts. Using a spatially structured cell-based model of a colonic crypt, we investigate the likelihood that the progeny of a mutated cell will dominate, or be sloughed out of, a crypt. Our approach is to perform multiple simulations, varying the spatial location of the initial mutation, and the proliferative and adhesive properties of the mutant cells, to obtain statistical distributions for the probability of their domination. Our simulations lead us to make a number of predictions. The process of monoclonal conversion always occurs, and does not require that the cell which initially gave rise to the population remains in the crypt. Mutations occurring more than one to two cells from the base of the crypt are unlikely to become the dominant clone. The probability of a mutant clone persisting in the crypt is sensitive to dysregulation of adhesion. By comparing simulation results with those from a simple one-dimensional stochastic model of population dynamics at the base of the crypt, we infer that this sensitivity is due to direct competition between wild-type and mutant cells at the base of the crypt. We also predict that increases in the extent of the spatial domain in which the mutant cells proliferate can give rise to counter-intuitive, non-linear changes to the probability of their fixation, due to effects that cannot be captured in simpler models.

A reaction-diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts

Background

Colon crypts, a single sheet of epithelia cells, consist of a periodic pattern of stem cells, transit-amplifying cells, and terminally differentiated cells that constantly renew and turnover. Experimental evidence suggests that Wnt signaling promotes and regulates stem cell division, differentiation, and possible cell migrations while intestinal BMP signaling inhibits stem cell self-renewal and repression in crypt formation. As more molecular details on Wnt and BMP in crypts are being discovered, little is still known about how complex interactions among Wnt, BMP, and different types of cells, and surrounding environments may lead to de novo formation of multiple crypts or how such interactions affect regeneration and stability of crypts.

Results

We present a mathematical model that contains Wnt and BMP, a cell lineage, and their feedback regulations to study formation, regeneration, and stability of multiple crypts. The computational explorations and linear stability analysis of the model suggest a reaction–diffusion mechanism, which exhibits a short-range activation of Wnt plus a long-range inhibition with modulation of BMP signals in a growing tissue of cell lineage, can account for spontaneous formation of multiple crypts with the spatial and temporal pattern observed in experiments. Through this mechanism, the model can recapitulate some distinctive and important experimental findings such as crypt regeneration and crypt multiplication. BMP is important in maintaining stability of crypts and loss of BMP usually leads to crypt multiplication with a fingering pattern.

Conclusions

The study provides a mechanism for de novo formation of multiple intestinal crypts and demonstrates a synergetic role of Wnt and BMP in regeneration and stability of intestinal crypts. The proposed model presents a robust framework for studying spatial and temporal dynamics of cell lineages in growing tissues driven by multiple signaling molecules.

Classifying general nonlinear force laws in cell-based models via the continuum limit

Although discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.