Order-of-Mutation Effects on Cancer Progression: Models for Myeloproliferative Neoplasm.

In some patients with myeloproliferative neoplasms (MPN), two genetic mutations are often found: JAK2 V617F and one in the TET2 gene. Whether one mutation is present influences how the other subsequent mutation will affect the regulation of gene expression. In other words, when a patient carries both mutations, the order of when they first arose has been shown to influence disease progression and prognosis. We propose a nonlinear ordinary differential equation, the Moran process, and Markov chain models to explain the non-additive and non-commutative mutation effects on recent clinical observations of gene expression patterns, proportions of cells with different mutations, and ages at diagnosis of MPN. Combined, these observations are used to shape our modeling framework. Our key proposal is that bistability in gene expression provides a natural explanation for many observed order-of-mutation effects. We also propose potential experimental measurements that can be used to confirm or refute predictions of our models.

Realizing the promise of Project Optimus: Challenges and emerging opportunities for dose optimization in oncology drug development.

Project Optimus is a US Food and Drug Administration Oncology Center of Excellence initiative aimed at reforming the dose selection and optimization paradigm in oncology drug development. This project seeks to bring together pharmaceutical companies, international regulatory agencies, academic institutions, patient advocates, and other stakeholders. Although there is much promise in this initiative, there are several challenges that need to be addressed, including multidimensionality of the dose optimization problem in oncology, the heterogeneity of cancer and patients, importance of evaluating long-term tolerability beyond dose-limiting toxicities, and the lack of reliable biomarkers for long-term efficacy. Through the lens of Totality of Evidence and with the mindset of model-informed drug development, we offer insights into dose optimization by building a quantitative knowledge base integrating diverse sources of data and leveraging quantitative modeling tools to build evidence for drug dosage considering exposure, disease biology, efficacy, toxicity, and patient factors. We believe that rational dose optimization can be achieved in oncology drug development, improving patient outcomes by maximizing therapeutic benefit while minimizing toxicity.

Order-of-mutation effects on cancer progression: models for myeloproliferative neoplasm.

In some patients with myeloproliferative neoplasms (MPN), two genetic mutations are often found, JAK2 V617F and one in the TET2 gene. Whether or not one mutation is present will influence how the other subsequent mutation affects the regulation of gene expression. When both mutations are present, the order of their occurrence has been shown to influence disease progression and prognosis. We propose a nonlinear ordinary differential equation (ODE), Moran process, and Markov chain models to explain the non-additive and non-commutative mutation effects on recent clinical observations of gene expression patterns, proportions of cells with different mutations, and ages at diagnosis of MPN. These observations consistently shape our modeling framework. Our key proposal is that bistability in gene expression provides a natural explanation for many observed order-of-mutation effects. We also propose potential experimental measurements that can be used to confirm or refute predictions of our models.

Order-of-mutation effects on cancer progression: models for myeloproliferative neoplasm.

In some patients with myeloproliferative neoplasms (MPN), two genetic mutations are often found, JAK2 V617F and one in the TET2 gene. Whether or not one mutation is present will influence how the other subsequent mutation affects the regulation of gene expression. When both mutations are present, the order of their occurrence has been shown to influence disease progression and prognosis. We propose a nonlinear ordinary differential equation (ODE), Moran process, and Markov chain models to explain the non-additive and non-commutative mutation effects on recent clinical observations of gene expression patterns, proportions of cells with different mutations, and ages at diagnosis of MPN. These observations consistently shape our modeling framework. Our key proposal is that bistability in gene expression provides a natural explanation for many observed order-of-mutation effects. We also propose potential experimental measurements that can be used to confirm or refute predictions of our models.

Sources of inter-individual variability leading to significant changes in anti-PD-1 and anti-PD-L1 efficacy identified in mouse tumor models using a QSP framework.

While anti-PD-1 and anti-PD-L1 [anti-PD-(L)1] monotherapies are effective treatments for many types of cancer, high variability in patient responses is observed in clinical trials. Understanding the sources of response variability can help prospectively identify potential responsive patient populations. Preclinical data may offer insights to this point and, in combination with modeling, may be predictive of sources of variability and their impact on efficacy. Herein, a quantitative systems pharmacology (QSP) model of anti-PD-(L)1 was developed to account for the known pharmacokinetic properties of anti-PD-(L)1 antibodies, their impact on CD8+ T cell activation and influx into the tumor microenvironment, and subsequent anti-tumor effects in CT26 tumor syngeneic mouse model. The QSP model was sufficient to describe the variability inherent in the anti-tumor responses post anti-PD-(L)1 treatments. Local sensitivity analysis identified tumor cell proliferation rate, PD-1 expression on CD8+ T cells, PD-L1 expression on tumor cells, and the binding affinity of PD-1:PD-L1 as strong influencers of tumor growth. It also suggested that treatment-mediated tumor growth inhibition is sensitive to T cell properties including the CD8+ T cell proliferation half-life, CD8+ T cell half-life, cytotoxic T-lymphocyte (CTL)-mediated tumor cell killing rate, and maximum rate of CD8+ T cell influx into the tumor microenvironment. Each of these parameters alone could not predict anti-PD-(L)1 treatment response but they could shift an individual mouse's treatment response when perturbed. The presented preclinical QSP modeling framework provides a path to incorporate potential sources of response variability in human translation modeling of anti-PD-(L)1.

Multi-method global sensitivity analysis of mathematical models.

Increasingly-sophisticated parameter-sensitivity analysis techniques continue to be developed, and each technique comes with its own set of advantages and disadvantages. Selecting which parameter-sensitivity method to use for a particular model, however, is not a straightforward task. In this work, we present a multi-method framework that incorporates three global sensitivity analysis methods: two variance-based methods and one derivative-based method. The two variance-based methods are Sobol's method and MeFAST. The derivative-based method is known as DGSM (Derivative-based Global Sensitivity Measures). MeFAST (Multi test eFAST) is a new parameter sensitivity analysis implementation we built upon the eFAST (Extended Fourier Amplitude Sensitivity Test) algorithm. The improvements incorporated into MeFAST address some important aspects of prior eFAST implementations. We present an intuitive description of each implemented algorithm along with MATLAB codes and a guide to tuning algorithm hyper-parameters for better efficiency. We demonstrate the full methodology and workflow using two example mathematical models of different complexity: the first is a model of HIV disease progression and the second is a model of tumor growth. The computational framework we provide generates graphics for visualizing and comparing the results of all three sensitivity analysis algorithms (DGSM, Sobol, and MeFAST). This algorithm output comparison tool allows one to make a more informed decision when assessing which parameters most importantly influence model outcomes.

Roadmap on plasticity and epigenetics in cancer.

The role of plasticity and epigenetics in shaping cancer evolution and response to therapy has taken center stage with recent technological advances including single cell sequencing. This roadmap article is focused on state-of-the-art mathematical and experimental approaches to interrogate plasticity in cancer, and addresses the following themes and questions: is there a formal overarching framework that encompasses both non-genetic plasticity and mutation-driven somatic evolution? How do we measure and model the role of the microenvironment in influencing/controlling non-genetic plasticity? How can we experimentally study non-genetic plasticity? Which mathematical techniques are required or best suited? What are the clinical and practical applications and implications of these concepts?

Machine learning and statistical prediction of patient quality-of-life after prostate radiation therapy.

Thanks to advancements in diagnosis and treatment, prostate cancer patients have high long-term survival rates. Currently, an important goal is to preserve quality of life during and after treatment. The relationship between the radiation a patient receives and the subsequent side effects he experiences is complex and difficult to model or predict. Here, we use machine learning algorithms and statistical models to explore the connection between radiation treatment and post-treatment gastro-urinary function. Since only a limited number of patient datasets are currently available, we used image flipping and curvature-based interpolation methods to generate more data to leverage transfer learning. Using interpolated and augmented data, we trained a convolutional autoencoder network to obtain near-optimal starting points for the weights. A convolutional neural network then analyzed the relationship between patient-reported quality-of-life and radiation doses to the bladder and rectum. We also used analysis of variance and logistic regression to explore organ sensitivity to radiation and to develop dosage thresholds for each organ region. Our findings show no statistically significant association between the bladder and quality-of-life scores. However, we found a statistically significant association between the radiation applied to posterior and anterior rectal regions and changes in quality of life. Finally, we estimated radiation therapy dose thresholds for each organ. Our analysis connects machine learning methods with organ sensitivity, thus providing a framework for informing cancer patient care using patient reported quality-of-life metrics.

A Mathematical Model for DC Vaccine Treatment of Type I Diabetes.

Type I diabetes (T1D) is an autoimmune disease that can be managed, but for which there is currently no cure. Recent discoveries, particularly in mouse models, indicate that targeted modulation of the immune response has the potential to move an individual from a diabetic to a long-term, if not permanent, healthy state. In this paper we develop a single compartment mathematical model that captures the dynamics of dendritic cells (DC and tDC), T cells (effector and regulatory), and macrophages in the development of type I diabetes. The model supports the hypothesis that differences in macrophage clearance rates play a significant role in determining whether or not an individual is likely to become diabetic subsequent to a significant immune challenge. With this model we are able to explore the effects of strengthening the anti-inflammatory component of the immune system in a vulnerable individual. Simulations indicate that there are windows of opportunity in which treatment intervention is more likely to be beneficial in protecting an individual from entering a diabetic state. This model framework can be used as a foundation for modeling future T1D treatments as they are developed.

Methods for determining key components in a mathematical model for tumor-immune dynamics in multiple myeloma.

In this work, we analyze a mathematical model we introduced previously for the dynamics of multiple myeloma and the immune system. We focus on four main aspects: (1) obtaining and justifying ranges and values for all parameters in the model; (2) determining a subset of parameters to which the model is most sensitive; (3) determining which parameters in this subset can be uniquely estimated given certain types of data; and (4) exploring the model numerically. Using global sensitivity analysis techniques, we found that the model is most sensitive to certain growth, loss, and efficacy parameters. This analysis provides the foundation for a future application of the model: prediction of optimal combination regimens in patients with multiple myeloma.

Mathematical modeling of spatiotemporal protein localization patterns in C. crescentus bacteria: A mechanism for asymmetric FtsZ ring positioning.

We examine the localization patterns of ParA, ParB, PopZ, and MipZ, which are key division proteins in C. crescentus bacteria. While Par and PopZ proteins have been implicated in the physical segregation of the replicated chromosome, MipZ dimers control the placement of the cell division plane by preventing FtsZ proteins from assembling into a Z-ring. MipZ proteins generate bipolar gradients that are sensitive to Par protein localization, however, it is not understood how the MipZ gradient is shaped so as to allow for the correct Z-ring placement during asymmetric cell division in C. crescentus. In this paper, we develop and analyze a mathematical model that incorporates the known interactions between Par, PopZ, and MipZ proteins and use it to test mechanisms for MipZ gradient formation. Using our model, we show that gradient-dependent ParB advection velocities in conjunction with a ParA polar recycling mechanism are sufficient to maintain a robust new pole-directed ParA dimer gradient during segregation. A "saturation of binding site" hypothesis limiting access of ParA and MipZ to the ParB complex is then necessary and sufficient to generate time-averaged bipolar MipZ protein gradients with minima that are skewed toward ParA gradient peaks at the new pole, in agreement with data. By analyzing reduced versions of the model, we show the existence of oscillatory ParA localization regimes provided that cytoplasmic PopZ oligomers interact with ParA and ParA is over-expressed. We use our model to study mechanisms by which these protein patterns may simultaneously direct proper chromosome segregation and division site placement in C. crescentus.

Stronger net posterior cortical forces and asymmetric microtubule arrays produce simultaneous centration and rotation of the pronuclear complex in the early Caenorhabditis elegans embryo.

Positioning of microtubule-organizing centers (MTOCs) incorporates biochemical and mechanical cues for proper alignment of the mitotic spindle and cell division site. Current experimental and theoretical studies in the early Caenorhabditis elegans embryo assume remarkable changes in the origin and polarity of forces acting on the MTOCs. These changes must occur over a few minutes, between initial centration and rotation of the pronuclear complex and entry into mitosis, and the models do not replicate in vivo timing of centration and rotation. Here we propose a model that incorporates asymmetry in the microtubule arrays generated by each MTOC, which we demonstrate with in vivo measurements, and a similar asymmetric force profile to that required for posterior-directed spindle displacement during mitosis. We find that these asymmetries are capable of and important for recapitulating the simultaneous centration and rotation of the pronuclear complex observed in vivo. The combination of theoretical and experimental evidence provided here offers a unified framework for the spatial organization and forces needed for pronuclear centration, rotation, and spindle displacement in the early C. elegans embryo.

Using Active Learning to Teach Concepts and Methods in Quantitative Biology.

This article provides a summary of the ideas discussed at the 2015 Annual Meeting of the Society for Integrative and Comparative Biology society-wide symposium on Leading Students and Faculty to Quantitative Biology through Active Learning. It also includes a brief review of the recent advancements in incorporating active learning approaches into quantitative biology classrooms. We begin with an overview of recent literature that shows that active learning can improve students' outcomes in Science, Technology, Engineering and Math Education disciplines. We then discuss how this approach can be particularly useful when teaching topics in quantitative biology. Next, we describe some of the recent initiatives to develop hands-on activities in quantitative biology at both the graduate and the undergraduate levels. Throughout the article we provide resources for educators who wish to integrate active learning and technology into their classrooms.

A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases.

The organization of the axonal cytoskeleton is a key determinant of the normal function of an axon, which is a long thin projection of a neuron. Under normal conditions two axonal cytoskeletal polymers, microtubules and neurofilaments, align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurotoxic and neurodegenerative disorders, microtubules and neurofilaments segregate apart from each other, with microtubules and membranous organelles clustered centrally and neurofilaments displaced to the periphery. This striking segregation precedes the abnormal and excessive neurofilament accumulation in these diseases, which in turn leads to focal axonal swellings. While neurofilament accumulation suggests an impairment of neurofilament transport along axons, the underlying mechanism of their segregation from microtubules remains poorly understood for over 30 years. To address this question, we developed a stochastic multiscale model for the cross-sectional distribution of microtubules and neurofilaments in axons. The model describes microtubules, neurofilaments and organelles as interacting particles in a 2D cross-section, and is built upon molecular processes that occur on a time scale of seconds or shorter. It incorporates the longitudinal transport of neurofilaments and organelles through this domain by allowing stochastic arrival and departure of these cargoes, and integrates the dynamic interactions of these cargoes with microtubules mediated by molecular motors. Simulations of the model demonstrate that organelles can pull nearby microtubules together, and in the absence of neurofilament transport, this mechanism gradually segregates microtubules from neurofilaments on a time scale of hours, similar to that observed in toxic neuropathies. This suggests that the microtubule-neurofilament segregation can be a consequence of the selective impairment of neurofilament transport. The model generates the experimentally testable prediction that the rate and extent of segregation will be dependent on the sizes of the moving organelles as well as the density of their traffic.

Mathematical modeling of bacterial track-altering motors: Track cleaving through burnt-bridge ratchets.

The generation of directed movement of cellular components frequently requires the rectification of Brownian motion. Molecular motor enzymes that use ATP to walk on filamentous tracks are typically involved in cell transport, however, a track-altering motor can arise when an enzyme interacts with and alters its track. In Caulobacter crescentus and other bacteria, an active DNA partitioning (Par) apparatus is employed to segregate replicated chromosome regions to specific locations in dividing cells. The Par apparatus is composed of two proteins: ParA, an ATPase that can form polymeric structures on the nucleoid, and ParB, a protein that can bind and destabilize ParA structures. It has been proposed that the ParB-mediated alteration of ParA structures could be responsible for generating the directed movement of DNA during bacterial division. How precisely these actions are coordinated and translated into directed movement is not clear. In this paper we consider the C. crescentus segregation apparatus as an example of a track altering motor that operates using a so-called burnt-bridge mechanism. We develop and analyze mathematical models that examine how diffusion and ATP-hydrolysis-mediated monomer removal (or cleaving) can be combined to generate directed movement. Using a mean first passage approach, we analytically calculate the effective ParA track-cleaving velocities, effective diffusion coefficient, and other higher moments for the movement a ParB protein cluster that breaks monomers away at random locations on a single ParA track. Our model results indicate that cleaving velocities and effective diffusion constants are sensitive to ParB-induced ATP hydrolysis rates. Our analytical results are in excellent agreement with stochastic simulation results.

Stochastic modelling of chromosomal segregation: errors can introduce correction.

Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious effects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two poles, there are frequent errors in kinetochore-microtubule attachment. Erroneous attachments are classified into two categories, syntelic and merotelic. In this paper, we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that merotely can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.

Distribution of lifetimes of kinetochore-microtubule attachments: interplay of energy landscape, molecular motors and microtubule (de-)polymerization.

Before a cell divides into two daughter cells, chromosomes are replicated resulting in two sister chromosomes embracing each other. Each sister chromosome is bound to a separate proteinous structure, called kinetochore (kt), that captures the tip of a filamentous protein, called microtubule (MT). Two oppositely oriented MTs pull the two kts attached to two sister chromosomes, thereby pulling the two sisters away from each other. Here we theoretically study an even simpler system, namely an isolated kt coupled to a single MT; this system mimics an in vitro experiment where a single kt-MT attachment is reconstituted using purified extracts from budding yeast. Our models not only account for the experimentally observed 'catch-bond-like' behavior of the kt-MT coupling, but also make new predictions on the probability distribution of the lifetimes of the attachments. In principle, our new predictions can be tested by analyzing the data collected in the in vitro experiments, provided that the experiment is repeated a sufficiently large number of times. Our theory provides a deep insight into the effects of (a) size, (b) energetics, and (c) stochastic kinetics of the kt-MT coupling on the distribution of the lifetimes of these attachments.

A mathematical model of force generation by flexible kinetochore-microtubule attachments.

Important mechanical events during mitosis are facilitated by the generation of force by chromosomal kinetochore sites that attach to dynamic microtubule tips. Several theoretical models have been proposed for how these sites generate force, and molecular diffusion of kinetochore components has been proposed as a key component that facilitates kinetochore function. However, these models do not explicitly take into account the recently observed flexibility of kinetochore components and variations in microtubule shape under load. In this paper, we develop a mathematical model for kinetochore-microtubule connections that directly incorporates these two important components, namely, flexible kinetochore binder elements, and the effects of tension load on the shape of shortening microtubule tips. We compare our results with existing biased diffusion models and explore the role of protein flexibility inforce generation at the kinetochore-microtubule junctions. Our model results suggest that kinetochore component flexibility and microtubule shape variation under load significantly diminish the need for high diffusivity (or weak specific binding) of kinetochore components; optimal kinetochore binder stiffness regimes are predicted by our model. Based on our model results, we suggest that the underlying principles of biased diffusion paradigm need to be reinterpreted.

A mathematical model of ParA filament-mediated chromosome movement in Caulobacter crescentus.

Caulobacter crescentus uses the dynamic interactions between ParA and ParB proteins to segregate copies of its circular chromosome. In this paper, we develop two mathematical models of the movement of the circular chromosome of this bacterium during division. In the first model, posed as a set of stochastic differential equations (SDE), we propose that a simple biased diffusion mechanism for ParB/ParA interactions can reproduce the observed patterns of ParB and ParA localization in the cell. The second model, posed as a set of nonlinear partial differential equations, is a continuous treatment of the problem where we use results from the SDE model to describe ParB/ParA interactions and we also track ParA monomer dynamics in the cytoplasm. For both models, we show that if ParB complexes bind weakly and nonspecifically to ParA filaments, then they can closely track and move with the edge of a shrinking ParA filament bundle. Unidirectional chromosome movement occurs when ParB complexes have a passive role in depolymerizing ParA filaments. Finally, we show that tight control of ParA filament dynamics is essential for proper segregation.

A mechanomolecular model for the movement of chromosomes during mitosis driven by a minimal kinetochore bicyclic cascade.

During mitosis chromosomes use a complex network of dynamic microtubules to find the cell equator in preparation for division signals. The roles of cellular chemical signals in mechanisms driving mitotic chromosomal movements are not well understood. In this paper we propose a mathematical model of this process which incorporates a molecular scale model of kinetochore-microtubule interactions into a negative feedback loop between spindle forces and local kinetochore biochemical reactions. This system allows kinetochore biochemical reactions to control and coordinate chromosome movement thus providing a direct connection between mechanical signals and mitosis chemical species. Our feedback control model can recreate chromosome movement from prometaphase to anaphase in good agreement with experimental data.