Kill rates by immune cells: Ratio-dependent, or mass action?

We describe a cell-based fixed-lattice model to simulate immune cell and tumor cell interaction involving MHC recognition, and FasL vs perforin lysis. We are motivated by open questions about the mechanisms behind observed kill rates of tumor cells by different types of effector cells. These mechanisms play a big role in the effectiveness of many cancer immunotherapies. The model is a stochastic cellular automaton on a hexagonal grid.

Predicting the impact of placing an overdose prevention site in Philadelphia: a mathematical modeling approach.

BACKGROUND: Fatal overdoses from opioid use and substance disorders are increasing at an alarming rate. One proposed harm reduction strategy for reducing overdose fatalities is to place overdose prevention sites-commonly known as safe injection facilities-in proximity of locations with the highest rates of overdose. As urban centers in the USA are tackling legal hurdles and community skepticism around the introduction and location of these sites, it becomes increasingly important to assess the magnitude of the effect that these services might have on public health. METHODS: We developed a mathematical model to describe the movement of people who used opioids to an overdose prevention site in order to understand the impact that the facility would have on overdoses, fatalities, and user education and treatment/recovery. The discrete-time, stochastic model is able to describe a range of user behaviors, including the effects from how far they need to travel to the site. We calibrated the model to overdose data from Philadelphia and ran simulations to describe the effect of placing a site in the Kensington neighborhood. RESULTS: In Philadelphia, which has a non-uniform racial population distribution, choice of site placement can determine which demographic groups are most helped. In our simulations, placement of the site in the Kensington neighborhood resulted in White opioid users being more likely to benefit from the site's services. Overdoses that occur onsite can be reversed. Our results predict that for every 30 stations in the overdose prevention site, 6 per year of these would have resulted in fatalities if they had occurred outside of the overdose prevention site. Additionally, we estimate that fatalities will decrease further when referrals from the OPS to treatment are considered. CONCLUSIONS: Mathematical modeling was used to predict the impact of placing an overdose prevention site in the Kensington neighborhood of Philadelphia. To fully understand the impact of site placement, both direct and indirect effects must be included in the analysis. Introducing more than one site and distributing sites equally across neighborhoods with different racial and demographic characteristics would have the broadest public health impact. Cities and locales can use mathematical modeling to help quantify the predicted impact of placing an overdose prevention site in a particular location.

Variability of Betweenness Centrality and Its Effect on Identifying Essential Genes.

This paper begins to build a theoretical framework that would enable the pharmaceutical industry to use network complexity measures as a way to identify drug targets. The variability of a betweenness measure for a network node is examined through different methods of network perturbation. Our results indicate a robustness of betweenness centrality in the identification of target genes.

Dynamic pricing of network goods with boundedly rational consumers.

We present a model of dynamic monopoly pricing for a good that displays network effects. In contrast with the standard notion of a rational-expectations equilibrium, we model consumers as boundedly rational and unable either to pay immediate attention to each price change or to make accurate forecasts of the adoption of the network good. Our analysis shows that the seller's optimal price trajectory has the following structure: The price is low when the user base is below a target level, is high when the user base is above the target, and is set to keep the user base stationary once the target level has been attained. We show that this pricing policy is robust to a number of extensions, which include the product's user base evolving over time and consumers basing their choices on a mixture of a myopic and a "stubborn" expectation of adoption. Our results differ significantly from those that would be predicted by a model based on rational-expectations equilibrium and are more consistent with the pricing of network goods observed in practice.

Signaled drug delivery and transport across the blood-brain barrier.

We use a mathematical model to describe the delivery of a drug to a specific region of the brain. The drug is carried by liposomes that can release their cargo by application of focused ultrasound (US). Thereupon, the drug is absorbed through the endothelial cells that line the brain capillaries and form the physiologically important blood-brain barrier (BBB). We present a compartmental model of a capillary that is able to capture the complex binding and transport processes the drug undergoes in the blood plasma and at the BBB. We apply this model to the delivery of levodopa (L-dopa, used to treat Parkinson's disease) and doxorubicin (an anticancer agent). The goal is to optimize the delivery of drug while at the same time minimizing possible side effects of the US.

Kinetics of bile salt binding to liposomes revealed by carboxyfluorescein release and mathematical modeling.

We propose a mathematical model for the release of carboxyfluorescein from liposomes whose membrane permeability is modified by the binding of different bile salts to the leaflets of the lipid bilayer. We find that the permeability of the liposomal bilayer depends on the difference in the concentrations of bile salt in the inner and outer leaflets and is only minimally influenced by the total concentration of bile salt in the bilayer. Deoxycholate and cholate are found to behave similarly in enhancing permeability for limited times, whereas the novel bile salt, 12-monoketocholate, flips from the outer to inner leaflet slowly, thereby enhancing membrane permeability for a prolonged time.

Introduction of Plasmid to the Murine Gut via Consumption of an Escherichia coli Carrier and Examining the Impact of Bacterial Dosing and Antibiotics on Persistence.

Purpose: We examine the impacts of dosing strategies of plasmids on bacterial communities in the murine gut by measuring the quantity of plasmids in mouse feces. Methods: We fed mice carrier bacteria, E. coli, that contain plasmids with both a reporter gene and an antibiotic resistant gene. We varied the quantity of the plasmid-carrying bacteria and the length of time the mice consumed the bacteria. We also pretreated the gut with broad-spectrum antibiotics and used continuous antibiotic treatment to investigate selection pressure. We collected bacteria from fecal pellets to quantify the number of plasmid-carrying bacteria via plate assay. Results: Dosing regimens with plasmid-carrying bacteria resulted in a significantly increased duration of persistence of the plasmid within the gut when supplemented continuously with kanamycin during as well as after completion of bacterial dosing. The carrier bacteria concentration influenced the short-term abundance of carrier bacteria. Conclusion: We evaluated the persistence of plasmid-carrying bacteria in the murine gut over time using varying dosage strategies. In future work, we will study how bacterial diversity in the gut impacts the degree of plasmid transfer and the prevalence of plasmid-carrying bacteria over time. Lay Summary: Observing how plasmids persist within the gut can help us understand how newly introduced genes, including antibiotic resistance, are transmitted within the gut microbiome. In our experiments, mice were given bacteria containing a genetically engineered plasmid and were examined for the persistence of the plasmid in the gut. We found long-term persistence of the plasmid in the gut when administering antibiotics during and following dosing of the mice with bacteria carrying the plasmid. The use of higher concentrations of carrier bacteria influenced the short-term abundance of the plasmid-carrying bacteria in the gut. Description of Future Works: Building on evidence from these initial studies that persistence of plasmids within the gut can be regulated by the dosage strategy, we will explore future studies and models of gene uptake in the context of spatial and taxonomic control and further determine if dosing strategies alter the compositional diversity of the gut microbiome.

Modeling the interaction of polymeric nanoparticles functionalized with cell penetrating peptides at the nano-bio interface.

The interaction of nanoparticles with Caco-2 monolayers in cell culture underpins our predictions of the uptake of nanoformulations in vivo for drug delivery. Cell-penetrating peptides (CPP), such as oligoarginine, are currently of interest to enhance cellular uptake of bioactives and nanoparticles. This paper assesses the cellular association of poly(ethyl-cyanoacrylate) nanoparticles functionalized with di-arginine-histidine (RRH) in a Caco-2 cell model. We applied a computational model of particokinetics, In vitro Sedimentation, Diffusion and Dosimetry (ISDD) to predict the accumulation of nanoparticles on the cell surface. An important finding is that the proportion of nanoparticles associated with cells was less than 5 %. This has important implications for interpreting nanoparticle uptake in vitro. RRH-decoration does not appear to alter nanoparticle deposition, but increases association of nanoparticles with Caco-2 cells. Immediate deposition of nanoparticles on the cell surface was apparent and similar between formulations, but underestimated by the ISDD model. Key to understanding the nano-bio interface for drug delivery, nanoparticles that reach the cells were not necessarily absorbed by them, but can become detached. This variable of nanoparticle release from cells was incorporated into a new mathematical model presented here.

Pairing food and drink: A physiological model of blood ethanol levels for a variety of drinking behaviors.

We present a blood ethanol concentration compartment model which utilizes an animal's ethanol intake, food intake, and weight to predict the animal's blood ethanol concentration at any given time. By incorporating the food digestion process into the model we can predict blood ethanol concentration levels over time for a variety of drinking and eating scenarios. The model is calibrated and validated using data from cohorts of male monkeys, and is able to capture blood ethanol concentration kinetics of the monkeys from a variety of drinking behavior classifications.

Increasing the speed of frequency-domain, homodyne thermoreflectance imaging.

Charge coupled device (CCD)-based thermoreflectance imaging using a "4-bucket" lock-in imaging algorithm is a well-established, powerful method for obtaining high spatial and thermal resolution two-dimensional thermal maps of optoelectronic, electronic, and micro-electro-mechanical systems devices. However, the technique is relatively slow, limiting broad commercial adoption. In this work, we examine the underlying limit on the image acquisition speed using the conventional "4-bucket" algorithm and show that the straightforward extension to an n-bucket technique by faster sampling does not address the underlying statistical bias in the data analysis and hence does not reduce the image acquisition time. Instead, we develop a modified "enhanced n-bucket" algorithm that halves the image acquisition time for every doubling of the number of buckets. We derive detailed statistical models of the algorithms and confirm both the models and the resulting speed enhancement experimentally, resulting in a practical means of significantly enhancing the speed and utility of CCD-based frequency domain, homodyne thermoreflectance imaging.

Time for a Drink? A Mathematical Model of Non-human Primate Alcohol Consumption.

We simulate a non-human primate's alcohol drinking pattern in order to better understand temporal patterning of alcoholic drinks that can lead to the excessive intakes associated with alcohol use disorder. A stochastic mathematical model of alcohol consumption pattern is developed, where model parameters are calibrated to an individual monkey's drinking history. The model predicts a time series that simulates a monkey's alcohol intake in time, and we analyze this drinking pattern to understand the variations in day and night drinking, the lengths of drinks (intake in 5 or more consecutive secs), and lengths of bouts (1 or more drinks per 5 min occasion). This time series can predict a lifetime categorical drinking level (light, binge, heavy, or very heavy), thus correlating an individual monkey's parameters with distinct long term drinking classifications.

Graph complexity analysis identifies an ETV5 tumor-specific network in human and murine low-grade glioma.

Conventional differential expression analyses have been successfully employed to identify genes whose levels change across experimental conditions. One limitation of this approach is the inability to discover central regulators that control gene expression networks. In addition, while methods for identifying central nodes in a network are widely implemented, the bioinformatics validation process and the theoretical error estimates that reflect the uncertainty in each step of the analysis are rarely considered. Using the betweenness centrality measure, we identified Etv5 as a potential tissue-level regulator in murine neurofibromatosis type 1 (Nf1) low-grade brain tumors (optic gliomas). As such, the expression of Etv5 and Etv5 target genes were increased in multiple independently-generated mouse optic glioma models relative to non-neoplastic (normal healthy) optic nerves, as well as in the cognate human tumors (pilocytic astrocytoma) relative to normal human brain. Importantly, differential Etv5 and Etv5 network expression was not directly the result of Nf1 gene dysfunction in specific cell types, but rather reflects a property of the tumor as an aggregate tissue. Moreover, this differential Etv5 expression was independently validated at the RNA and protein levels. Taken together, the combined use of network analysis, differential RNA expression findings, and experimental validation highlights the potential of the computational network approach to provide new insights into tumor biology.

A validated mathematical model of cell-mediated immune response to tumor growth.

Mathematical models of tumor-immune interactions provide an analytic framework in which to address specific questions about tumor-immune dynamics. We present a new mathematical model that describes tumor-immune interactions, focusing on the role of natural killer (NK) and CD8+ T cells in tumor surveillance, with the goal of understanding the dynamics of immune-mediated tumor rejection. The model describes tumor-immune cell interactions using a system of differential equations. The functions describing tumor-immune growth, response, and interaction rates, as well as associated variables, are developed using a least-squares method combined with a numerical differential equations solver. Parameter estimates and model validations use data from published mouse and human studies. Specifically, CD8+ T-tumor and NK-tumor lysis data from chromium release assays as well as in vivo tumor growth data are used. A variable sensitivity analysis is done on the model. The new functional forms developed show that there is a clear distinction between the dynamics of NK and CD8+ T cells. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge are able to reproduce data from the published studies. A sensitivity analysis reveals that the variable to which the model is most sensitive is patient specific, and can be measured with a chromium release assay. The variable sensitivity analysis suggests that the model can predict which patients may positively respond to treatment. Computer simulations highlight the importance of CD8+ T-cell activation in cancer therapy.

Accounting for unintended binding events in the analysis of quartz crystal microbalance kinetic data.

Previous methods for analyzing protein-ligand binding events using the quartz crystal microbalance with dissipation monitoring (QCM-D) fail to account for unintended binding that inevitably occurs during surface measurements and obscure kinetic information. In this article, we present a system of differential equations that accounts for both reversible and irreversible unintended interactions. This model is tested on three protein-ligand systems, each of which has different features, to establish the feasibility of using the QCM-D for protein binding analysis. Based on this analysis, we were able to obtain kinetic information for the intended interaction that is consistent with those obtained in literature via bulk-phase methods. In the appendix, we include a method for decoupling these from the intended binding events and extracting relevant affinity information.

A model of dendritic cell therapy for melanoma.

Dendritic cells are a promising immunotherapy tool for boosting an individual's antigen-specific immune response to cancer. We develop a mathematical model using differential and delay-differential equations to describe the interactions between dendritic cells, effector-immune cells, and tumor cells. We account for the trafficking of immune cells between lymph, blood, and tumor compartments. Our model reflects experimental results both for dendritic cell trafficking and for immune suppression of tumor growth in mice. In addition, in silico experiments suggest more effective immunotherapy treatment protocols can be achieved by modifying dose location and schedule. A sensitivity analysis of the model reveals which patient-specific parameters have the greatest impact on treatment efficacy.

A multiple time-scale computational model of a tumor and its micro environment.

Experimental evidence suggests that a tumor's environment may be critical to designing successful therapeutic protocols: Modeling interactions between a tumor and its environment could improve our understanding of tumor growth and inform approaches to treatment. This paper describes an efficient, flexible, hybrid cellular automaton-based implementation of numerical solutions to multiple time-scale reaction-diffusion equations, applied to a model of tumor proliferation. The growth and maintenance of cells in our simulation depend on the rate of cellular energy (ATP) metabolized from nearby nutrients such as glucose and oxygen. Nutrient consumption rates are functions of local pH as well as local concentrations of oxygen and other fuels. The diffusion of these nutrients is modeled using a novel variation of random-walk techniques. Furthermore, we detail the effects of three boundary update rules on simulations, describing their effects on computational efficiency and biological realism. Qualitative and quantitative results from simulations provide insight on how tumor growth is affected by various environmental changes such as micro-vessel density or lower pH, both of high interest in current cancer research.

A model for the evolution of extremely fragmented macronuclei in ciliates.

While all ciliates possess nuclear dimorphism, several ciliates - like those in the classes Phyllopharyngea, Spirotrichea, and Armophorea - have an extreme macronuclear organization. Their extensively fragmented macronuclei contain upwards of 20,000 chromosomes, each with upwards of thousands of copies. These features have evolved independently on multiple occasions throughout ciliate evolutionary history, and currently no models explain these structures in an evolutionary context. In this paper, we propose that competition between two forces - the limitation and avoidance of chromosomal imbalances as a ciliate undergoes successive asexual divisions, and the costs of replicating massive genomes - is sufficient to explain this particular nuclear structure. We present a simulation of ciliate cell evolution under control of these forces, allowing certain features of the population to change over time. Over a wide range of parameters, we observe the repeated emergence of this unusual genomic organization found in nature. Although much remains to be understood about the evolution of macronuclear genome organization, our results show that the proposed model is a plausible explanation for the emergence of these extremely fragmented, highly polyploid genomes.

Best practices in mathematical modeling.

Mathematical modeling is a vehicle that allows for explanation and prediction of natural phenomena. In this chapter we present guidelines and best practices for developing and implementing mathematical models, using cancer growth, chemotherapy, and immunotherapy modeling as examples.

Team research at the biology-mathematics interface: project management perspectives.

The success of interdisciplinary research teams depends largely upon skills related to team performance. We evaluated student and team performance for undergraduate biology and mathematics students who participated in summer research projects conducted in off-campus laboratories. The student teams were composed of a student with a mathematics background and an experimentally oriented biology student. The team mentors typically ranked the students' performance very good to excellent over a range of attributes that included creativity and ability to conduct independent research. However, the research teams experienced problems meeting prespecified deadlines due to poor time and project management skills. Because time and project management skills can be readily taught and moreover typically reflect good research practices, simple modifications should be made to undergraduate curricula so that the promise of initiatives, such as MATH-BIO 2010, can be implemented.