In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). The Universal PINN approach (UPINN) adds a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of the neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of UPINNs even when provided with very few measurements of noisy data in both the ODE and PDE regime.

The over-expression of the Bcl-2 protein is a common feature of many solid cancers and hematological malignancies, and it is typically associated with poor prognosis and resistance to chemotherapy. Bcl-2-specific inhibitors, such as venetoclax, have recently been approved for the treatment of chronic lymphocytic leukemia and small lymphocytic lymphoma, and they are showing promise in clinical trials as a targeted therapy for patients with relapsed or refractory acute myeloid leukemia (AML). However, successful treatment of AML with Bcl-2-specific inhibitors is often followed by the rapid development of drug resistance. An emerging paradigm for overcoming drug resistance in cancer treatment is through the targeting of mitochondrial energetics and metabolism. In AML in particular, it was recently observed that inhibition of mitochondrial translation via administration of the antibiotic tedizolid significantly affects mitochondrial bioenergetics, activating the integrated stress response (ISR) and subsequently sensitizing drug-resistant AML cells to venetoclax. Here we develop an integrative systems biology approach to acquire a deeper understanding of the molecular mechanisms behind this process, and in particular, of the specific role of the ISR in the commitment of cells to apoptosis. Our multi-scale mathematical model couples the ISR to the intrinsic apoptosis pathway in venetoclax-resistant AML cells, includes the metabolic effects of treatment, and integrates RNA, protein level, and cellular viability data. Using the mathematical model, we identify the dominant mechanisms by which ISR activation helps to overcome venetoclax resistance, and we study the temporal sequencing of combination treatment to determine the most efficient and robust combination treatment protocol.

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

Hypoxia-activated prodrugs (HAPs) present a conceptually elegant approach to not only overcome, but better yet, exploit intra-tumoural hypoxia. Despite being successful in vitro and in vivo, HAPs are yet to achieve successful results in clinical settings. It has been hypothesised that this lack of clinical success can, in part, be explained by the insufficiently stringent clinical screening selection of determining which tumours are suitable for HAP treatments. Taking a mathematical modelling approach, we investigate how tumour properties and HAP-radiation scheduling influence treatment outcomes in simulated tumours. The following key results are demonstrated in silico: (i) HAP and ionising radiation (IR) monotherapies may attack tumours in dissimilar, and complementary, ways. (ii) HAP-IR scheduling may impact treatment efficacy. (iii) HAPs may function as IR treatment intensifiers. (iv) The spatio-temporal intra-tumoural oxygen landscape may impact HAP efficacy. Our in silico framework is based on an on-lattice, hybrid, multiscale cellular automaton spanning three spatial dimensions. The mathematical model for tumour spheroid growth is parameterised by multicellular tumour spheroid (MCTS) data.

Tumour hypoxia is a well-studied phenomenon with implications in cancer progression, treatment resistance, and patient survival.While a clear adverse prognosticator, hypoxia is also a theoretically ideal target for guided drug delivery.This idea has lead to the development of hypoxia-activated prodrugs (HAPs): a class of chemotherapeutics which remain inactive in the body until metabolized within hypoxic regions.In theory, these drugs have the potential for increased tumour selectivity and have therefore been the focus of numerous preclinical studies.Unfortunately, HAPs have had mixed results in clinical trials, necessitating further study in order to harness their therapeutic potential.One possible avenue for the improvement of HAPs is through the selective application of anti angiogenic agents (AAs) to improve drug delivery.Such techniques have been used in combination with other conventional chemotherapeutics to great effect in many studies.A further benefit is theoretically achieved through nanocell administration of the combination, though this idea has not been the subject of any experimental or mathematical studies to date.In the following, a mathematical model is outlined and used to compare the predicted efficacies of separate vs. nanocell administration for AAs and HAPs in tumours.The model is experimentally motivated, both in mathematical form and parameter values.Preliminary results of the model are highlighted throughout which qualitatively agree with existing experimental evidence.The novel prediction of our model is an improvement in the efficacy of AA/HAP combination therapies when administered through nanocells as opposed to separately.While this study specifically models treatment on glioblastoma, similar analyses could be performed for other vascularized tumours, making the results potentially applicable to a range of tumour types. Author summaryTumour hypoxia is a well-documented phenomenon with adverse effects for the progression of the cancer.Accordingly, various therapeutic strategies have emerged in recent years to combat its effects.Herein, we present an experimentally-motivated mathematical model used to assess the feasibility of the therapeutic combination of anti angiogenic agents with hypoxia-activated prodrugs.Analysis of the combination therapy shows that

The mean conditional fixation time of a mutant is an important measure of stochastic population dynamics, widely studied in ecology and evolution. Here, we investigate the effect of spatial randomness on the mean conditional fixation time of mutants in a constant population of cells, N. Specifically, we assume that fitness values of wild type cells and mutants at different locations come from given probability distributions and do not change in time. We study spatial arrangements of cells on regular graphs with different degrees, from the circle to the complete graph, and vary assumptions on the fitness probability distributions. Some examples include: identical probability distributions for wild types and mutants; cases when only one of the cell types has random fitness values while the other has deterministic fitness; and cases where the mutants are advantaged or disadvantaged. Using analytical calculations and stochastic numerical simulations, we find that randomness has a strong impact on fixation time. In the case of complete graphs, randomness accelerates mutant fixation for all population sizes, and in the case of circular graphs, randomness delays mutant fixation for N larger than a threshold value (for small values of N, different behaviors are observed depending on the fitness distribution functions). These results emphasize fundamental differences in population dynamics under different assumptions on cell connectedness. They are explained by the existence of randomly occurring "dead zones" that can significantly delay fixation on networks with low connectivity; and by the existence of randomly occurring "lucky zones" that can facilitate fixation on networks of high connectivity. Results for death-birth and birth-death formulations of the Moran process, as well as for the (haploid) Wright Fisher model are presented.

Tumor-induced angiogenesis leads to the development of leaky tumor vessels devoid of structural and morphological integrity.Due to angiogenesis, elevated interstitial fluid pressure (IFP) and low blood perfusion emerge as common properties of the tumor microenvironment that act as barriers for drug delivery.In order to overcome these barriers, normalization of vasculature is considered to be a viable option.However, insight is needed into the phenomenon of normalization and in which conditions it can realize its promise.In order to explore the effect of microenvironmental conditions and drug scheduling on normalization benefit, we build a mathematical model that incorporates tumor growth, angiogenesis and IFP.We administer various theoretical combinations of antiangiogenic agents and cytotoxic nanoparticles through heterogeneous vasculature that displays a similar morphology to tumor vasculature.We observe differences in drug extravasation that depend on the scheduling of combined therapy; for concurrent therapy, total drug extravasation is increased but in adjuvant therapy, drugs can penetrate into deeper regions of tumor. Author summaryTumor vessels being very different from their normal counterparts are leaky and lack organization that sustains blood circulation.As a result, insufficient blood supply and high fluid pressure begin to appear inside the tumor which leads to a reduced delivery of drugs within the tumor, especially in tumor center.A treatment strategy that utilizes antivascular drugs is observed to revert these alterations in tumor vessels, making them more normal.This approach is suggested to improve drug delivery by enhancing physical transport of drugs.In this paper, we build a mathematical model to simulate tumor and vessel growth as well as fluid pressure inside the tumor.This framework enables us to simulate drug treatment scenarios on tumors.We use this model to find whether the delivery of the chemotherapy drugs is enhanced by application of anti-vascular drugs by making vessels more normal.Our simulations show that anti-vascular drug not only enhances the