Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology—defined here simply as the use of mathematics in cancer research—complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

β-Lactamase inhibition is an important clinical strategy in overcoming β-lactamase-mediated resistance to β-lactam antibiotics in Gram negative bacteria. A new β-lactamase inhibitor, avibactam, is entering the clinical arena and promising to be a major step forward in our antibiotic armamentarium. Avibactam has remarkable broad-spectrum activity in being able to inhibit classes A, C, and some class D β-lactamases. We present here structural investigations into class A β-lactamase inhibition by avibactam as we report the crystal structures of SHV-1, the chromosomal penicillinase of Klebsiella pneumoniae, and KPC-2, an acquired carbapenemase found in the same pathogen, complexed with avibactam. The 1.80 Å KPC-2 and 1.42 Å resolution SHV-1 β-lactamase avibactam complex structures reveal avibactam covalently bonded to the catalytic S70 residue. Analysis of the interactions and chair-shaped conformation of avibactam bound to the active sites of KPC-2 and SHV-1 provides structural insights into recently laboratory-generated amino acid substitutions that result in resistance to avibactam in KPC-2 and SHV-1. Furthermore, we observed several important differences in the interactions with amino acid residues, in particular that avibactam forms hydrogen bonds to S130 in KPC-2 but not in SHV-1, that can possibly explain some of the different kinetic constants of inhibition. Our observations provide a possible reason for the ability of KPC-2 β-lactamase to slowly desulfate avibactam with a potential role for the stereochemistry around the N1 atom of avibactam and/or the presence of an active site water molecule that could aid in avibactam desulfation, an unexpected consequence of novel inhibition chemistry.

The pace and unpredictability of evolution are critically relevant in a variety of modern challenges: combating drug resistance in pathogens and cancer, understanding how species respond to environmental perturbations like climate change, and developing artificial selection approaches for agriculture. Great progress has been made in quantitative modeling of evolution using fitness landscapes, allowing a degree of prediction for future evolutionary histories. Yet fine-grained control of the speed and the distributions of these trajectories remains elusive. We propose an approach to achieve this using ideas originally developed in a completely different context -counterdiabatic driving to control the behavior of quantum states for applications like quantum computing and manipulating ultra-cold atoms. Implementing these ideas for the first time in a biological context, we show how a set of external control parameters (i.e. varying drug concentrations / types, temperature, nutrients) can guide the probability distribution of genotypes in a population along a specified path and time interval. This level of control, allowing empirical optimization of evolutionary speed and trajectories, has myriad potential applications, from enhancing adaptive therapies for diseases, to the development of thermotolerant crops in preparation for climate change, to accelerating bioengineering methods built on evolutionary models, like directed evolution of biomolecules.The quest to control evolutionary processes in areas like agriculture and medicine predates our understanding of evolution itself. Recent years have seen growing research efforts toward this goal, driven by rapid progress in quantifying genetic changes across a population 1-3 as well as a global rise in challenging problems like therapeutic drug resistance 4-6 . New approaches that have arisen in response include prospective therapies that steer evolution of pathogens toward maximized drug sensitivity 7, 8 , typically requiring multiple rounds of selective pressures and subsequent evolution under them. Since we cannot predict the exact progression of mutations that occur in the course of the treatment, the best we can hope for is to achieve control over probability distributions of evolutionary outcomes. However, our lack of precise control over the timing of these outcomes poses a major practical impediment to engineering the course of evolution. This naturally raises a question: Rather than being at the mercy of evolution's unpredictability and pace, what if we could simultaneously control the speed and the distribution of genotypes over time?Controlling an inherently stochastic process like evolution has close parallels to problems in other disciplines.Quantum information protocols crucially depend on coherent control over the time evolution of quantum states under external driving 9, 10 , in many cases requiring that a system remain in an instantaneous ground state of a time-varying Hamiltonian in applications like cold atom transport 11 and quantum adiabatic computat...