Parkinson’s disease has been traditionally thought of as a dopaminergic (DA) disease in which cells of the substantia nigra pars compacta (SNc) die. However, accumulating evidence implies an important role for the serotonergic (5HT) system in Parkinson’s disease. We use a mathematical model to investigate the consequences of levodopa therapy on the serotonergic system and on the pulsatile release of DA from dopaminergic and serotonergic terminals in the striatum. We compute the time course of DA release in the striatum from both 5HT and DA neurons and show how the time course changes as more and more SNc cells die. This enables us to explain the shortening of the therapeutic time window for the efficacy of levodopa as Parkinson’s disease progresses. Finally, we study the effects 5HT1a and 5HT1b autoreceptor agonists and explain why they have a synergistic effect and why they lengthen the therapeutic time window for LD therapy. Our results are consistent with and help explain results in the experimental literature and provide new predictions that can be tested experimentally.

The use of delay differential equations (DDEs) to study biological phenomena has a long history, when the rate of change of model variables depends their previous history. Today, DDEs occupy a central place in models of infectious disease dynamics, epidemiology, ecology and tumor growth. In this talk, I will present a delayed partial differential equation (PDE) model of tumor growth and treatment. The model accounts for cell cycle arrest and cell death induced by chemotherapy, and explicitly includes intracellular signaling pathways relevant to drug action. The model is simplified to give a 1D hyperbolic PDE, which is further reduced to a nonlinear, non-autonomous DDE by projecting along characteristics. Necessary and sufficient conditions for the global stability of the cancer-free equilibrium are derived and conditions under which the system evolves to periodic solutions are determined. This has clinical implications since it leads to a lower bound for the amount of therapy required to affect a cure. Finally, I will present a clinical application of the model, by applying it to the treatment of ovarian cancers. Two types of drugs are considered – platinum-based chemotherapeutic agents that are the current standard of care for most solid tumors, and small molecule cell death inducers that are currently under development. The model is calibrated versus in vitro experimental results, and is then used to predict optimal doses and administration time scheduling for the treatment of a tumor growing in vivo.

In a 2009 paper Yu, Etheridge and Cuthbertson proposed a model that was intended to address two questions. The first was a question related to Muller's ratchet, "What ratio of mutations must be beneficial for the meanfitness of a population to increase in time?" The second question was related to the Hill-Robertson effect, "If many beneficial mutations are introduced into a population, how much will competition slow the rate of adaptation?" They introduced a model of an asexually reproducing population of fixed size N and mutation rate mu and conjectured that the rate of adaptation is O(logN/(log logN)^2) for large N so long as there is some positive ratio of beneficial mutations. I will present an outline of my proof of this conjecture.

Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical work rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next. We have introduced a broad class of Markov models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We have characterized the behavior of standard population-genetic quantities across this family of generalized models. The generalized population models can produce startling phenomena that differ qualitatively from classical behavior -- such as assured fixation of a new mutant despite the presence of genetic drift. We have derived the forward-time continuum limits of the generalized processes, analogous to Kimura's diffusion limit of the Wright-Fisher process. Finally, we have shown that some of these exotic models are more likely than the Wright-Fisher model itself, given empirical data on genetic variation in Drosophila populations. Joint work with Ricky Der and Charlie Epstein.

Mechanistic mathematical models are important tools for understanding the processes that shape ecological systems. Models have been used to describe life cycles of individuals, population dynamics, behavior, and more. However, in order for these models to reach their full potential as both tools for understanding and for prediction we must be able to link modeled quantities to data and infer model parameters. However, general methods of parameter inference for many of these models, including Individual Based Models (IBMs) or their component models, are not available. In addition, some models include components that are unmeasurable or poorly known, which can impact parameter inference and thus prediction. Here I discuss two examples of ecological models of these types. First is a bioenergetic model of individual growth and reproduction in a dynamic environment. This example highlights how input mis-specification can affect inference, and the consequences for prediction for both individuals and populations. The second example uses an example of an IBM developed to describe the spread of Chytridiomycosis in a population of frogs. This case study shows how one can perform inference for IBMs that exhibit certain characteristics with a traditional likelihood-based approach.

I consider two kinds of biological novelty using shape of the Drosophila wing as a focal phenotype. Quantitative novelty is the evolution of more or less of the same elements already present in the ancestor, or evolution within one topological space. Qualitative novelty is gain of elements not present in the ancestor, or perhaps the loss of such elements, making a transition from one topological space to another. A quantitative genetic approach allows us to studying the major determinants of quantitative novelty, standing genetic variation and mutational variation. I will present measurements of both mutation and standing variation for a multivariate phenotype, the fly wing. Mathematical and statistical challenges in quantitative novelty involve the identification of subspaces within a topological space with genetic variation, and their relationship to the subspaces found in other samples. Qualitative novelty is easy to identify among species, but its evolution is difficult to study, as by definition it is absent in the ancestor. The process of development, where qualitatively different structures are progressively introduced during most life cycles, provides a framework to understand the evolution of novelty. Developmental studies show that continuous changes in gene expression precede transitions to a new shape topology. Gene expression state within a shape is more complex object that shape itself, adding dimensions that may allow us to identify regions of one shape topology that are neighbors to other topologies.

The chemotaxis network of bacteria such as E. coli is remarkable for its sensitivity to minute relative changes in chemical concentrations in the environment. Indeed, E. coli cells can detect concentration changes corresponding to only ~3 molecules in the volume of a cell. Much of this acute sensitivity can be traced to the collective behavior of teams of chemoreceptors on the cell surface. Instead of receptors switching individually between active and inactive configurations, teams of 6-20 receptors switch on and off, and bind or unbind ligand, collectively. Similar to the binding and unbinding of oxygen molecules by tetramers of hemoglobin, the result is a sigmoidal binding curve. Coupled with a system for adaptation that tunes the operating point to the steep region of this sigmoidal curve, the advantage for chemotaxis is gain – i.e., small relative changes in chemical concentrations are transduced into large relative changes in signaling activity (specifically, the rate of phosphorylation of the response regulator CheY).  However, something is troubling about this simple explanation: in addition to providing gain, the coupling of receptors into teams also increases noise, and the net result is a decrease in the signal-to-noise ratio of the network. Why then are chemoreceptors observed to form cooperative teams? We present a novel hypothesis that the run-and-tumble chemotactic strategy of bacteria leads to a “noise threshold”, below which noise does not significantly decrease chemotactic velocity, but above which noise dramatically decreases this velocity.

This talk provides some fundamentals of statistical techniques for data on non-Euclidean spaces. Such data occur in the analysis of shape of geometrical objects, e.g. in applications studying biological growth. Naturally, shape is modeled on a manifold quotient (e.g. unit size objects) under a Lie group action (e.g. translations and rotations) which can be given a manifold structure, possibly with singularities. We show how this scenario allows for one and two sample tests as well as principal component analysis.

In an experimental study of single enzyme reactions, it has been proposed that the rate constants of the enzymatic reactions fluctuate randomly, according to a given distribution. To quantify the uncertainty arising from random rate constants, it is necessary to investigate how one can simulate such a biochemical system. To do this, we will take the Gillespie's stochastic simulation algorithm for simulation the evolution of the state of a chemical system, and study a modification of the algorithm that incorporates the random rate constants, using in part the Metropolis-Hastings algorithm to enact the distribution on the random rate constants. This modified algorithm, when applied to the single enzyme reaction system, produces simulation outputs that are corroborated by the experimental results. This project is in its early stages, and it is hoped that it can subsequently be used as a tool for the estimation or calibration of parameters in the system using experimental data.

Graphs are becoming a favorite mathematical object for representation of data. Yet, statistical pattern recognition has focused almost entirely on vector valued data in Euclidean space. Graphs, however, live in graph space, which is non-Euclidean. Thus, most inference techniques are not even defined for graph valued data. Previous work in the classification of graph-valued data typically follows one of two recipes. (1) Vectorize the adjacency matrices of the graphs, and apply standard machine learning techniques. (2) Compute some number of graph invariants (e.g., clustering coefficient, or degree distribution) for each graph, and then apply standard machine learning techniques. We follow a different recipe based in the probabilistic theory of pattern recognition. First, we define a joint graph-class model. Given this model, we derive classifiers which we prove are consistent; that is, they converge to the Bayes optimal classifier. Specifically, we build two consistent classifiers for graph valued data, a parametric and a non-parametric version. In a sense, these classifiers span the spectrum of complexity, the former is consistent for graphs sampled from relatively simple random graph distributions, the latter is consistent for graphs sampled from (nearly) any random graph distribution. Although both classifiers assume that all our graphs have labeled vertices, we generalize these results to also incorporate unlabeled graphs, as well as weighted and multigraphs. We apply these graph classifiers to human brain data. Specifically, using diffusion MRI, we can obtain large brain-graphs (10,000 vertices) for each subject, where vertices correspond to voxels. We then coarsen the graphs spatially to obtain smaller (70 vertex) graphs per subject. Using <50 subjects, we are able to achieve nearly 85% classification accuracy, with results interpretable to neurobiologists with regard to the brain regions of interest.

A multiscale approach to characterize macromolecular dynamics and functions The understanding of emerging collective behaviors in biomolecular complexes represents a major challenge in modern biophysics. As a first step toward the study of such processes we have applied multi-resolution nonlinear dimensionality reduction and diffusion analysis to obtain reliable low-dimensional representations and models for the dynamics of apparently high-dimensional complex systems such as proteins in a biological environment. The results clearly show that the proposed methods can efficiently find low dimensional representations of a complex process such as protein folding.

Circadian clocks persist with a constant period (~24-hour) even after a significant change of the expression level of clock genes. To study the biochemical mechanisms of timekeeping, we develop the most accurate mathematical model of mammalian intracellular timekeeping, as well as a simplified model amenable to mathematical analysis. This modeling work raises interesting questions about existence and uniqueness of models given knowledge of their solutions. Although much is known about cellular circadian timekeeping, little is known about how these rhythms are sustained with a constant period. Here, we show how a universal motif of circadian timekeeping, where repressors bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that, even in the presence of large changes in gene expression levels, an additional slow negative feedback loop keeps this stoichiometry in balance and maintains oscillations with a fixed period. These results explain why the network structure found naturally in circadian clocks can generate ~24-hour oscillations in many conditions.

It is common to use a multitype branching process to model the accumulation of mutations that leads to cancer progression, metastasis, and resistance to treatment. In this talk I will describe results about the time until the first type k (cell with k mutations) and the growth of the type k population obtained in joint work with Stephen Moseley, and their use in evaluating possible screening strategies for ovarian cancer, work in progress with Duke undergraduate Kaveh Danesh. The point process representation of the limit, which is a one-sided stable law, together with results from 10-60 years ago leads to remarkable explicit formulas for Simpson's index and the size of the largest clone. These results are important in understanding tumor diversity which can present serious obstacles to treatment. The last topic is joint work with Jasmine Foo, Kevin Leder, John Mayberry, and Franziska Michor

A preliminary examination of a very rich data set consisting of a detailed survey of individuals in male-male sex networks in South India. The motivation for the study is to understand the spread of HIV in male-male sex networks in South India. The data contains survey information from participants, as well as their cell phone contacts and incomplete information on the contacts by participants. We provide predictive models of attributes of contacts given participant attributes, as well as predictive models of the attributes, such as sexual position. We study how model parameters vary as a function of connectedness of individuals and how modeling network interactions has an effect on the model.

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. One motivation for this work is to provide analytical solutions to help validate the accuracy of numerical methods for elastic membranes in flow. A second motivation is to clarify the steady response of capsules in some canonical flows. Finally, we investigate the formation of finite-time cusp singularities, of which there are only a few examples in interfacial Stokes flow, and where none involve elastic interfaces. This is joint work with Michael Booty and Michael Higley.

Prior to cell division, chromosomes are segregated by mitotic spindle. This molecular machine self-assembles remarkably fast and accurately in an elegant process of search and capture, during which dynamic polymers from two centers grow and shrink rapidly and repeatedly until a contact with chromosomes is established. At the same time, the chromosomes interact with each other. I will show computer simulation and experimental microscopy results illustrating how cell deploys a number of redundant mechanisms optimizing the assembly by keeping it accurate yet rapid. Very similar strategies are used to assemble Golgi apparatus. The talk will illustrate how computer modeling assists experiment in unraveling mysteries of cell dynamics.

A ureteric stent is a slender polymer tube that can be placed within the ureter (the muscular tube that conveys urine from the kidney to the bladder) to relieve a blockage due, for example, to a kidney stone in transit, or to external pressure from a tumor. A urinary catheter can be placed similarly within the urethra (the muscular tube conveying urine from the bladder out of the body), either again to relieve a blockage, or to allow control of urination in incontinent patients or those recovering from surgery. Several clinical complications are associated with each of these biomedical devices. Both become encrusted, over time, with salts that precipitate out from the urine. Such encrustation is often associated with infection and the presence of bacterial biofilm on the device and, if severe, can make removal of the device difficult and painful. Ureteric stents are also associated with urinary reflux: retrograde flow of urine back towards the kidney. This arises because the stent prevents proper function of the sphincter between ureter and bladder that normally closes off when bladder pressure rises. Such reflux can expose the kidney to dangerously high pressures, and increase the risk of renal infection, both of which can lead to long-term damage. This talk will highlight aspects of our interdisciplinary work on such problems. We present mathematical models of the reflux and encrustation processes and consider the implications for device design and clinical practice.

Viscoelastic interactions of Myxococcus xanthus cells in a low-density domain close to the edge of a swarm have been recently studied in [1] using a combination of a cell-based three-dimensional Subcellular Element (SCE) model [1,2] and cell-tracking experiments. The model takes into account the flexible nature of M. xanthus as well as the effects of adhesion between cells arising from the interaction of the capsular polysaccharide covering two cells in contact with each other. New image and dynamic cell curvature analysis algorithms were used to track and measure the change in cell shapes that occur as flexible cells undergo significant bending during collisions resulting in direct calibration of the model parameters. It will be shown in this talk that flexibility of cells and the adhesive cell–cell and cell–substrate interactions of M. xanthus together with cell to aspect-ratio and directional reversals [3], play an important role in smooth cell gliding and more efficient swarming. In the second part of the talk results of the analysis of the three dimensional structures of fibrin networks, with and without cells, reconstructed from two-dimensional z-stacks of confocal microscopy sections using novel image analysis algorithms, will be presented. These images were used to establish microstructure-based models for studying the relationship between the structural features and the mechanical properties of the fibrin networks in blood clots. The change in the fibrin network alignment under applied strain and the elastic modulus values will be shown to agree well with the experimental data [4]. 1. C.W. Harvey, F. Morcos, C.R. Sweet, D. Kaiser, S. Chatterjee, X. Lu, D. Chen and M. Alber [2011], Study of elastic collisions of M. xanthus in swarms, Physical Biology 8, 026016. 2. C.R. Sweet, S. Chatterjee, Z. Xu, K. Bisordi, E.D. Rosen and M. Alber [2011], Modeling Platelet-Blood Flow Interaction Using Subcellular Element Langevin Method, J R Soc Interface, 2011 May 18. [Epub ahead of print], doi: 10.1098/rsif.2011.0180. 3. Y. Wu, Y. Jiang, D. Kaiser and M. Alber [2009], Periodic reversal of direction allows Myxobacteria to swarm, Proc. Natl. Acad. Sci. USA 106 4 1222-1227. 4. E. Kim, O.V. Kim, K.R. Machlus, X. Liu, T. Kupaev, J. Lioi, A.S. Wolberg, D.Z. Chen, E.D. Rosen, Z. Xu and M. Alber [2011], Soft Matter 7, 4983-4992.

ABSTRACT: The past decade has seen a number of engineering innovations that make construction of devices of micro- and even nanometric dimensions feasible. Hence, there is a growing interest in exploring new and efficient ways to generate propulsion at these small scales. Here we explore optimization of one particular type of low Reynolds number propulsion mechanism – flagella. Beyond the general challenges associated with optimization, there are a number of issues that are unique to swimming at low Reynolds numbers. At small scales, the fluid equations of motion are linear and time-reversible, hence reciprocal motion – i.e., strokes that are symmetric with respect to time reversal – cannot generate any net translation (a limitation commonly referred to as the Scallop Theorem). One possible way to break this symmetry is through carefully chosen morphologies and kinematics. One symmetry-breaking solution commonly employed by eukaryotic microorganisms is to select nonreciprocal stroke patterns by actively generating torques at fixed intervals along the organism. Hence, we will address the question: For a given morphology, what are the optimal kinematics? In this talk we present optimal stroke patterns using biologically inspired geometries such as single-tailed spermatozoa and the double-tail morphology of Chlamydomonas, a genus of green alga widely considered to be a model system in molecular biology.

Certain neurons can, in isolation, generate a bursting rhythm, in which phases of active spiking alternative repetitively with phases of quiescence. This behavior is itself mathematically interesting, and neurons with this capability have been found in the mammalian respiratory brain stem, suggesting that they might drive the respiratory rhythm. In this talk, I will survey some mathematical and computational work that runs counter to this suggestion. The methods involved include slow-fast decomposition and associated bifurcation analysis in single-neuron and few-neuron ODE models as well as a genetic algorithm applied to larger network models. The larger network results may have general implications for networks of nodes with heterogeneous dynamics, coupled in small-world, scale-free, and other architectures

Eukaryotic cell microenvironment (inner and outer) is composed in large parts from fluids that interact with solid and elastic bodies, whereas it is the cell cytoplasm, cytoskeleton and basal membrane; the interstitial fluid interpenetrating the stroma and tumor cells; or blood flow carrying the immune or circulating tumor cells. I will discuss the use of two fluid-structure interactions methods, the immersed boundary and the regularized Stokeslets, in applications dealing with the tumor development and treatment. First model operates on the cellular scale and will be used to model various cell processes, such as cell growth, division or death, during the cellular self-organization into a normal mammary acinus, a 3D in vitro structure recapitulating the morphology of breast cysts (acini). I will discuss model development, parameterization and tuning with the experimental data, as well as their subsequent use to investigate the link between morphogenesis of epithelial mutants and molecular alterations of tumor cells. Second model acts on the tissue level, and will be used to investigate the relation between tumor tissue structure and efficacy of anticancer drugs in the context of interstitial fluid flow. I will present simulation results showing non-linear relation between tumor tissue structure and effectiveness of drug penetration. I will also discuss how tumor tissue metabolic state(its oxygenation and acidity) becomes modified due to actions of chemotherapeutic drugs leading to the emergence of tumor zones with potentially drug-resistant cells and/or to tumor areas that are not exposed to drugs at all. Both of these phenomena can contribute to the moderateclinical success of many anticancer drugs.

Cartilage physiology is regulated by a single population of specialized cells called chondrocytes. The chondroyctes are sparsely distributed within the extracellular matrix (ECM) and maintain a state of homeostasis in healthy tissue. ECM degeneration due to osteoarthritis can lead to compete degradation of cartilage surfaces, necessitating total joint replacement. Chondrocytes can be utilized to regenerate cartilage via tissue engineering approaches in which these cells are seeded in biocompatible and degradable biopolymer or hydrogel scaffolds. In such systems, biosynthetic activity of the cells in response to their non-native environment results in regeneration and accumulation of ECM constituents concurrent with degradation of the surrounding scaffold material. In this talk, mixture models are presented for interactions between biosynthesis of ECM constituents and ECM linking in cell-seeded scaffolds. Both ODE-based (temporal) models for evolution of average apparent densities and PDE-based (spatio-temporal) models will be presented for variables including unlinked ECM, linked ECM and scaffold. Model extensions accounting for cell proliferation will also be discussed. Of particular interest are model predictions for the evolution of solid phase apparent density, which is correlated with the compressive elastic modulus of the tissue construct. These models provide a quantitative framework for assessing and optimizing the design of engineered cell-scaffold systems and guiding strategies for articular cartilage tissue engineering.

Fluid mechanics plays a crucial role in many cellular processes. One example is the external fluid mechanics of motile cells such as bacteria, spermatozoa, algae, and essentially half of the microorganisms on earth. The most commonly-studied organisms exploit the bending or rotation of a small number of flagella (short whip-like organelles, length scale from a few to tens of microns) to create fluid-based propulsion. Ciliated microorganisms swim by exploiting the coordinated surface beating of many cilia (which are short flagella) distributed along their surface. After a short introduction to the fundamentals of fluid-based locomotion on small scales, we pose separate optimization problems addressing the optimal geometries and locomotion gaits of low-Reynolds-number swimmers. First, we characterize the optimal dynamics of simple flapping swimmers with two degrees of freedom. Second, we derive analytically and computationally the optimal waveform of an elastic flagellum, such as the one employed by eukaryotic cells for propulsion. Third, we investigate the optimal shapes of helical propellers, and use our results to help rationalize the shape selection mechanism in bacterial flagella. Finally, we characterize the optimal locomotion by surface distortions of blunt swimmers, and demonstrate the appearance of waves, reminiscent of the metachronal waves displayed by ciliated organisms.

Inside every eukaryotic cell is the nucleus, organelles, and the surrounding cytoplasm, which typically accounts for 50% of the cell volume. The cytoplasm is a complex mixture of water, protein, and a dynamic polymer network. Cells use cytoplasmic streaming to transmit chemical signals, to distribute nutrients, and to generate forces involved in locomotion. In this talk we present two different models related to cytoplasmic streaming in amoeboid cells. In the first part of the talk, we present a computational model to describe the dynamics of blebbing, which occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. The model is used to explore the relative roles in bleb dynamics of cytoplasmic viscosity, permeability of the cytoskeleton, and elasticity of the membrane and cytoskeleton. In the second part of the talk we examine how flow-induced instabilities of cytoplasm are related to the structural organization of the giant amoeboid cell Physarum polycephalum. We use a multiphase flow model that treats both the cytosol and cytoskeleton as fluids each with its own material properties and internal forces, and we discuss instabilities of the sol/gel mixture that produce flow channels within the gel. We analyze a reduced model and offer a new and general explanation for how fluid flow is involved in cytoskeletal reorganization.

One of the challenges in biology is relating biochemical reactions that occur at the protein nanoparticle size of 1-100 nm to large scale effects on the cell or tissue scale of 0.01-10 mm. The cytoskeleton is a remarkable example with actin polymerization/depolymerization leading to locomotion, metastasis or apoptosis. This talk presents a recently developed multiscale model that captures large-scale effects produced by changes in biochemical reactions. The model is a computational algorithm that determines effective continuum properties of a homogenized cytoskeleton model by concurrent microscopic simulation. Concepts from information theory and optimal transport are applied to link disparate scales in a computationally efficient manner. One of the interesting aspects of this approach is the combination of standard computational modeling techniques (finite volume, numerical stochastic ODEs) with statistical concepts and learning theory.

The jellyfish has been the subject of numerous mathematical and physical studies ranging from the discovery of reentry phenomenon in electrophysiology to the development of axisymmetric methods for solving fluid-structure interaction problems. In this presentation, we develop and test mathematical models describing the pulsing dynamics and the resulting fluid flow generated by the upside down jellyfish, Cassiopea. The kinematics of contraction and distributions of pulse frequencies were obtained from videos and used as inputs into numerical simulations. Particle image velocimetry was used to obtain spatially and temporally resolved flow fields experimentally. The immersed boundary method was then used to solve the fluid-structure interaction problem and explore how changes in morphology and pulsing dynamics alter the resulting fluid flow. Unlike pelagic (swimming) jellyfish, there is no evidence of the formation of a train of vortex rings. Instead, significant mixing occurs around and directly above the oral arms and secondary mouths. We found good agreement between the numerical simulations and experiments, suggesting that the presence of porous oral arms induce net horizontal flow towards the bell and mixing.

Biological projectiles range from a 10-micrometer spore to a 1-meter leaping mammal. Pre-launch accelerations scale inversely with length, with that of the smallest projectile approaching a million times gravity. These projectiles follow Borelli's rule, that all jumpers should jump to the same height. Nonetheless, his rationale is wrong on at least two accounts. For one thing, it presumes a muscular engine operating with no energy storage, often far from the case. For another, it ignores drag, critical for small projectiles, which operate in an overwhelmingly drag-dominated rather than gravity- dominated domain and whose optimal trajectories look decidedly unfamiliar. But the rule can be given quite a different—and more general—basis. And a simple dimensionless index helps us anticipate best launch angles and path lengths, these latter illustrated with a simple computer simulation.