Simple Mathematical Laws Research Articles

Growth exponents reflect evolutionary processes and treatment response in brain metastases

Tumor growth is the result of the interplay of complex biological processes in huge numbers of individual cells living in changing environments. Effective simple mathematical laws have been shown to describe tumor growth in vitro, or simple animal models with bounded-growth dynamics accurately. However, results for the growth of human cancers in patients are scarce. Our study mined a large dataset of 1133 brain metastases (BMs) with longitudinal imaging follow-up to find growth laws for untreated BMs and recurrent treated BMs. Untreated BMs showed high growth exponents, most likely related to the underlying evolutionary dynamics, with experimental tumors in mice resembling accurately the disease. Recurrent BMs growth exponents were smaller, most probably due to a reduction in tumor heterogeneity after treatment, which may limit the tumor evolutionary capabilities. In silico simulations using a stochastic discrete mesoscopic model with basic evolutionary dynamics led to results in line with the observed data.

Simple mathematical law benchmarks human confrontations

Many high-profile societal problems involve an individual or group repeatedly attacking another – from child-parent disputes, sexual violence against women, civil unrest, violent conflicts and acts of terror, to current cyber-attacks on national infrastructure and ultrafast cyber-trades attacking stockholders. There is an urgent need to quantify the likely severity and timing of such future acts, shed light on likely perpetrators, and identify intervention strategies. Here we present a combined analysis of multiple datasets across all these domains which account for >100,000 events, and show that a simple mathematical law can benchmark them all. We derive this benchmark and interpret it, using a minimal mechanistic model grounded by state-of-the-art fieldwork. Our findings provide quantitative predictions concerning future attacks; a tool to help detect common perpetrators and abnormal behaviors; insight into the trajectory of a ‘lone wolf'; identification of a critical threshold for spreading a message or idea among perpetrators; an intervention strategy to erode the most lethal clusters; and more broadly, a quantitative starting point for cross-disciplinary theorizing about human aggression at the individual and group level, in both real and online worlds.

The strength of genetic interactions scales weakly with mutational effects

BackgroundGenetic interactions pervade every aspect of biology, from evolutionary theory, where they determine the accessibility of evolutionary paths, to medicine, where they can contribute to complex genetic diseases. Until very recently, studies on epistatic interactions have been based on a handful of mutations, providing at best anecdotal evidence about the frequency and the typical strength of genetic interactions. In this study, we analyze a publicly available dataset that contains the growth rates of over five million double knockout mutants of the yeast Saccharomyces cerevisiae.ResultsWe discuss a geometric definition of epistasis that reveals a simple and surprisingly weak scaling law for the characteristic strength of genetic interactions as a function of the effects of the mutations being combined. We then utilized this scaling to quantify the roughness of naturally occurring fitness landscapes. Finally, we show how the observed roughness differs from what is predicted by Fisher's geometric model of epistasis, and discuss the consequences for evolutionary dynamics.ConclusionsAlthough epistatic interactions between specific genes remain largely unpredictable, the statistical properties of an ensemble of interactions can display conspicuous regularities and be described by simple mathematical laws. By exploiting the amount of data produced by modern high-throughput techniques, it is now possible to thoroughly test the predictions of theoretical models of genetic interactions and to build informed computational models of evolution on realistic fitness landscapes.

System economics: Overcoming the pitfalls of forecasting models via a multidisciplinary approach

In areas from medicine to climate change to economics, we are faced with huge challenges and a need for accurate forecasts, yet our ability to predict the future has been found wanting. The basic problem is that complex systems such as the atmosphere or the economy can not be reduced to simple mathematical laws and modeled accordingly. The equations in numerical models are therefore only approximations to reality, and are often highly sensitive to external influences and small changes in parameterisation — they can be made to fit past data, but are less good at prediction. Since decisions are usually based on our best models of the future, how can we proceed? This paper draws a comparison between two apparently different fields: biology and economics. In biology, drug development is a highly inefficient and expensive process, which in the past has relied heavily on trial and error. Institutions such as pharmaceutical companies and universities are now radically changing their approach and adopting techniques from the new field of systems biology to integrate information from disparate sources and improve the development process. A similar revolution is required in economics if models are to reflect the nature of human economic activity and provide useful tools for policy makers. We outline the main foundations for a theory of systems economics.

Mass by numbers

A highly precise calculation of the masses of strongly interacting particles, based on fundamental theory, is testament to the age-old verity that physical reality embodies simple mathematical laws.

Black swans and white eagles: on mathematics and finance

It is often objected that we cannot use mathematical methods in finance because (1) finance is dominated by unpredictable unique events (the black swans), (2) there are qualitative effects that cannot be quantified, and (3) the laws themselves of finance keep on changing. In this paper we discuss these three objections, offering arguments to reject them. We begin by reviewing the development of the physical sciences, pointing out parallels that are relevant for our discussion. Modern science has abandoned the objective of describing reality and has adopted an operational point of view that regards physical laws as tools to connect observations. Modern science is no longer deterministic, but has accepted a fundamental uncertainty in physical laws both at micro and macroscopic levels. Unpredictable pivotal events exist in the physical sciences as well in finance but this does not lead us to question the use of mathematics in the physical sciences. On the contrary, using principles of safe design, we try to understand how to avoid and contain unpredictability. Financial markets are manmade artifacts with, as actors, a large number of interacting agents. If we so wish, we can reduce the level of uncertainty present in markets: But if we try to do so describing financial markets with simple mathematical laws, we find that these laws are not stable but change over time, eventually with sudden structural breaks. This makes the use of mathematical finance difficult but not impossible. We can forecast human decision-making processes, crucial in forecasting financial markets, at the statistical level in aggregate. From an operational point of view, we have the tools to understand and describe the behavior of large number of interacting agents. At the present stage of development of our science, we need to use the mathematics of adaptive systems, changing mathematical models in function of different market states. However, reductionism to a small number of basic laws remains a fundamental objective of financial economics as it is in the physical sciences.

Computational neuroscience grand challenges - a humble attempt at future forecast

Computational Neuroscience is a still nascent field, aspiring to imitate the great success of theoretical physics. The crucial advance of theoretical physics that transformed our understanding of the material world around us beyond recognition was the demonstration that various physical processes, no matter how complicated and rich they may appear to an observer, are governed by a well-defined set of simple principles that are formulated with the precise language of mathematical equations. Are there simple mathematical laws that capture the fundamental laws that govern the operation of neural systems? Will we one day understand the brain to a similar degree as we now understand the material world? If so, we will be able to quantitatively predict complicated cognitive behaviours under arbitrary external and internal conditions and build good brain emulators, very much as we can predict the trajectory of an asteroid or design a bridge with a given safety level. A mere acceptance of such a possibility should not be taken for granted however, as it might put us in a shaky position with respect to some deeply entrenched (and hotly debated) concepts, such as e.g. free will and related issues. This simple consideration illustrates in my view that even the ultimate goals of computational neuroscience should continually be debated and refined. Theoretical studies of the brain are pursued along several disjoint scientific approaches. We have rather accurate mathematical models of ion channels, neurons and synapses, at the level approaching physics. These are formulated with the language of dynamical systems theory, and have reasonable predictive capacity. Extending this approach to larger systems, giving rise to neural network theory, resulted in some profound and imaginative ideas about such complex cognitive phenomena as learning and memory, object recognition and spatial navigation. Predictive capacity of neural network models is however much more limited, and there are still profound uncertainties about their validity, explained among other things by still inadequate experimental techniques of measuring activity of large neuronal ensembles and anatomy of neuronal interactions. Moreover, neural networks models still have a rather poor repertoire of cognitive behaviours that they were applied to. Another approach is to view the brain as a biological implementation of a complex computer, and correspondingly to study it with the tools of computer science. In a sense, a researcher tries to formulate a certain computational task that the brain is performing, and then tries to imagine which algorithm he/she would use to perform the task on a computer. Neuronal architectures may be then used to put certain constrains on the possible algorithm, depending on the particular problem in question. Some examples of a successful application of this approach are the reinforcement theory of learning, computer vision and Bayesian inference framework for the information processing in the brain. Despite of significant achievements, this approach still appears to be largely detached from more biologically inspired modelling. Yet another direction could be called ‘data-driven approach’, who's proponents are mainly concerned with developing sophisticated techniques for analyzing multidimensional data that is generated by various experimental tools such as optical imaging and fMRI. What are the goals that computational neuroscience can realistically achieve in the observable future? I believe the first challenge is to see how far the neural networks theory can be pushed to get close to explaining high-level brain functions. Any cognitive process, vision as an example, involves multi-stage stream of processing with progressive specialized brain areas having similar overall architecture but different aspects of information representation and processing. Neural networks for the most part were confined to functionally uniform neuronal populations. The second challenge that computational neuroscience will have to meet in order to make a progress will be to partially break with the tradition of trying to explain all the richness of the brain with networks of simple elements and complex architectures. This tradition was probably a necessary first step that allowed the researchers to set the stage for the new field and borrow the techniques and intuitions developed in physics, but it resulted in a certain degree of ignorance about the incredible complexity of the neuronal hardware in the brain. It appears reasonable to predict that the full-fledged neuronal network theory will have to seriously consider the effects of neuromodulation, neuronal and synaptic adaptation on various spatial and time scales, incredible diversity of neuron types, the role of glia cells, etc etc etc. Finally, different theoretical approaches mentioned above will have to be more closely integrated into a unified theoretical framework. Thus, theoreticians who come from different scientific traditions not only should make bridges to their experimental colleagues, they should also talk much more to each other, which sometimes, paradoxically, appears to be even more challenging.

Variations in Posttonsillectomy Hemorrhage Rates Are Scale Invariant

Scale invariance is a property of scientific laws or objects that change in a prescribed fashion if measurements are scaled, and is often represented by a power-law relationship. Power laws suggest that events of a large magnitude will be rare, while small events will be much more common, and that a simple mathematical law relates "severity" with frequency. Scale invariance has been demonstrated in scientific fields including physics, social science, and economics. The authors use the complication of a posttonsillectomy hemorrhage to test whether this property is a feature of surgical complications. Non-identifiable data were obtained regarding posttonsillectomy hemorrhage and subcategorized by calendar month, and the percentage rate of posttonsillectomy hemorrhage was calculated. The data were then transformed using a logarithmic function. This transformed data were plotted and a linear regression analysis was performed. The 13-year period studied included 6,381 tonsillectomy procedures. The logarithm of the frequency of a given rate range of posttonsillectomy hemorrhage (y) was linearly related to the logarithm of the geometric mean of the rate range (x). The best-fit straight line was y = -1.3996x + 2.0624 with R2 = 0.851, n = 10, r = 0.922, and P < .001. The authors found that the incidence of posttonsillectomy hemorrhage is scale invariant. The practical implication is that the observation of rare incidences of large hemorrhage rates may not be due to a unique circumstance or a particular operative fault. To reduce the incidence of extreme rates of postoperative hemorrhage, a review of the entire process of tonsillectomy would be required. Scale-invariance analysis may represent a novel tool that should be considered when reviewing surgical complications.

Accounting for temporal variation in soil hydrological properties when simulating surface runoff on tilled plots

Tillage operations are known to greatly influence local overland flow, infiltration and depressional storage by altering soil hydraulic properties and soil surface roughness. The calibration of runoff models for tilled fields is not identical to that of untilled fields, as it has to take into consideration the temporal variability of parameters due to the transient nature of surface crusts. In this paper, we seek the application of a rainfall-runoff model and the development of a calibration methodology to take into account the impact of tillage on overland flow simulation at the scale of a tilled plot (3240 m 2) located in southern France. The selected model couples the (Morel-Seytoux, H.J., 1978. Derivation of equations for variable rainfall infiltration. Water Resources Research. 14(4), 561-568). Infiltration equation to a transfer function based on the diffusive wave equation. The parameters to be calibrated are the hydraulic conductivity at natural saturation K s, the surface detention S d and the lag time ω. A two-step calibration procedure is presented. First, eleven rainfall-runoff events are calibrated individually and the variability of the calibrated parameters are analysed. The individually calibrated K s values decrease monotonously according to the total amount of rainfall since tillage. No clear relationship is observed between the two parameters S d and ω, and the date of tillage. However, the lag time ω increases inversely with the peakflow of the events. Fairly good agreement is observed between the simulated and measured hydrographs of the calibration set. Simple mathematical laws describing the evolution of K s and ω are selected, while S d is considered constant. The second step involves the collective calibration of the law of evolution of each parameter on the whole calibration set. This procedure is calibrated on 11 events and validated on ten runoff inducing and four non-runoff inducing rainfall events. The suggested calibration methodology seems robust and can be transposed to other gauged sites.

Math connects vascular network to the universe

Endothelial cells (ECs) plated on a surface of Matrigel, a basement membrane matrix, spontaneously form a capillary network that is very similar to capillary beds formed in vivo. Now, Guido Serini, Federico Bussolino (University of Torino, Torino, Italy), Davide Ambrosi, Andrea Gamba (Polytechnic of Torino, Torino, Italy) and colleagues have used a combination of mathematical modeling and in vitro experiments to demonstrate that the key parameters in the pattern formation are the density of ECs and the biochemical properties of the chemoattractant. Figure Endothelial cells move toward increasing concentrations (blue to red) of VEGF-A. The group found that if they disrupted either variable in the cell culture system then the capillary network formed was aberrant. For example, with fewer than 100 cells/mm2, the cells form a continuous network, but more than 200 cells/mm2 led to a Swiss cheese–like mat. And when the researchers swamped out the natural VEGF-A chemoattractant gradient, which forms as a result of secretion from the ECs, the cells fail to aggregate. All of the results could be accurately modeled using a nonlinear diffusion equation, called the adhesion model, which implies that the behavior of ECs and the pattern follow simple mathematical laws. And by combining the two approaches, the group was able to define specific characteristics about the chemoattractant, such as its diffusion coefficient, that they otherwise would not have known. But perhaps most exciting, says Bussolino, is that the same type of equation can be used to describe the formation of many patterns in nature such as bacterial colony formation and Dictyostelium morphogenesis—and, oddly enough, the events that occurred just after the big bang that gave rise to the universe. ▪ Reference: Serini, G., et al. 2003. EMBO J. 22:1771–1779. [PubMed]

Regression and functional relations

SummaryRegression is frequently abused in soil research. Its proper use is for statistical prediction. It may also be used to calculate equations for calibration A regression equation may be used to express a functional relation between two soil variables that are thought to be dated by some simple mathematical law but only where one of the variables is known exactly. In most other circumstances regression is inappropriate. Where departures from a functional relation are a result of errors of measurement and sampling fluctuation it should be replaced by a structural analysis to find the best equation. Where the underlying relation is truly bivariate it should be described as such.

A graphical treatment of the tone reproduction problem in television systems

The impracticability of analysing the tone reproduction properties of a television system in terms of simple mathematical laws is discussed. A construction for a tone reproduction chart is presented which enables a realistic assessment of the overall performance of a system to be made, subjective effects being taken into account. The use of the chart is demonstrated by examining the effects of altering certain parameters of the system.

On the selection of subsequences

The problem we shall discuss in the following is concerned with the possibility of extracting from a given sequence a subsequence whose terms obey some relatively simple mathematical law. As a prototype of such questions one may take the prime representing function determined by W. H. Mills [3 ] which has the property of defining within the sequence of all prime numbers a subsequence whose terms have a mathematical expression of simple type. We shall establish a general theorem of this kind which makes it possible to formulate selection laws for a great variety of sequences when only quite weak conditions are prescribed in advance. Among the various applications of this result let us mention especially its bearing upon the so-called frequency definition of probability. Here, as one may recall, the natural symbol for the repetition of an event is taken to be a sequence of signs s and f, denoting success and failure. Such a sequence can equally well be described by a series of increasing integers

The Form of the Atoms in Relation to their Spectra

SINCE Balmer's important discovery in 1885 that it is possible to calculate the wave-lengths of the first nine lines of the hydrogen spectrum by means of a simple formula, the existence of series of lines, obeying simple mathematical laws, has been established in the case of the spectra of several other elements, notably by the researches of Rydberg and of Kayser and Runge. Among the various attempts that have been made to account for these series of lines, and, in general, for the different spectra, the most promising seems to be that of Prof. F. von Lindemann, of Munich, who in some recent papers1 has more fully developed an idea that he was first led to enunciate in 1888, after the publication of Lord Kelvin's Baltimore lectures on molecular dynamics. Prof. von Lindemann's method consists, not in deriving an empirical relationship between the wavelengths or frequencies of the spectral lines, but in investigating mathematically the possible waves which a hypothetical atom can send out into the luminiferous ether.