Array programming with NumPy.

Array programming provides a powerful, compact and expressive syntax for accessing, manipulating and operating on data in vectors, matrices and higher-dimensional arrays. NumPy is the primary array programming library for the Python language. It has an essential role in research analysis pipelines in fields as diverse as physics, chemistry, astronomy, geoscience, biology, psychology, materials science, engineering, finance and economics. For example, in astronomy, NumPy was an important part of the software stack used in the discovery of gravitational waves1 and in the first imaging of a black hole2. Here we review how a few fundamental array concepts lead to a simple and powerful programming paradigm for organizing, exploring and analysing scientific data. NumPy is the foundation upon which the scientific Python ecosystem is constructed. It is so pervasive that several projects, targeting audiences with specialized needs, have developed their own NumPy-like interfaces and array objects. Owing to its central position in the ecosystem, NumPy increasingly acts as an interoperability layer between such array computation libraries and, together with its application programming interface (API), provides a flexible framework to support the next decade of scientific and industrial analysis.

Author Correction: SciPy 1.0: fundamental algorithms for scientific computing in Python.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

SciPy 1.0: fundamental algorithms for scientific computing in Python.

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

Restricted ADP movement in cardiomyocytes: Cytosolic diffusion obstacles are complemented with a small number of open mitochondrial voltage-dependent anion channels.

Adequate intracellular energy transfer is crucial for proper cardiac function. In energy starved failing hearts, partial restoration of energy transfer can rescue mechanical performance. There are two types of diffusion obstacles that interfere with energy transfer from mitochondria to ATPases: mitochondrial outer membrane (MOM) with voltage-dependent anion channel (VDAC) permeable to small hydrophilic molecules and cytoplasmatic diffusion barriers grouping ATP-producers and -consumers. So far, there is no method developed to clearly distinguish the contributions of cytoplasmatic barriers and MOM to the overall diffusion restriction. Furthermore, the number of open VDACs in vivo remains unknown. The aim of this work was to establish the partitioning of intracellular diffusion obstacles in cardiomyocytes. We studied the response of mitochondrial oxidative phosphorylation of permeabilized rat cardiomyocytes to changes in extracellular ADP by recording 3D image stacks of NADH autofluorescence. Using cell-specific mathematical models, we determined the permeability of MOM and cytoplasmatic barriers. We found that only ~2% of VDACs are accessible to cytosolic ADP and cytoplasmatic diffusion barriers reduce the apparent diffusion coefficient by 6-10×. In cardiomyocytes, diffusion barriers in the cytoplasm and by the MOM restrict ADP/ATP diffusion to similar extents suggesting a major role of both barriers in energy transfer and other intracellular processes.

Real-time determination of sarcomere length of a single cardiomyocyte during contraction.

Sarcomere length of a cardiomyocyte is an important control parameter for physiology studies on a single cell level; for instance, its accurate determination in real time is essential for performing single cardiomyocyte contraction experiments. The aim of this work is to develop an efficient and accurate method for estimating a mean sarcomere length of a contracting cardiomyocyte using microscopy images as an input. The novelty in developed method lies in 1) using unbiased measure of similarities to eliminate systematic errors from conventional autocorrelation function (ACF)-based methods when applied to region of interest of an image, 2) using a semianalytical, seminumerical approach for evaluating the similarity measure to take into account spatial dependence of neighboring image pixels, and 3) using a detrend algorithm to extract the sarcomere striation pattern content from the microscopy images. The developed sarcomere length estimation procedure has superior computational efficiency and estimation accuracy compared with the conventional ACF and spectral analysis-based methods using fast Fourier transform. As shown by analyzing synthetic images with the known periodicity, the estimates obtained by the developed method are more accurate at the subpixel level than ones obtained using ACF analysis. When applied in practice on rat cardiomyocytes, our method was found to be robust to the choice of the region of interest that may 1) include projections of carbon fibers and nucleus, 2) have uneven background, and 3) be slightly disoriented with respect to average direction of sarcomere striation pattern. The developed method is implemented in open-source software.

Molecular dynamics simulations of creatine kinase and adenine nucleotide translocase in mitochondrial membrane patch.

Interaction between mitochondrial creatine kinase (MtCK) and adenine nucleotide translocase (ANT) can play an important role in determining energy transfer pathways in the cell. Although the functional coupling between MtCK and ANT has been demonstrated, the precise mechanism of the coupling is not clear. To study the details of the coupling, we turned to molecular dynamics simulations. We introduce a new coarse-grained molecular dynamics model of a patch of the mitochondrial inner membrane containing a transmembrane ANT and an MtCK above the membrane. The membrane model consists of three major types of lipids (phosphatidylcholine, phosphatidylethanolamine, and cardiolipin) in a roughly 2:1:1 molar ratio. A thermodynamics-based coarse-grained force field, termed MARTINI, has been used together with the GROMACS molecular dynamics package for all simulated systems in this work. Several physical properties of the system are reproduced by the model and are in agreement with known data. This includes membrane thickness, dimension of the proteins, and diffusion constants. We have studied the binding of MtCK to the membrane and demonstrated the effect of cardiolipin on the stabilization of the binding. In addition, our simulations predict which part of the MtCK protein sequence interacts with the membrane. Taken together, the model has been verified by dynamical and structural data and can be used as the basis for further studies.

Sensitivity analysis of flux determination in heart by H2 ¹8O -provided labeling using a dynamic Isotopologue model of energy transfer pathways.

To characterize intracellular energy transfer in the heart, two organ-level methods have frequently been employed: ³¹P − NMR inversion and saturation transfer, and dynamic ¹8O labeling. Creatine kinase (CK) fluxes obtained by following oxygen labeling have been considerably smaller than the fluxes determined by ³¹P − NMR saturation transfer. It has been proposed that dynamic ¹8O labeling determines net flux through CK shuttle, whereas ³¹P − NMR saturation transfer measures total unidirectional flux. However, to our knowledge, no sensitivity analysis of flux determination by oxygen labeling has been performed, limiting our ability to compare flux distributions predicted by different methods. Here we analyze oxygen labeling in a physiological heart phosphotransfer network with active CK and adenylate kinase (AdK) shuttles and establish which fluxes determine the labeling state. A mathematical model consisting of a system of ordinary differential equations was composed describing ¹8O enrichment in each phosphoryl group and inorganic phosphate. By varying flux distributions in the model and calculating the labeling, we analyzed labeling sensitivity to different fluxes in the heart. We observed that the labeling state is predominantly sensitive to total unidirectional CK and AdK fluxes and not to net fluxes. We conclude that measuring dynamic incorporation of ¹8O into the high-energy phosphotransfer network in heart does not permit unambiguous determination of energetic fluxes with a higher magnitude than the ATP synthase rate when the bidirectionality of fluxes is taken into account. Our analysis suggests that the flux distributions obtained using dynamic ¹8O labeling, after removing the net flux assumption, are comparable with those from ³¹P − NMR inversion and saturation transfer.

Analysis of molecular movement reveals latticelike obstructions to diffusion in heart muscle cells.

Intracellular diffusion in muscle cells is known to be restricted. Although characteristics and localization of these restrictions is yet to be elucidated, it has been established that ischemia-reperfusion injury reduces the overall diffusion restriction. Here we apply an extended version of raster image correlation spectroscopy to determine directional anisotropy and coefficients of diffusion in rat cardiomyocytes. Our experimental results indicate that diffusion of a smaller molecule (1127 MW fluorescently labeled ATTO633-ATP) is restricted more than that of a larger one (10,000 MW Alexa647-dextran), when comparing diffusion in cardiomyocytes to that in solution. We attempt to provide a resolution to this counterintuitive result by applying a quantitative stochastic model of diffusion. Modeling results suggest the presence of periodic intracellular barriers situated ∼1 µm apart having very low permeabilities and a small effect of molecular crowding in volumes between the barriers. Such intracellular structuring could restrict diffusion of molecules of energy metabolism, reactive oxygen species, and apoptotic signals, enacting a significant role in normally functioning cardiomyocytes as well as in pathological conditions of the heart.

Bidirectionality and compartmentation of metabolic fluxes are revealed in the dynamics of isotopomer networks.

Isotope labeling is one of the few methods of revealing the in vivo bidirectionality and compartmentalization of metabolic fluxes within metabolic networks. We argue that a shift from steady state to dynamic isotopomer analysis is required to deal with these cellular complexities and provide a review of dynamic studies of compartmentalized energy fluxes in eukaryotic cells including cardiac muscle, plants, and astrocytes. Knowledge of complex metabolic behaviour on a molecular level is prerequisite for the intelligent design of genetically modified organisms able to realize their potential of revolutionizing food, energy, and pharmaceutical production. We describe techniques to explore the bidirectionality and compartmentalization of metabolic fluxes using information contained in the isotopic transient, and discuss the integration of kinetic models with MFA. The flux parameters of an example metabolic network were optimized to examine the compartmentalization of metabolites and and the bidirectionality of fluxes in the TCA cycle of Saccharomyces uvarum for steady-state respiratory growth.

Cross-Bridge Group Ensembles Describing Cooperativity in Thermodynamically Consistent Way.

The aim of this work is to incorporate cooperativity into Huxley-type cross-bridge model in thermodynamically consistent way. While the Huxley-type models assume that cross-bridges act independently from each other, we take into account that each cross-bridge is influenced by its neighbors and cooperativity is induced by tropomyosin movement. For that, we introduce ensembles of cross-bridge groups connected by elastic tropomyosin. By taking into account that the mechanical displacement of tropomyosin induces free energy change of the cross-bridge group ensemble, we develop the formalism for thermodynamically consistent description of the cooperativity in muscle contraction. An example model was composed to test the approach. The model parameters were found by optimization from the linear relation between oxygen consumption and stress-strain area as well as experimentally measured stress dynamics of rat trabecula. We have found a good agreement between the optimized model solution and experimental data. Simulations also showed that it is possible to study cooperativity with the approach developed in this work.

Tight coupling of Na+/K+-ATPase with glycolysis demonstrated in permeabilized rat cardiomyocytes.

The effective integrated organization of processes in cardiac cells is achieved, in part, by the functional compartmentation of energy transfer processes. Earlier, using permeabilized cardiomyocytes, we demonstrated the existence of tight coupling between some of cardiomyocyte ATPases and glycolysis in rat. In this work, we studied contribution of two membrane ATPases and whether they are coupled to glycolysis--sarcoplasmic reticulum Ca2+ ATPase (SERCA) and plasmalemma Na+/K+-ATPase (NKA). While SERCA activity was minor in this preparation in the absence of calcium, major role of NKA was revealed accounting to ∼30% of the total ATPase activity which demonstrates that permeabilized cell preparation can be used to study this pump. To elucidate the contribution of NKA in the pool of ATPases, a series of kinetic measurements was performed in cells where NKA had been inhibited by 2 mM ouabain. In these cells, we recorded: ADP- and ATP-kinetics of respiration, competition for ADP between mitochondria and pyruvate kinase (PK), ADP-kinetics of endogenous PK, and ATP-kinetics of total ATPases. The experimental data was analyzed using a series of mathematical models with varying compartmentation levels. The results show that NKA is tightly coupled to glycolysis with undetectable flux of ATP between mitochondria and NKA. Such tight coupling of NKA to PK is in line with its increased importance in the pathological states of the heart when the substrate preference shifts to glucose.

Symbolic flux analysis for genome-scale metabolic networks.

BACKGROUND: With the advent of genomic technology, the size of metabolic networks that are subject to analysis is growing. A common task when analyzing metabolic networks is to find all possible steady state regimes. There are several technical issues that have to be addressed when analyzing large metabolic networks including accumulation of numerical errors and presentation of the solution to the researcher. One way to resolve those technical issues is to analyze the network using symbolic methods. The aim of this paper is to develop a routine that symbolically finds the steady state solutions of large metabolic networks. RESULTS: A symbolic Gauss-Jordan elimination routine was developed for analyzing large metabolic networks. This routine was tested by finding the steady state solutions for a number of curated stoichiometric matrices with the largest having about 4000 reactions. The routine was able to find the solution with a computational time similar to the time used by a numerical singular value decomposition routine. As an advantage of symbolic solution, a set of independent fluxes can be suggested by the researcher leading to the formation of a desired flux basis describing the steady state solution of the network. These independent fluxes can be constrained using experimental data. We demonstrate the application of constraints by calculating a flux distribution for the central metabolic and amino acid biosynthesis pathways of yeast. CONCLUSIONS: We were able to find symbolic solutions for the steady state flux distribution of large metabolic networks. The ability to choose a flux basis was found to be useful in the constraint process and provides a strong argument for using symbolic Gauss-Jordan elimination in place of singular value decomposition.