Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution behaves asymptotically like the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briefly review first the main theoretical facts, describe the main ingredients of the algorithm, highlight the essential data structures, and illustrate the results by one- and two-dimensional numerical examples including comparisons with an adaptive finite element scheme.

Connecting the dots between the protein sequence and its function is of fundamental interest for protein engineers. In-silico methods are useful in this quest especially when structural information is not available. In this study we propose a mutant library screening tool called iSAR (innovative Sequence Activity Relationship) that relies on the physicochemical properties of the amino acids, digital signal processing and partial least squares regression to uncover these sequence-function correlations. We show that the digitalized representation of the protein sequence in the form of a Fourier spectrum can be used as an efficient descriptor to model the sequence-activity relationship of proteins. The iSAR methodology that we have developed identifies high fitness mutants from mutant libraries relying on physicochemical properties of the amino acids, digital signal processing and regression techniques. iSAR correlates variations caused by mutations in spectra with biological activity/fitness. It takes into account the impact of mutations on the whole spectrum and does not focus on local fitness alone. The utility of the method is illustrated on 4 datasets: cytochrome P450 for thermostability, TNF-alpha for binding affinity, GLP-2 for potency and enterotoxins for thermostability. The choice of the datasets has been made such as to illustrate the ability of the method to perform when limited training data is available and also when novel mutations appear in the test set, that have not been featured in the training set. The combination of Fast Fourier Transform and Partial Least Squares regression is efficient in capturing the effects of mutations on the function of the protein. iSAR is a fast algorithm which can be implemented with limited computational resources and can make effective predictions even if the training set is limited in size.

In directed evolution experiments, it is at stake to have methods to screen efficiently the mutant libraries. We propose a web-based tool that implements an established in silico method for the rational screening of mutant libraries. The method, known as ProSAR, attempts to link sequence data to activity. The method uses statistical models trained on small experimental datasets provided by the user. These can integrate potential epistatic interactions between mutations and be used in many diverse biological contexts. It drastically improves the search for leading mutants. The tool is freely available to non-commercial users at http://bo-protscience.fr/prosar/.

The selection of optimal enzyme concentration in multienzyme cascade reactions for the highest product yield in practice is very expensive and time-consuming process. The modelling of biological pathways is a difficult process because of the complexity of the system. The mathematical modelling of the system using an analytical approach depends on the many parameters of enzymes which rely on tedious and expensive experiments. The artificial neural network (ANN) method has been successively applied in different fields of science to perform complex functions. In this study, ANN models were trained to predict the flux for the upper part of glycolysis as inferred by NADH consumption, using four enzyme concentrations i.e., phosphoglucoisomerase, phosphofructokinase, fructose-bisphosphate-aldolase, triose-phosphate-isomerase. Out of three ANN algorithms, the neuralnet package with two activation functions, “logistic” and “tanh” were implemented. The prediction of the flux was very efficient: RMSE and R² were 0.847, 0.93 and 0.804, 0.94 respectively for logistic and tanh functions using a cross validation procedure. This study showed that a systemic approach such as ANN could be used for accurate prediction of the flux through the metabolic pathway. This could help to save a lot of time and costs, particularly from an industrial perspective. The R-code is available at: https://github.com/DSIMB/ANN-Glycolysis-Flux-Prediction.

The metabolic engineering of pathways has been used extensively to produce molecules of interest on an industrial scale. Methods like gene regulation or substrate channeling helped to improve the desired product yield. Cell-free systems are used to overcome the weaknesses of engineered strains. One of the challenges in a cell-free system is selecting the optimized enzyme concentration for optimal yield. Here, a machine learning approach is used to select the enzyme concentration for the upper part of glycolysis. The artificial neural network approach (ANN) is known to be inefficient in extrapolating predictions outside the box: high predicted values will bump into a sort of “glass ceiling”. In order to explore this “glass ceiling” space, we developed a new methodology named glass ceiling ANN (GC-ANN). Principal component analysis (PCA) and data classification methods are used to derive a rule for a high flux, and ANN to predict the flux through the pathway using the input data of 121 balances of four enzymes in the upper part of glycolysis. The outcomes of this study are i. in silico selection of optimum enzyme concentrations for a maximum flux through the pathway and ii. experimental in vitro validation of the “out-of-the-box” fluxes predicted using this new approach. Surprisingly, flux improvements of up to 63% were obtained. Gratifyingly, these improvements are coupled with a cost decrease of up to 25% for the assay.

Metabolic pathway modeling plays an increasing role in drug design by allowing better understanding of the underlying regulation and controlling networks in the metabolism of living organisms. However, despite rapid progress in this area, pathway modeling can become a real nightmare for researchers, notably when few experimental data are available or when the pathway is highly complex. Here, three different approaches were developed to model the second part of glycolysis of E. histolytica as an application example, and have succeeded in predicting the final pathway flux: one including detailed kinetic information (white-box), another with an added adjustment term (grey-box) and the last one using an artificial neural network method (black-box). Afterwards, each model was used for metabolic control analysis and flux control coefficient determination. The first two enzymes of this pathway are identified as the key enzymes playing a role in flux control. This study revealed the significance of the three methods for building suitable models adjusted to the available data in the field of metabolic pathway modeling, and could be useful to biologists and modelers.