Interval prediction is crucial in decision-making processes across many domains. Although significant progress has been made in existing interval prediction methods, they still face several challenges, such as assumptions about data distribution, fixed interval widths, limitations of gradient-based optimization algorithm, crossing of upper and lower bounds, and insufficient consideration of multi-scale spatial-temporal patterns. To address these issues, we propose a Multi-Scale Deep Interval Prediction Network (MSDIPN). Specifically, a MultiScale Spatio-Temporal Self-Attention Mechanism is introduced to capture spatio-temporal dependencies across different spatial scales. Additionally, a Temporal Self-Attention Mechanism module is constructed to extract temporal dependencies of historical variables across varying lag phases. Then a Global Self-Attention Mechanism module is designed to address representation degradation using residual connections and self-attention mechanisms. To overcome limitations related to distributional assumptions, fixed interval widths, and crossing problems, an Improved LUBE module is developed as the output module for generating prediction intervals (PIs) of time series data. Furthermore, a gradientbased PIs loss function is designed to address the optimization issue of MSDIPN by integrating a smooth approximation function with a pinball loss function. We validate the effectiveness of the proposed algorithm using five real-world datasets, demonstrating its superiority over traditional models

Feature selection has been extensively applied to identify cancer genes using omics data. Although substantial studies have been conducted to search for cancer genes, the available rich knowledge on various cancers is seldom used as prior information in feature selection. This paper proposes a two-stage prior LASSO (TSPLASSO) method, which represents an early attempt in designing feature selection algorithms using prior information. The first stage performs gene selection via linear regression with LASSO. Candidate genes that are correlated with known cancer genes are retained for subsequent analysis. The second stage establishes a logistic regression model with LASSO to realize final cancer gene selection and sample classification. The key advantages of TSPLASSO include the successive consideration of prior cancer genes and binary sample types as response variables in stages one and two, respectively. In addition, the TSPLASSO performs sample classification and variable selection simultaneously. Compared with six state-of-the-art algorithms, numerical simulations in six real-world datasets show that TSPLASSO can improve the accuracy of variable selection by 5%-400% in the three bulk sequencing datasets and the scRNA-seq dataset; and the performance is robust against data noise and variations of prior cancer genes. The TSPLASSO provides an efficient, stable and practical algorithm for exploring biomedcial and health informatics from omics data.

The insulin-like growth factor (IGF) receptor signaling pathway plays a crucial role in regulating growth, development, and metabolism in the human body. It encompasses a group of signalling molecules, receptors, and downstream signaling pathways that are involved in various physiological processes. Given the complexity of molecular processes within this pathway, mathematical models have emerged as potent tools for dissecting the dynamic behaviors of this signaling pathways. To address the heterogeneity of cellular signalling observed in single-cell studies, the nonlinear mixed-effects model has been used recently to generate different simulations that represent the molecular dynamics in different cells. In this study, we use a nonlinear mixed-effects ordinary differential equation model, based on an established ordinary differential equation model, to generate simulations for the IGFR signaling pathway. Simulations shows that, when the AKT and mitogen-activated protein (MAP) kinase have different activities, other kinases in the pathway have various dynamic properties. We also test the influence of two inhibitors for the MAP kinase pathway and PI3K/AKT pathway separately. Simulation results are consistent with the regulatory mechanisms of each pathway as well as the cross-talk between the MAP kinase pathway and AKT pathway. This study provides an alternative method to study the functions of model parameter variations on the dynamics of complex systems.

Stochastic modelling is an important method to investigate the functions of noise in a wide range of biological systems. However, the parameter inference for stochastic models is still a challenging problem partially due to the large computing time required for stochastic simulations. To address this issue, we propose a novel early-rejection method by using sequential stochastic simulations. We first show that a large number of stochastic simulations are required to obtain reliable inference results. Instead of generating a large number of simulations for each parameter sample, we propose to generate these simulations in a number of stages. The simulation process will go to the next stage only if the accuracy of simulations at the current stage satisfies a given error criterion. We propose a formula to determine the error criterion and use a stochastic differential equation model to examine the effects of different criteria. Three biochemical network models are used to evaluate the efficiency and accuracy of the proposed method. Numerical results suggest the proposed early-rejection method achieves substantial improvement in the efficiency for the inference of stochastic models.

The advances of single-cell technologies provide un-precedented opportunities to investigate heterogeneity of cellular processes. Although population modelling is a powerful tool to describe different cellular dynamics in a number of cells, it is a challenge and computational burden to infer different sets of parameters in the mathematical models based on the observations in different cells. To address this issue, this work proposes a population Monte Carlo framework to increase the acceptance rates in the approximate Bayesian computation algorithms. We first test a prior distribution and distinct tolerance threshold sequence for the dataset of the first cell. After applying the Bayesian inference methods, we obtain the estimated model parameters of the first cell. This inferred parameter set will be used to construct the prior distribution and tolerance threshold sequence of the following cells. This adaptive approach will be repeatedly applied for the inference of the following cells. The accuracy and efficacy of the proposed algorithms is rigorously examined by using two mathematical models for genetic networks. Numerical results shows that the proposed algorithm improve the efficiency substantially and provide a powerful tool to obtain more accurate inference results for large-scale regulatory networks.

Online learning methods are designed to establish timely predictive models for machine learning problems. The methods for online learning of nonlinear systems are usually developed in the reproducing kernel Hilbert space (RKHS) associated with Gaussian kernel in which the kernel bandwidth is manually selected and remains steady during the entire modeling process in most cases. This setting may make the learning model rigid and inappropriate for complex data streams. Since the bandwidth appears in a nonlinear term of the kernel model, it raises substantial challenges in the development of learning methods with an adaptive bandwidth. In this article, we propose a novel approach to address this important open issue. By a carefully casted linearization scheme, the nonlinear learning problem is reasonably transformed into a state feedback control problem for a series of controllable systems. Then, by employing optimal control techniques, an effective algorithm is developed, and the parameters in the learning model including kernel bandwidth can be efficiently updated in a real-time manner. By taking advantage of the particular structure of the Gaussian kernel model, a theoretical analysis on the convergence and rationality of the proposed method is also provided. Compared with the kernel algorithms with a fixed bandwidth, our novel learning framework can not only achieve adaptive learning results with a better prediction accuracy but also show performance that is more robust with a faster convergence speed. Encouraging numerical results are provided to demonstrate the advantages of our new method.

We want to find a tree where the path length between any two vertices on this tree is as close as possible to their corresponding distance in the complete weighted graph of vertices upon which the tree is built.We use the residual sum of squares as the optimality criterion to formulate this problem, and use the Cholesky decomposition to solve the system of linear equations to optimize weights of a given tree.We also use two metaheuristics, namely Simulated Annealing (SA) and Iterated Local Search (ILS) to optimize the tree structure.Our results suggest that SA and ILS both perform well at finding the optimal tree structure when the dispersion of distances in the complete graph is large.However, when the dispersion of distances is small, only ILS has a solid performance.

It is widely accepted that experimental data often include noise because of the limitation in experimental conditions. In addition, biological systems inside the cells also contain uncertainty due to small copy molecular numbers. To address this issue, it was proposed that experimental data include both real system state and a noise term whose variance is a constant. An additional assumption is that the observation data of different variables are independent to each other. However, recent research works showed that noise in experimental data might not be the white noise. In addition, the observed values of different variables may be correlated. This work designs a new algorithm to infer the unknown model parameters based on noisy data. The innovation of this method includes a new noise model, in which the variance of noise is dependent on the system state, and a copula particle filter algorithm that uses the copula density functions to describe the dependence of different variables. The proposed algorithm is evaluated by using two deterministic models for gene networks and a stochastic model. Numerical results show that the accuracy of our proposed method is better than that of the widely used Liu-West filter and copula particle filter algorithms.

In this paper, we propose a novel kernel method for the online identification of stochastic nonlinear spatiotemporal dynamical systems using the robust control approach. By the difference method, the stochastic spatiotemporal (SST) systems driven by multiplicative noise are first transformed into a class of multi-input-multi-output-partially linear kernel models (PLKMs) with heterogeneous random terms. With the help of techniques established for reproducing kernel Hilbert space, the online learning problem is reasonably considered as an output feedback control problem for a group of time varying linear dynamical systems. We develop an effective algorithm to address the learning problem of PLKM and SST systems by employing the model predictive control theory. Compared with the existing learning methods, the new one can achieve adaptive, robust, and fast convergent online modeling performance for the spatiotemporal dynamics with multiplicative noise, which greatly facilitates the characterization of physical characteristics of the system. Moreover, this investigation for the first time addresses the learning problems for SST systems with novel robust control techniques, which can provide some novel insights into the design of kernel machine learning methods from the perspective of optimal control theory. Numerical studies for benchmark systems are presented to illustrate the effectiveness and efficiency of our new method.

The six papers in this special section were presented at the IEEE BIBM 2015 conference that was held in Washington, D.C., November 9-12, 2015. The scientific program highlighted five themes to provide breadth, depth, and synergy for research collaboration: (1) genomics and molecular structure, function, and evolution; (2) computational systems biology; (3) medical informatics and translational bioinformatics; (4) cross-cutting computational methods and bioinformatics infrastructures; and (5) healthcare informatics,

Cancer of the pancreas is a highly lethal disease and has an extremely poor prognosis. Mathematical modelling and computer simulations have been proposed as important tool to predictor initiation and progression of cancer diseases, which are very important in cancer study. Among these studies, it is widely assumed that the gene mutation rate is unchanged, which is not realistic based on recently biological and medical studies. In this work, we present a new approach using non-constant mutation rate and hence reveal several important biological parameters of cancer progression. Under more realistic assumptions regarding gene mutation and a more reasonable mutation rate, our proposed model and calculated results may provide insights into dynamics of cancer metastasis and have clinic implications.

Biological experimental data normally contain noise due to probabilistic character of biochemical reactions and environmental fluctuations. It has been widely assumed that the observed data is the corresponding system states plus a white noise whose variance is a constant. In addition, all observations are assumed to be independent to each other. However, recently biological studies suggested that the randomness in experimental data may depend on system state and observations of different variables may be highly correlated. To address these issues, this work proposes a new model in which the variance of noise is a function of system state. We design a copula particle filter algorithm that is characterized by using copula density functions in place of multivariable normal density functions. The combination of the noise model and copula particle filter leads to a novel algorithm whose performance is rigorously evaluated by inferring unknown parameters in mathematical models. We test three linear/nonlinear functions to fit the noisy data and numerical results suggest that the nonlinear sigmoid function is the best function to represent the state-dependent variance of noise. Our proposed method has better accuracy of estimated parameters than the widely use the Liu-West filter and copula particle filter algorithm. Numerical results suggest that our proposed method is superior to the other two widely filter algorithms and is very effective for parameter estimation in biochemical network models under noisy dependent observations.

Welcome to the 2016 IEEE International Conference on Bioinformatics and Biomedicine (IEEE BIBM 2016) being held in the Shenzhen, China from December 15–18, 2016. On behalf of the IEEE BIBM 2016 Organizing Team, we would like to thank you for your participation and hope you enjoy the conference.

Single cell experimental studies provide an unprecedented opportunity to examine the heterogeneity of molecular processes in different cells. However, the reconstruction of a sequence of changes in molecular processes and development of regulatory networks using single cell data are still challenging problems in bioinformatics and systems biology. In this work we propose an integrated framework to infer genetic regulatory networks using single cell experimental data. We first use the Wanderlust algorithm to construct the pseudo-trajectory of gene expression activities. Due to noise in the expression data, a Gauss process regression method is employed to produce a smoothly trajectory. Our integrated approach includes both a top-down approach (i.e. the GENIE3 algorithm) to infer the network structure and a bottom-up approach (i.e. differential equation model) to reverse-engineering the regulatory network. Using the gene network of hematopoietic development in the mouse embryo as the test problem, we developed a dynamic model for a network of nine genes. Our results suggest that the proposed integrated framework is an effective approach to reconstruct regulatory networks from single cell data.

Metastasis is responsible for 90% of cancer deaths from solid tumors. Although it is well known that metastasis is the result of a series of biological hurdles, it is still a challenge to calculate the probability of metastasis at diagnosis. Recently a new method has been developed based on metastasis formation to investigate the progression of pancreatic cancer. However, the exact solution of this model was difficult to find and hence there is an issue regarding the prediction accuracy of this model. To remedy it, we present in this work an alternative formula which can be easily solved by employing a simple programming and our value is consistent with those predicted by the published model.

Investigating the dynamics of genetic regulatory networks through high throughput experimental data, such as microarray gene expression profiles, is very important but challenging. One of the major hindrances in building detailed mathematical models for genetic regulation is the large number of unknown model parameters. To tackle this problem, a new integrated method is proposed by combining both the top-down and bottom-up approaches. Firstly, a top-down approach, using probability graphical models, is employed to predict the network structure of DNA repair pathway that involves p53 regulation. Then, a bottom-up approach, using differential equation models, is applied to study the detailed genetic regulations based on either a fully-connected regulatory network or gene networks inferred with the top-down approach. Optimal network is selected based on model simulation error and robustness property. Overall, the proposed new integrated method is efficient for studying large dynamical genetic regulations.

BACKGROUND: The advances of systems biology have raised a large number of sophisticated mathematical models for describing the dynamic property of complex biological systems. One of the major steps in developing mathematical models is to estimate unknown parameters of the model based on experimentally measured quantities. However, experimental conditions limit the amount of data that is available for mathematical modelling. The number of unknown parameters in mathematical models may be larger than the number of observation data. The imbalance between the number of experimental data and number of unknown parameters makes reverse-engineering problems particularly challenging. RESULTS: To address the issue of inadequate experimental data, we propose a continuous optimization approach for making reliable inference of model parameters. This approach first uses a spline interpolation to generate continuous functions of system dynamics as well as the first and second order derivatives of continuous functions. The expanded dataset is the basis to infer unknown model parameters using various continuous optimization criteria, including the error of simulation only, error of both simulation and the first derivative, or error of simulation as well as the first and second derivatives. We use three case studies to demonstrate the accuracy and reliability of the proposed new approach. Compared with the corresponding discrete criteria using experimental data at the measurement time points only, numerical results of the ERK kinase activation module show that the continuous absolute-error criteria using both function and high order derivatives generate estimates with better accuracy. This result is also supported by the second and third case studies for the G1/S transition network and the MAP kinase pathway, respectively. This suggests that the continuous absolute-error criteria lead to more accurate estimates than the corresponding discrete criteria. We also study the robustness property of these three models to examine the reliability of estimates. Simulation results show that the models with estimated parameters using continuous fitness functions have better robustness properties than those using the corresponding discrete fitness functions. CONCLUSIONS: The inference studies and robustness analysis suggest that the proposed continuous optimization criteria are effective and robust for estimating unknown parameters in mathematical models.

BACKGROUND: Mathematical modeling is an important tool in systems biology to study the dynamic property of complex biological systems. However, one of the major challenges in systems biology is how to infer unknown parameters in mathematical models based on the experimental data sets, in particular, when the data are sparse and the regulatory network is stochastic. RESULTS: To address this issue, this work proposed a new algorithm to estimate parameters in stochastic models using simulated likelihood density in the framework of approximate Bayesian computation. Two stochastic models were used to demonstrate the efficiency and effectiveness of the proposed method. In addition, we designed another algorithm based on a novel objective function to measure the accuracy of stochastic simulations. CONCLUSIONS: Simulation results suggest that the usage of simulated likelihood density improves the accuracy of estimates substantially. When the error is measured at each observation time point individually, the estimated parameters have better accuracy than those obtained by a published method in which the error is measured using simulations over the entire observation time period.

Tumorigenesis can be regarded as an evolutionary process, in which the transformation of a normal cell into a tumor cell involves a number of limiting genetic and epigenetic events. To study the progression process, a time scheme has been presented for colorectal cancer through micro-adenoma, small-adenoma, large-adenoma early carcinoma, advanced carcinoma and metastasis processes by an extensive clinical investigation. In addition, a mathematical model has been designed to describe this biological process. It is a challenge to calculate the time required for the first mutation occurs and to determine the influence of tumor size on the mutation rate. In this work we present a general framework to remedy the shortcoming of existing models. By matching the clinical cancer time scheme, we determine the values of a number of parameters, including the selective advantage of cancer cells and initial mutation rate for individual patients. The averaged values of doubling time and selective advantage coefficient generated by our model are consistent with the predictions made by the published models.

To study the dynamic properties of complex biological systems, mathematical modeling has been used widely in systems biology. Apart from the well-established knowledge for modeling techniques, there are still some difficulties while understanding the dynamics in system biology. One of the major challenges is how to infer unknown parameters in mathematical models based on the experimentally observed data sets. This is extremely difficult when the experimental data are sparse and the biological systems are stochastic. To tackle this problem, in this work we revised one computation method for inference called approximate Bayesian computation (ABC) and conducted extensive computing tests to examine the influence of a number of factors on the performance of ABC. Based on simulation results, we found that the number of stochastic simulations and step size of the observation data have substantial influence on the estimation accuracy. We applied the ABC method to two stochastic systems to test the efficiency and effectiveness of the ABC and obtained promising approximation for the unknown parameters in the systems. This work raised a number of important issues for designing effective inference methods for estimating rate constants in biochemical reaction systems.

The development of simple mathematical model for representing complicated real-life chemical reaction systems has been a fundamental issue in computational biology and bioinformatics. In particular, the accurate description of chemical events with multi-step chemical reactions has been regarded as an essential problem in chemistry and biophysics. To model chemical reaction systems in a manageable way, multi-step chemical reactions were normally simplified into a one-step reaction. In recent years, a number of modelling approaches have been attempted to use simplified model to describe multi-step chemical reactions accurately. In this work, we proposed a two-variable model to describe chemical events with multi-step chemical reactions. We introduced a new concept to represent the location of molecules in the multi-step reactions, and use it as the second indicator of the system dynamics. The accuracy of the proposed new model was evaluated via using a deterministic model. The proposed model has been applied to study the mRNA degradation process. Numerical simulations of the designed simplified models matched the simulations of multi-step chemical reactions very well.

Message RNA (mRNA) is the template for protein synthesis. It carries information from DNA in the nucleus to the ribosome sites of protein synthesis in the cell. The turnover process of mRNA is a chemical event with multiple small step reactions; and the degradation of mRNA molecules is an important step in gene expression. A number of mathematical models have been proposed to study the dynamics of mRNA turnover, ranging from a one-step first order reaction model to the linear multicomponent models. Although the linear multicomponent models provide detailed dynamics of mRNA degradation, the simple first-order reaction model has been widely used in mathematical modeling of genetic regulatory networks. To illustrate the difference between these models, we first considered a stochastic model based on the multicomponent model. Then a simpler linear chain stochastic model was proposed to approximate the linear multicomponent model. We also discussed the delayed one-step reaction models with different types of time delay, including the constant delay, exponentially distributed delay and Erlang distributed delay. The comparison study suggested that the one-step reaction models failed to realize the dynamics of mRNA turnover accurately. Therefore more sophisticated one-step reaction models are needed to study the dynamics of mRNA degradation.

BACKGROUND: The availability of various "omics" datasets creates a prospect of performing the study of genome-wide genetic regulatory networks. However, one of the major challenges of using mathematical models to infer genetic regulation from microarray datasets is the lack of information for protein concentrations and activities. Most of the previous researches were based on an assumption that the mRNA levels of a gene are consistent with its protein activities, though it is not always the case. Therefore, a more sophisticated modelling framework together with the corresponding inference methods is needed to accurately estimate genetic regulation from "omics" datasets. RESULTS: This work developed a novel approach, which is based on a nonlinear mathematical model, to infer genetic regulation from microarray gene expression data. By using the p53 network as a test system, we used the nonlinear model to estimate the activities of transcription factor (TF) p53 from the expression levels of its target genes, and to identify the activation/inhibition status of p53 to its target genes. The predicted top 317 putative p53 target genes were supported by DNA sequence analysis. A comparison between our prediction and the other published predictions of p53 targets suggests that most of putative p53 targets may share a common depleted or enriched sequence signal on their upstream non-coding region. CONCLUSIONS: The proposed quantitative model can not only be used to infer the regulatory relationship between TF and its down-stream genes, but also be applied to estimate the protein activities of TF from the expression levels of its target genes.

MOTIVATION: Kinetic rate in gene expression is a key measurement of the stability of gene products and gives important information for the reconstruction of genetic regulatory networks. Recent developments in experimental technologies have made it possible to measure the numbers of transcripts and protein molecules in single cells. Although estimation methods based on deterministic models have been proposed aimed at evaluating kinetic rates from experimental observations, these methods cannot tackle noise in gene expression that may arise from discrete processes of gene expression, small numbers of mRNA transcript, fluctuations in the activity of transcriptional factors and variability in the experimental environment. RESULTS: In this paper, we develop effective methods for estimating kinetic rates in genetic regulatory networks. The simulated maximum likelihood method is used to evaluate parameters in stochastic models described by either stochastic differential equations or discrete biochemical reactions. Different types of non-parametric density functions are used to measure the transitional probability of experimental observations. For stochastic models described by biochemical reactions, we propose to use the simulated frequency distribution to evaluate the transitional density based on the discrete nature of stochastic simulations. The genetic optimization algorithm is used as an efficient tool to search for optimal reaction rates. Numerical results indicate that the proposed methods can give robust estimations of kinetic rates with good accuracy.

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

Recent advances in gene-expression profiling technologies provide large amounts of gene expression data. This raises the possibility for a functional understanding of genome dynamics by means of mathematical modelling. As gene expression involves intrinsic noise, stochastic models are essential for better descriptions of gene regulatory networks. However, stochastic modelling for large scale gene expression data sets is still in the very early developmental stage. In this paper we present some stochastic models by introducing stochastic processes into neural network models that can describe intermediate regulation for large scale gene networks. Poisson random variables are used to represent chance events in the processes of synthesis and degradation. For expression data with normalized concentrations, exponential or normal random variables are used to realize fluctuations. Using a network with three genes, we show how to use stochastic simulations for studying robustness and stability properties of gene expression patterns under the influence of noise, and how to use stochastic models to predict statistical distributions of expression levels in population of cells. The discussion suggest that stochastic neural network models can give better description of gene regulatory networks and provide criteria for measuring the reasonableness o mathematical models.