Method for identification of sensitive nodes in Boolean models of biological networks

Biological systems are often represented as Boolean networks and analysed to identify sensitive nodes which on perturbation disproportionately change a predefined output. There exist different kinds of perturbation methods: perturbation of function, perturbation of state and perturbation in update scheme. Nodes may have defects in interpretation of the inputs from other nodes and calculation of the node output. To simulate these defects and systematically assess their effect on the system output, two new function perturbations, referred to as ‘not of function’ and ‘function of not’, are introduced. In the former, the inputs are assumed to be correctly interpreted but the output of the update rule is perturbed; and in the latter, each input is perturbed but the correct update rule is applied. These and previously used perturbation methods were applied to two existing Boolean models, namely the human melanogenesis signalling network and the fly segment polarity network. Through mathematical simulations, it was found that these methods successfully identified nodes earlier found to be sensitive using other methods, and were also able to identify sensitive nodes which were previously unreported.Inspec keywords: Boolean functions, perturbation theory, physiological modelsOther keywords: Boolean models, biological networks, perturbation methods, human melanogenesis signalling network, fly segment polarity network     

Digital clocks: simple Boolean models can quantitatively describe circadian systems

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock''s oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.     

Optimal control policy for probabilistic Boolean networks with hard constraints

It is well known that the control/intervention of some genes in a genetic regulatory network is useful for avoiding undesirable states associated with some diseases like cancer. For this purpose, both optimal finitehorizon control and infinite-horizon control policies have been proposed. Boolean networks (BNs) and its extension probabilistic Boolean networks (PBNs) as useful and effective tools for modelling gene regulatory systems have received much attention in the biophysics community. The control problem for these models has been studied widely. The optimal control problem in a PBN can be formulated as a probabilistic dynamic programming problem. In the previous studies, the optimal control problems did not take into account the hard constraints, i.e. to include an upper bound for the number of controls that can be applied to the captured PBN. This is important as more treatments may bring more side effects and the patients may not bear too many treatments. A formulation for the optimal finite-horizon control problem with hard constraints introduced by the authors. This model is state independent and the objective function is only dependent on the distance between the desirable states and the terminal states. An approximation method is also given to reduce the computational cost in solving the problem. Experimental results are given to demonstrate the efficiency of our proposed formulations and methods.     

Detecting small attractors of large Boolean networks by function‐reduction‐based strategy

Boolean networks (BNs) are widely used to model gene regulatory networks and to design therapeutic intervention strategies to affect the long‐term behaviour of systems. A central aim of Boolean‐network analysis is to find attractors that correspond to various cellular states, such as cell types or the stage of cell differentiation. This problem is NP‐hard and various algorithms have been used to tackle it with considerable success. The idea is that a singleton attractor corresponds to n consistent subsequences in the truth table. To find these subsequences, the authors gradually reduce the entire truth table of Boolean functions by extending a partial gene activity profile (GAP). Not only does this process delete inconsistent subsequences in truth tables, it also directly determines values for some nodes not extended, which means it can abandon the partial GAPs that cannot lead to an attractor as early as possible. The results of simulation show that the proposed algorithm can detect small attractors with length p = 4 in BNs of up to 200 nodes with average indegree K = 2.Inspec keywords: Boolean functions, genetics, cellular biophysicsOther keywords: detecting small attractors, function‐reduction‐based strategy, model gene regulatory networks, therapeutic intervention strategies, Boolean‐network analysis, cellular states, NP‐hard, singleton attractor, Boolean functions, partial gene activity profile, cell differentiation     

Integer programming‐based method for observability of singleton attractors in Boolean networks

Boolean network (BN) is a popular mathematical model for revealing the behaviour of a genetic regulatory network. Furthermore, observability, an important network feature, plays a significant role in understanding the underlying network. Several studies have been done on analysis of observability of BNs and complex networks. However, the observability of attractor cycles, which can serve as biomarker detection, has not yet been addressed in the literature. This is an important, interesting and challenging problem that deserves a detailed study. In this study, a novel problem was first proposed on attractor observability in BNs. Identification of the minimum set of consecutive nodes can be used to discriminate different attractors. Furthermore, it can serve as a biomarker for different disease types (represented as different attractor cycles). Then a novel integer programming method was developed to identify the desired set of nodes. The proposed approach is demonstrated and verified by numerical examples. The computational results further illustrates that the proposed model is effective and efficient.Inspec keywords: integer programming, Boolean algebra, complex networks, diseasesOther keywords: disease, consecutive nodes, biomarker detection, attractor cycles, complex networks, genetic regulatory network, mathematical model, Boolean networks, singleton attractors, integer programming‐based method     

On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.     

Stochastic dynamic modeling of short gene expression time-series data

In this paper, the expectation maximization (EM) algorithm is applied for modeling the gene regulatory network from gene time-series data. The gene regulatory network is viewed as a stochastic dynamic model, which consists of the noisy gene measurement from microarray and the gene regulation first-order autoregressive (AR) stochastic dynamic process. By using the EM algorithm, both the model parameters and the actual values of the gene expression levels can be identified simultaneously. Moreover, the algorithm can deal with the sparse parameter identification and the noisy data in an efficient way. It is also shown that the EM algorithm can handle the microarrary gene expression data with large number of variables but a small number of observations. The gene expression stochastic dynamic models for four real-world gene expression data sets are constructed to demonstrate the advantages of the introduced algorithm. Several indices are proposed to evaluate the models of inferred gene regulatory networks, and the relevant biological properties are discussed.     

Construction of Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix: A Maximum Entropy Rate Approach

Modeling genetic regulatory networks is an important problem in genomic research. Boolean Networks (BNs) and their extensions Probabilistic Boolean Networks (PBNs) have been proposed for modeling genetic regulatory interactions. In a PBN, its steady-state distribution gives very important information about the long-run behavior of the whole network. However, one is also interested in system synthesis which requires the construction of networks. The inverse problem is ill-posed and challenging, as there may be many networks or no network having the given properties, and the size of the problem is huge. The construction of PBNs from a given transition-probability matrix and a given set of BNs is an inverse problem of huge size. We propose a maximum entropy approach for the above problem. Newton’s method in conjunction with the Conjugate Gradient (CG) method is then applied to solving the inverse problem. We investigate the convergence rate of the proposed method. Numerical examples are also given to demonstrate the effectiveness of our proposed method.     

Distribution and enumeration of attractors in probabilistic boolean networks

Many mathematical models for gene regulatory networks have been proposed. In this study, the authors study attractors in probabilistic Boolean networks (PBNs). They study the expected number of singleton attractors in a PBN and show that it is (2 - (1=2)/sup L-1)/sup n/, where n is the number of nodes in a PBN and L is the number of Boolean functions assigned to each node. In the case of L = 2, this number is simplified into 1.5/sup n/. It is an interesting result because it is known that the expected number of singleton attractors in a Boolean network (BN) is 1. Then, we present algorithms for identifying singleton and small attractors and perform both theoretical and computational analyses on their average case time complexities. For example, the average case time complexities for identifying singleton attractors of a PBN with L = 2 and L = 3 are O(1.601/sup n/) and O(1.763/sup n/), respectively. The results of computational experiments suggest that these algorithms are much more efficient than the naive algorithm that examines all possible 2/sup n/ states.     

Optimization of properties of fly ash aggregates for high-strength lightweight concrete production

The optimization of properties of lightweight fly ash aggregates for suitability in high-strength lightweight fly ash concrete production was investigated using response surface methodology (RSM). Design-Expert software was used to establish the design matrix and to analyze the experimental data. The relationships between the sintering parameters (temperature, binder content and binder type) and experimentally obtained three responses (specific gravity, water absorption and crushing strength) were established. Also, the optimization capabilities in Design-Expert software were used to optimize the sintering process. Historical data design technique under RSM was performed to optimize the input parameter interactions which showed the best conditions for preparation of fly ash pellets. According to the obtained results, the developed models are statistically accurate and can be used for further analysis. The experimental values agreed with the predicted ones, thus indicating suitability of the model employed and the success of RSM in optimizing the sintering conditions.     

Determination of critical network interactions: an augmented Boolean pseudo-dynamics approach

Network theory has established that highly connected nodes in regulatory networks (hubs) show a strong correlation with criticality in network function. Although topological analysis is fully capable of identifying network hubs, it does not provide an objective method for ranking the importance of a particular node by relating its contribution to the overall network response. Towards this end, the authors have developed an augmented Boolean pseudo-dynamics approach to a priori determine the critical network interactions in biological interaction networks. The approach utilises network topology and dynamic state information to determine the set of active pathways. The active pathways are used in conjunction with the key cellular properties of efficiency and robustness, to rank the network interactions based on their importance in the sustenance of network function. To demonstrate the utility of the approach, the authors consider the well characterised guard cell signalling network in plant cells. An integrated analysis of the network revealed the critical mechanisms resulting in stomata closure in the presence and absence of abscisic acid, in excellent agreement with published results.     

Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method

The ordinary differential equation (ODE) and partial differential equation (PDE) image- processing methods have been applied to reduce noise and enhance the contrast of electronic speckle pattern interferometry fringe patterns. We evaluate the performance of a few representative PDE denoising models quantitatively with two parameters called image fidelity and speckle index, and then we choose a good denoising model. Combining this denoising model with the ODE enhancement method, we make it possible to perform contrast enhancement and denoising simultaneously. Second, we introduce the delta-mollification method to smooth the unwrapped phase map. Finally, based on PDE image processing, delta mollification and some traditional techniques, an approach of phase extraction from a single fringe pattern is tested for computer-simulated and experimentally obtained fringe patterns. The method works well under a high noise level and limited visibility and can extract accurate phase values.     

On approximate stochastic control in genetic regulatory networks

The control of probabilistic Boolean networks as a model of genetic regulatory networks is formulated as an optimal stochastic control problem and has been solved using dynamic programming; however, the proposed methods fail when the number of genes in the network goes beyond a small number. There are two dimensionality problems. First, the complexity of optimal stochastic control exponentially increases with the number of genes. Second, the complexity of estimating the probability distributions specifying the model increases exponentially with the number of genes. We propose an approximate stochastic control method based on reinforcement learning that mitigates the curses of dimensionality and provides polynomial time complexity. Using a simulator, the proposed method eliminates the complexity of estimating the probability distributions and, because the method is a model-free method, it eliminates the impediment of model estimation. The method can be applied on networks for which dynamic programming cannot be used owing to computational limitations. Experimental results demonstrate that the performance of the method is close to optimal stochastic control.     

Phase transitions and assortativity in models of gene regulatory networks evolved under different selection processes

We study a simplified model of gene regulatory network evolution in which links (regulatory interactions) are added via various selection rules that are based on the structural and dynamical features of the network nodes (genes). Similar to well-studied models of ‘explosive’ percolation, in our approach, links are selectively added so as to delay the transition to large-scale damage propagation, i.e. to make the network robust to small perturbations of gene states. We find that when selection depends only on structure, evolved networks are resistant to widespread damage propagation, even without knowledge of individual gene propensities for becoming ‘damaged’. We also observe that networks evolved to avoid damage propagation tend towards disassortativity (i.e. directed links preferentially connect high degree ‘source’ genes to low degree ‘target’ genes and vice versa). We compare our simulations to reconstructed gene regulatory networks for several different species, with genes and links added over evolutionary time, and we find a similar bias towards disassortativity in the reconstructed networks.     

Gene regulatory network discovery using pairwise Granger causality

Discovery of gene regulatory network from gene expression data can yield a useful insight to drug development. Among the methods applied to time‐series data, Granger causality (GC) has emerged as a powerful tool with several merits. Since gene expression data usually have a much larger number of genes than time points therefore a full model cannot be applied in a straightforward manner, GC is often applied to genes pairwisely. In this study, the authors first investigate with synthetic data how spurious causalities (false discoveries) may arise because of the use of pairwise rather than full‐model GC detection. Furthermore, spurious causalities may also arise if the order of the vector autoregressive model is not high enough. As a remedy, the authors demonstrate that model validation techniques can effectively reduce the number of false discoveries. Then, they apply pairwise GC with model validation to the real human HeLa cell‐cycle dataset. They find that Akaike information criterion is generally most suitable for determining model order, but precaution should be taken for extremely short time series. With the authors proposed implementation, degree distributions and network hubs are obtained and compared with existing results, giving a new observation that the hubs tend to act as sources rather than receivers of interactions.Inspec keywords: biology computing, cancer, causality, cellular biophysics, genetics, genomics, time seriesOther keywords: gene regulatory network discovery, pairwise Granger causality, gene expression data, drug development, time‐series data, synthetic data, spurious causalities, full‐model Granger causality detection, vector autoregressive model, real human HeLa cell‐cycle dataset, Akaike information criterion, degree distributions, network hubs     

ROBUST DESIGN USING COMPROMISE DECISION SUPPORT PROBLEMS

Compromise Decision Support Problems (DSPs) are used to model engineering decisions involving multiple trade-offs. In this paper, the focus is on how to apply such decision models in robust design. Suh's independence and information content axioms and Taguchi's signal to noise ratio are used as metrics for the assessment and improvement of the quality in this decision model. As an example, a compromise DSP for the robust design of an electrical network is used. Traditionally, in robust design, parameter and tolerance design are done sequentially and not concurrently. Furthermore, each time parameter and tolerance design are done in practice, the focus is usually on looking at one parameter at a time and not on looking at multiple parameters simultaneously. Using the electrical network as an example, it is shown how parameter and tolerance design involving multiple parameters can be performed concurrently.     

Physics-integrated deep learning for uncertainty quantification and reliability estimation of nonlinear dynamical systems

Assumptions and approximations made while analyzing any physical system induce modeling uncertainty, which, if left unchecked, can result in the erroneous analysis of the system under consideration. Additionally, the discrepancy in the exact knowledge of system parameters can further result in deviation from the ground truth. This paper explores Physics-integrated Variational Auto-Encoder (PVAE) to account for modeling and parametric uncertainties in partially known nonlinear dynamical systems. The PVAE under consideration has three main parts: encoder, latent space, and decoder. The complete PVAE architecture is employed during the training stage of the machine learning model, while only the decoder is used to make the final predictions. The encoder determines the correct parameter values for the known part of the model (in the form of a known ODE). The decoder is augmented with an ODE solver that solves the known part of the system and the estimated discrepancy together to reconstruct the measurements. To test the efficacy of the PVAE architecture, three case studies are carried out, each presenting unique challenges. The probability density functions obtained for the various systems’ responses demonstrate the efficacy of the PVAE architecture. Furthermore, reliability analysis has been carried out, and the results produced have been compared against those obtained from a multi-layered, densely connected forward neural network.