The complexity of linkage analysis with neural networks

As the focus of genome-wide scans for disease loci have shifted from simple Mendelian traits to genetically complex traits, researchers have begun to consider new alternative ways to detect linkage that will consider more than the marginal effects of a single disease locus at a time. One interesting new method is to train a neural network on a genome-wide data set in order to search for the best non-linear relationship between identity-by-descent sharing among affected siblings at markers and their disease status. We investigate here the repeatability of the neural network results from run to run, and show that the results obtained by multiple runs of the neural network method may differ quite a bit. This is most likely due to the fact that training a neural network involves minimizing an error function with a multitude of local minima.     

Dynamical complexity in cognitive neural networks

In the last twenty years an important effort in brain sciences, especially in cognitive science, has been the development of mathematical tool that can deal with the complexity of extensive recordings corresponding to the neuronal activity obtained from hundreds of neurons. We discuss here along with some historical issues, advantages and limitations of Artificial Neural Networks (ANN) that can help to understand how simple brain circuits work and whether ANN can be helpful to understand brain neural complexity.     

Modes and cuts in metabolic networks: complexity and algorithms

Constraint-based approaches recently brought new insight into our understanding of metabolism. By making very simple assumptions such as that the system is at steady-state and some reactions are irreversible, and without requiring kinetic parameters, general properties of the system can be derived. A central concept in this methodology is the notion of an elementary mode (EM for short) which represents a minimal functional subsystem. The computation of EMs still forms a limiting step in metabolic studies and several algorithms have been proposed to address this problem leading to increasingly faster methods. However, although a theoretical upper bound on the number of elementary modes that a network may possess has been established, surprisingly, the complexity of this problem has never been systematically studied. In this paper, we give a systematic overview of the complexity of optimisation problems related to modes. We first establish results regarding network consistency. Most consistency problems are easy, i.e., they can be solved in polynomial time. We then establish the complexity of finding and counting elementary modes. We show in particular that finding one elementary mode is easy but that this task becomes hard when a specific EM (i.e. an EM containing some specified reactions) is sought. We then show that counting the number of elementary modes is musical sharpP-complete. We emphasize that the easy problems can be solved using currently existing software packages. We then analyse the complexity of a closely related task which is the computation of so-called minimum reaction cut sets and we show that this problem is hard. We then present two positive results which both allow to avoid computing EMs as a prior to the computation of reaction cuts. The first one is a polynomial approximation algorithm for finding a minimum reaction cut set. The second one is a test for verifying whether a set of reactions constitutes a reaction cut; this test can be readily included in existing algorithms to improve their performance. Finally, we discuss the complexity of other cut-related problems.     

Reducing complexity in metabolic networks: making metabolic meshes manageable

The dynamics of complex systems can be effectively analyzed by judicious use of intrinsic time constants. Order of magnitude estimation based on time constants has been used successfully to examine the dynamic behavior of complicated processes. The main goal of this paper is to introduce this approach to the analysis of complex metabolic systems. Time constants and dynamic modes of motion are defined within the context of well-established linear algebra. The order of magnitude estimation is then introduced into the systemic framework. The main goals of the analysis are: to provide improved understanding of biochemical dynamics and their physiological significance, and to yield reduced dynamic models that are physiologically realistic but tractable for practical use.     

Rings and networks: the amazing complexity of FtsZ in chloroplasts

Bacteria have proteins that can form filaments and rings, and these are thought to be the evolutionary progenitors of actin and tubulin. Plant homologues of the most intensively studied bacterial FtsZ protein are nuclear-encoded by a small gene family, are plastid-bound and participate in the plastid division process. The hypothesis is put forward that FtsZ and other proteins form a filamentous network in plastids, a plastoskeleton, which keeps these organelles in shape and helps them to divide.     

Combinatorial complexity of pathway analysis in metabolic networks

Elementary flux mode analysis is a promising approach for a pathway-oriented perspective of metabolic networks. However, in larger networks it is hampered by the combinatorial explosion of possible routes. In this work we give some estimations on the combinatorial complexity including theoretical upper bounds for the number of elementary flux modes in a network of a given size. In a case study, we computed the elementary modes in the central metabolism of Escherichia coli while utilizing four different substrates. Interestingly, although the number of modes occurring in this complex network can exceed half a million, it is still far below the upper bound. Hence, to a certain extent, pathway analysis of central catabolism is feasible to assess network properties such as flexibility and functionality.     

Genome-wide identification of flowering time genes associated with vernalization and the regulatory flowering networks in Chinese cabbage

Flowering time (Ft) is the most important characteristic of Chinese cabbage with high leaf yields and late-flowering are favorable traits, while little knowledge on genes involved in Ft and the flowering mechanism in this crop. In this study, we conducted genome-wide RNA-seq analysis using an inbred Chinese cabbage ‘4004’ line in response to vernalization and compared the Ft gene expression with radish crop. A number of Ft genes which play roles in flowering pathways were performed quantitative RT-PCR analysis to verify the regulatory flowering gene network in Chinese cabbage. We found that a total of 223 Ft genes in Chinese cabbage, and 50 of these genes responded to vernalization. The majority of flowering enhancers were upregulated, whereas most flowering repressors were downregulated in response to vernalization as confirmed by RT-qPCR. Among the major Ft genes, the expression of BrCOL1-2, BrFT1/2, BrSOC1/2/3, BrFLC1/2/3/5, and BrMAF was strongly affected by vernalization. In reference to comparative RNA-seq profiling of Ft genes, Chinese cabbage and radish revealed substantially different vernalization response in particular GA flowering pathway. Thus, this study provides new insight into functional divergence in flowering pathways and the regulatory mechanisms in Brassicaceae crops. Further analysis of the major integrator genes between early and late-flowering inbred lines facilitates understanding flowering trait variation and molecular basis of flowering in Chinese cabbage.     

Robustness and complexity co-constructed in multimodal signalling networks

In animal communication, signals are frequently emitted using different channels (e.g. frequencies in a vocalization) and different modalities (e.g. gestures can accompany vocalizations). We explore two explanations that have been provided for multimodality: (i) selection for high information transfer through dedicated channels and (ii) increasing fault tolerance or robustness through multichannel signals. Robustness relates to an accurate decoding of a signal when parts of a signal are occluded. We show analytically in simple feed-forward neural networks that while a multichannel signal can solve the robustness problem, a multimodal signal does so more effectively because it can maximize the contribution made by each channel while minimizing the effects of exclusion. Multimodality refers to sets of channels where within each set information is highly correlated. We show that the robustness property ensures correlations among channels producing complex, associative networks as a by-product. We refer to this as the principle of robust overdesign. We discuss the biological implications of this for the evolution of combinatorial signalling systems; in particular, how robustness promotes enough redundancy to allow for a subsequent specialization of redundant components into novel signals.     

The water relations of Verna Lemon trees from flowering to the end of rapid fruit growth

Lemon plants (Citrus limonum L. cv. Verna) were grown in the field under two different flood irrigation treatments. The dry treatment received four irrigations per year (March, July, September and November) and the wet treatment one monthly. The amounts of water applied per year for dry and wet treatments were 340.0 mm and 1020.0 mm, respectively. The effects of the two treatments on certain aspects of the plant water relations during the period between flowering and the end of rapid fruit growth (critical period) were studied. Soil matric potential (ψm) and leaf water potential (ψi) values in the dry treatment revealed development of water stress during the experimental period. The water supply in the wet treatment seems sufficient to achieve the crop water requirements. The g₁ values in July were higher in the wet than dry treatments. Pronounced oscillations in g₁ from sunrise to afternoon were found especially in the dry treatment.     

Path lengths in protein-protein interaction networks and biological complexity

We investigated the biological significance of path lengths in 12 protein-protein interaction (PPI) networks. We put forward three predictions, based on the idea that biological complexity influences path lengths. First, at the network level, path lengths are generally longer in PPIs than in random networks. Second, this pattern is more pronounced in more complex organisms. Third, within a PPI network, path lengths of individual proteins are biologically significant. We found that in 11 of the 12 species, average path lengths in PPI networks are significantly longer than those in randomly rewired networks. The PPI network of the malaria parasite Plasmodium falciparum, however, does not exhibit deviation from rewired networks. Furthermore, eukaryotic PPIs exhibit significantly greater deviation from randomly rewired networks than prokaryotic PPIs. Thus our study highlights the potentially meaningful variation in path lengths of PPI networks. Moreover, node eccentricity, defined as the longest path from a protein to others, is significantly correlated with the levels of gene expression and dispensability in the yeast PPI network. We conclude that biological complexity influences both global and local properties of path lengths in PPI networks. Investigating variation of path lengths may provide new tools to analyze the evolution of functional modules in biological systems.     

Shadows of complexity: what biological networks reveal about epistasis and pleiotropy

Pleiotropy, in which one mutation causes multiple phenotypes, has traditionally been seen as a deviation from the conventional observation in which one gene affects one phenotype. Epistasis, or gene–gene interaction, has also been treated as an exception to the Mendelian one gene–one phenotype paradigm. This simplified perspective belies the pervasive complexity of biology and hinders progress toward a deeper understanding of biological systems. We assert that epistasis and pleiotropy are not isolated occurrences, but ubiquitous and inherent properties of biomolecular networks. These phenomena should not be treated as exceptions, but rather as fundamental components of genetic analyses. A systems level understanding of epistasis and pleiotropy is, therefore, critical to furthering our understanding of human genetics and its contribution to common human disease. Finally, graph theory offers an intuitive and powerful set of tools with which to study the network bases of these important genetic phenomena.     

BioJazz: in silico evolution of cellular networks with unbounded complexity using rule-based modeling

Systems biologists aim to decipher the structure and dynamics of signaling and regulatory networks underpinning cellular responses; synthetic biologists can use this insight to alter existing networks or engineer de novo ones. Both tasks will benefit from an understanding of which structural and dynamic features of networks can emerge from evolutionary processes, through which intermediary steps these arise, and whether they embody general design principles. As natural evolution at the level of network dynamics is difficult to study, in silico evolution of network models can provide important insights. However, current tools used for in silico evolution of network dynamics are limited to ad hoc computer simulations and models. Here we introduce BioJazz, an extendable, user-friendly tool for simulating the evolution of dynamic biochemical networks. Unlike previous tools for in silico evolution, BioJazz allows for the evolution of cellular networks with unbounded complexity by combining rule-based modeling with an encoding of networks that is akin to a genome. We show that BioJazz can be used to implement biologically realistic selective pressures and allows exploration of the space of network architectures and dynamics that implement prescribed physiological functions. BioJazz is provided as an open-source tool to facilitate its further development and use. Source code and user manuals are available at: http://oss-lab.github.io/biojazz and http://osslab.lifesci.warwick.ac.uk/BioJazz.aspx.