Session J44: Biological Networks

Network complexity: when interaction strengths matter more than topology

A typical cellular network has thousands of microscopic parameters that cannot all be equally relevant to the network function; yet discarding them and considering only the topology of interactions is unacceptably coarse. How much do quantitative details matter? We present a toy model where the appropriately mesoscopic level of description can be found exactly. We define a measure of network complexity and find that both the choice of topology and strength of interactions can affect the complexity dramatically, but optimizing interaction strengths typically has a stronger effect. We further show that a larger network is not automatically more complex; constructing a high-complexity network always requires a careful adjustment of the strengths of interactions. This suggests that the challenge of ``evolving a complex network'' does not reduce to making new connections and constructing a dense topology of interactions. Evolution acting on ``numbers on arrows'' (interaction strengths), even within the confines of a fixed topology, is a much more effective way of increasing complexity than adding new components and connections of some typical strength.   

Ising models of strongly coupled biological networks with multivariate interactions

Biological networks consist of a large number of variables that can be coupled by complex multivariate interactions. However, several neuroscience and cell biology experiments have reported that observed statistics of network states can be approximated surprisingly well by maximum entropy models that constrain correlations only within pairs of variables. We would like to verify if this reduction in complexity results from intricacies of biological organization, or if it is a more general attribute of these networks. We generate random networks with p-spin $(p>2)$ interactions, with N spins and M interaction terms. The probability distribution of the network states is then calculated and approximated with a maximum entropy model based on constraining pairwise spin correlations. Depending on the M/N ratio and the strength of the interaction terms, we observe a transition where the pairwise approximation is very good to a region where it fails. This resembles the sat-unsat transition in constraint satisfaction problems. We argue that the pairwise model works when the number of highly probable states is small. We argue that many biological systems must operate in a strongly constrained regime, and hence we expect the pairwise approximation to be accurate for a wide class of problems.   

Add HOC?: dendritic nonlinearities shape higher-than-pairwise correlations and improve coding in noisy (spiking) neural populations

Recent experiments with relatively large neural populations show significant higher-order correlations (HOC): the data are poorly fit by pair-wise maximum entropy models, but well-fit by higher-order models. We seek to understand how HOC are shaped by the properties of networks and of the neurons therein, and how these HOC affect population coding. In our presentation, we will demonstrate that dendritic non-linearities similar to those observed in physiology experiments are equivalent to beyond-pairwise interactions in a spin-glass-type statistical model: they can either increase, or decrease, the magnitude of the HOC relative to the pair-wise correlations. We will then discuss a population coding model with parameterized pairwise- and higher-order interactions, revealing the conditions under which the beyond-pairwise interactions (dendritic nonlinearities) can increase the mutual information between a given set of stimuli, and the population responses. For jointly Gaussian stimuli, coding performance can be slightly improved by shaping the output HOC via dendritic nonlinearities, if the neural firing rates are low. For skewed stimulus distributions, like the distribution of luminance values in natural images, the performance gains are much larger.   

Automated adaptive model inference to predict biological network dynamics

Dynamical models of cellular regulation often consist of large and intricate networks of interactions at the molecular scale. Since individual interaction parameters are usually difficult to measure, these parameters are often estimated implicitly, using statistical fits. This can lead to overfitting and degradation in the quality of models' predictions. Here we study phenomenological models that adapt their level of detail to the amount of available data, leading to accurate predictions even when microscopic details are not well understood. The model search is made computationally efficient by testing an ordered, nested set of models and by using a model class that can be solved using linear regression in log-space. We test the method on synthetic data and find that phenomenological models inferred this way often outperform detailed, ``correct'' molecular models in making predictions about responses of the system to signals yet unseen.   

Using dynamics to identify network topology

To elucidate the topology of a signaling pathway, generally, experimentalists manipulate a cell's molecular architecture, for example, by knocking out genes. Molecular biology techniques, though, are not only invasive and labor-intensive, they have also often been eluded by the complexity of biological networks, e.g., in the case of the gonadotropin-releasing hormone (GnRH) system. Inspired by the rapidly accumulating examples of oscillatory signaling in biology, we explored whether such dynamical stimuli can be used to discriminate different topologies of adaptive pathways, which are ubiquitous in biology. Responses to static inputs are nearly indistinguishable given strong measurement noise. Sine function stimuli, widely used in physics, are difficult to implement in standard microfluidics or optogenetics set-ups and do not simplify the mathematical analysis because of the nonlinearities in these systems. With periodic on-off pulses, which can be easily produced, however, simple adaptive circuit motifs and detailed models from the literature robustly reveal distinct output characteristics, which manifest in how the period of maximal output varies with pulse width. Our calculations provide a framework for using existing methods to discover difficult to reveal mechanisms. Furthermore, our results constrain the possible design principles of the presumed frequency decoders in biological systems where pulsatile signaling has recently been discovered.   

Characterization of genotype-phenotype mapping of biological networks reconciles robustness-evolvability paradox

Typical biological system is both highly robust and highly evolvable. Yet robustness appears against changes whereas evolvability for changes. The concurrence of these two seemingly incompatible features is a central paradox for contemporary evolutionary biology. Using a Boolean model of yeast cell cycle networks, we quantitatively determine (1) the genotype-phenotype mapping. Here genotype stands for the network structure and phenotype for its dynamics; (2) the precise topology of neutral network, i.e. the interconnecting network of networks of different structures but the same dynamics; and (3) the number of new phenotypes in the vicinity of a neutral network. Our results demonstrate that both biological genotype and phenotype are atypical. We next show via sampling that all neutral networks exhibit a similar topology that is simply connected, fractal and sloppy (stiff in certain dimensions but diffusive otherwise). This percolating nature of neutral network leads to a positive correlation between robustness and evolvability and hence resolves the paradox. A likely explanation for such a correlation is that higher robustness results in a larger neutral network, measured by its designability and radius of gyration, which in turn accesses more new phenotypes.   

Robustness of Network Measures to Link Errors

Researchers studying biological networks use a variety of measures to identify ``important'' nodes in their networks. However, the robustness of these measures in the presence of link inaccuracies stemming from noisy data has not been well characterized. Here we present two simple models of false and missing links and their effect on different commonly used centrality measures, focusing particularly on degree centrality, betweenness centrality, and dynamical importance. We show that, compared to degree centrality, betweenness centrality and dynamical importance are much more robust in the face of noise if the false positives are randomly distributed. When the noise has more structure, the differences in the robustness levels of the various metrics can change dramatically.   

Cascade-likeness is an intrinsic property of biological processes

A central theme in systems biology is to reveal the intricate relationship between structure, dynamics, and function of biological networks. The biological function is usually realized by the transformation of the relevant molecules through their interacting network. We name this trajectory of transformation as a biological process. In contrast to the structure-centric approach, we take a process-centric view to address such questions as what a biological process looks like and how it differs from an arbitrary process. As an example, we studied a simple Boolean model for the cell cycle process of budding yeast to characterize a large number of putative processes. This computational task was made possible by some highly efficient algorithms we developed. Our results demonstrated that the biological process is very robust and highly designable. Moreover, we uncovered two dynamical rules that dramatically enhance the robustness and designability. Finally, all processes in a system of small size were enumerated and highly designable processes are cascade-like. This implies that cascade-likeness is an intrinsic property of biological processes.   

Autonomous Boolean Models of Regulatory Networks

Autonomous Boolean network (ABN) models have been developed to represent directly the connectivity, logic, and timing of updates in regulatory networks. [1] An ABN is a Boolean network in which the sequence of updates of nodes is determined internally by time delay parameters associated with each link. We propose a method to convert a given ODE model into an ABN that is applicable when the ODE dynamics produces clearly separated high and low values at each node. The ODE parameters are mapped into ABN logic and delay parameters using only local information about each link. Using the example of Ingolia's ODE model of the regulatory network that maintains segment boundaries in the \textit{Drosophila} embryo [2], we show that the resulting ABN model captures both the biologically relevant outcomes and the transient dynamics of the ODE model, and that the ABN framework provides direct insights into the mechanism supporting the biological function. [1] X. Cheng, M. Sun, and J. E. S. Socolar, 2012, J. R. Soc. Interface, (DOI: 10.1098/rsif.2012.0574) [2] Ingolia NT., 2004, PLoS Biol. 2, 805-815. (DOI:10.1371/journal.pbio.0020123)   

Epigenetic landscapes explain partially reprogrammed cells and identify key reprogramming gene

A common metaphor for describing development is a rugged epigenetic landscape where cell fates are represented as attracting valleys resulting from a complex regulatory network. Here, we introduce a framework for explicitly constructing epigenetic landscapes that combines genomic data with techniques from physics, specifically Hopfield neural networks. Each cell fate is a dynamic attractor, yet cells can change fate in response to external signals. Our model suggests that partially reprogrammed cells (cells found in reprogramming experiments but not in vivo) are a natural consequence of high-dimensional landscapes and predicts that partially reprogrammed cells should be hybrids that coexpress genes from multiple cell fates. We verify this prediction by reanalyzing existing data sets. Our model reproduces known reprogramming protocols and identifies candidate transcription factors for reprogramming to novel cell fates, suggesting epigenetic landscapes are a powerful paradigm for understanding cellular identity.   

Phage-bacteria infection networks: From nestedness to modularity

Bacteriophages (viruses that infect bacteria) are the most abundant biological life-forms on Earth. However, very little is known regarding the structure of phage-bacteria infections. In a recent study we re-evaluated 38 prior studies and demonstrated that phage-bacteria infection networks tend to be statistically nested in small scale communities (Flores et al 2011). Nestedness is consistent with a hierarchy of infection and resistance within phages and bacteria, respectively. However, we predicted that at large scales, phage-bacteria infection networks should be typified by a modular structure. We evaluate and confirm this hypothesis using the most extensive study of phage-bacteria infections (Moebus and Nattkemper 1981). In this study, cross-infections were evaluated between 215 marine phages and 286 marine bacteria. We develop a novel multi-scale network analysis and find that the Moebus and Nattkemper (1981) study, is highly modular (at the whole network scale), yet also exhibits nestedness and modularity at the within-module scale. We examine the role of geography in driving these modular patterns and find evidence that phage-bacteria interactions can exhibit strong similarity despite large distances between sites.   

Continuum neural dynamics models for visual object identification

Visual object identification has remained one of the most challenging problems even after decades of research. Most of the current models of the visual cortex represent neurons as discrete elements in a largely feedforward network arrangement. They are generally very specific in the objects they can identify. We develop a continuum model of recurrent, nonlinear neural dynamics in the primary visual cortex, incorporating connectivity patterns and other ~experimentally observed features of the cortex. The model has an interesting correspondence to the Landau-DeGennes theory of a nematic liquid crystal in two dimensions. ~We use collective spatiotemporal excitations of the model cortex as a signal for segmentation of contiguous objects from the background clutter. The model is capable of suppressing clutter in images and filling in occluded elements of object contours, resulting in high-precision, high-recall identification of large objects from cluttered scenes.   

The parameter landscape of a mammalian circadian clock model

In mammals, an intricate system of feedback loops enables autonomous, robust oscillations synchronized with the daily light/dark cycle. Based on recent experimental evidence, we have developed a simplified dynamical model and parameterized it by compiling experimental data on the amplitude, phase, and average baseline of clock gene oscillations. Rather than identifying a single ``optimal'' parameter set, we used Monte Carlo sampling to explore the fitting landscape. The resulting ensemble of model parameter sets is highly anisotropic, with very large variances along some (non-trivial) linear combinations of parameters and very small variances along others. This suggests that our model exhibits ``sloppy'' features that have previously been identified in various multi-parameter fitting problems. We will discuss the implications of this model fitting behavior for the reliability of both individual parameter estimates and systems-level predictions of oscillator characteristics, as well as the impact of experimental constraints. The results of this study are likely to be important both for improved understanding of the mammalian circadian oscillator and as a test case for more general questions about the features of systems biology models.   

Effect of Transcranial Magnetic Stimulation on Neuronal Networks

The human brain contains around 100 billion nerve cells controlling our day to day activities. Consequently, brain disorders often result in impairments such as paralysis, loss of coordination and seizure. It has been said that 1 in 5 Americans suffer some diagnosable mental disorder. There is an urgent need to understand the disorders, prevent them and if possible, develop permanent cure for them. As a result, a significant amount of research activities is being directed towards brain research. Transcranial Magnetic Stimulation (TMS) is a promising tool for diagnosing and treating brain disorders. It is a non-invasive treatment method that produces a current flow in the brain which excites the neurons. Even though TMS has been verified to have advantageous effects on various brain related disorders, there have not been enough studies on the impact of TMS on cells. In this study, we are investigating the electrophysiological effects of TMS on one dimensional neuronal culture grown in a circular pathway. Electrical currents are produced on the neuronal networks depending on the directionality of the applied field. This aids in understanding how neuronal networks react under TMS treatment.   

Biochemical response and the effects of bariatric surgeries on type 2 diabetes

A general method is introduced for calculating the biochemical response to pharmaceuticals, surgeries, or other medical interventions. This method is then applied in a simple model of the response to Roux-en-Y gastric bypass (RYGB) surgery in obese diabetic patients. We specifically address the amazing fact that glycemia correction is usually achieved immediately after RYGB surgery, long before there is any appreciable weight loss. Many studies indicate that this result is not due merely to caloric restriction, and it is usually attributed to an increase in glucagon-like peptide 1 (GLP-1) levels observed after the surgery. However, our model indicates that this mechanism alone is not sufficient to explain either the largest declines in glucose levels or the measured declines in the homeostatic model assessment insulin resistance (HOMA-IR). The most robust additional mechanism would be production of a factor which opens an insulin-independent pathway for glucose transport into cells, perhaps related to the well-established insulin-independent pathway associated with exercise. Potential candidates include bradykinin, a 9 amino acid peptide. If such a substance were found to exist, it would offer hope for medications which mimic the immediate beneficial effect of RYGB surgery.