Table of contents 11 chapters Table of contents 11 chapters Nonsmooth critical point theory and quasilinear elliptic equations Pages Canino, Annamaria et al. Second order differential equations on manifolds and forced oscillations Pages Furi, Massimo.
Effects of delays on dynamics Pages Hale, Jack K. Existence principles for differential equations and systems of equations Pages Lee, John W. Continuation theorems and periodic solutions of ordinary differential equations Pages Mawhin, Jean.
Some applications of the topological degree to stability theory Pages Ortega, Rafael. The center manifold technique and complex dynamics of parabolic equations Pages Rybakowski, Krzysztof P. Positive solutions of semilinear elliptic boundary value problems Pages Schmitt, Klaus.
Show next xx. Read this book on SpringerLink. Recommended for you. More detailed simulations were run to quantitatively capture the manner in which the feed-forward dynamics vary with different lattice density, D , and the number of nodes, V , in the FFA-series. The equation derived shows considerable acceleration of the FF processes with the increase in density in agreement with the law of mass action, since density is proportional to a substrate concentration modeled as number of cells per unit lattice area.
The regressions best expressing the dependence of a and b on the lattice density D were. The last parameter to select was the number of source cells N. The number of cells of all other pathway constituents was kept equal to cells. The enzymes associated with each of the biochemical reaction steps were kept equal to 20 cells. All enzyme activities were also kept equal by applying the same CA probabilistic rule in order to extract the purely topological effects on the network dynamics Since the concentration, simulated as number of cells per unit lattice, was kept constant, N itself should not influence the correlation coefficient of the linear model relating the number of nodes in the pathway to the rate of the source-to-outcome conversion.
The feed-forward motif FFA shown in Fig. One may also consider double, triple, etc. Such structural motifs with more complex topologies may be obtained by merging two or more primary FF motifs. This approach may be of interest in predicting dynamic patterns in larger networks as combinations of well established patterns in small subnetworks motifs. We selected for our study several such complex cases of feed-forward patterns of PFFs Fig.
Feed-forward motifs with different network topologies. The first and the last member of the examined seven feed-forward motifs series are shown only. The series include all structures with intermediate number of nodes FFA series had networks with 3 to 9 nodes; all other series had 4—10 nodes. This number serves as an inverse measure of the FF motif dynamics the larger the number of iterations needed for arriving at a steady-state, the slower the overall process.
Linear dependence of the overall rate of feed-forward motifs on the number of motive nodes. Seven series Fig.
The central focus of our analyses was to study how network topology affects the dynamics of processes in different feed-forward motifs. In order to ensure that the networks analyzed were comparable to enable the identification of stable structure-dynamics patterns, we assumed that i the rate constants for all processes are equal, ii the initial conditions are chosen such that the source S is initially five times larger than each of the other species, and iii all enzyme activities are constant and equal. We constructed a chart containing all ten motifs having four nodes Fig.
Each network gives rise to a system of four linear ODEs, which can be solved explicitly. In the nonlinear case we performed numerical simulation with both irreversible and reversible first reaction steps. The comparison of the efficacy of performance of the ten four-node networks Fig. The linear ODE also ranks H and I as the fastest four-node topologies, adding a third structure G , not only showing the same conversion time 2.
The nonlinear ODE models with reversible and irreversible first steps produce identical ordering of the ten structures.
It coincides with the ordering of the first seven structures, described above by CA and linear ODEs, while suggesting that network I is the fastest, G and H having very close performance, H shown as slightly slower than G. The topological analysis of the nine networks revealed some useful patterns of their dynamics. Although the networks analyzed here are relatively simple, they could be of use when analyzing local topology in large complex networks. Several of the observed topodynamic patterns are described below. Note, that the bi-parallel motifs F and J do not obey this rate inequality.
The CA and the nonlinear ODE simulations showed these two motifs with different, although relatively close efficacy, the structure J being the slower one:. The property of isodynamicity is a surprising novel network pattern, which could warrant further detailed studies.
The aim of this book is a detailed study of topological effects related to continuity of the dependence of solutions on initial values and parameters. This allows us. Chapter 51 - Topological structures of ordinary differential equations. Author links open Some basic properties of the set Z are discussed in the chapter.
Any ring closure of a linear chain of conversion of a source substrate S to the target product T accelerates the transformation. Acceleration of the process is strongest when the feed-forward link directly connects the substrate to the target and is the smallest when the link connects the substrate with an intermediate product Figs.
Topological feed-forward transformations 1, 2, and 3 always accelerate processes described as a linear chain of events. Adding a second feed-forward edge always accelerates the processes in a feed-forward motif. The ring-closures described by this pattern are shown in Fig. In all cases, the standard deviation the number of CA iterations was found to be more than two orders of magnitude smaller than that number. This pattern goes beyond the simple topological patterns 1 and 2 shown above, which cannot discriminate between FFB and FFC series. Adding a second feed-forward edge double feed-forward motif , between any pair of nodes in the longer path of the FF loop, speeds up the dynamics of the source substrate conversion into the target product Fig.
These inequalities for the number of iterations, illustrated in Fig.
Feed-forward motifs with different network topologies. The optimal number of runs must be large enough to provide reliable statistics, and at the same time not excessively large so as to minimize the computation time. Guglielmi, V. We derive a model system that describes the dynamics of a single species over two patches with local dynamics governed by Nicholson's DDE and coupled by density dependent dispersals. Old Password.
Comparing the FFF and FFE series, one may generalize that the acceleration of substrate-to-target conversion is higher when the second FF-link starts in a node located on the longer source-target path and ends into the target node, rather than to start in the source node and end in another node before the target one.
However, the ODE models do not confirm this result with the linear model showing G and H to be isodynamic, whereas the two nonlinear models shows G as slightly more efficient than H. Therefore, adding a third feed-forward link does not necessarily result in acceleration and no stable trend exists. Reversing the direction of one or more links in a feed-forward motif to turn it into a bi-parallel and tri-parallel one increases the network efficacy.
At the same number of nodes, the feed-forward motif is slower than the bi-parallel motif. The topology producing the fastest dynamics is that of the tri-parallel motif I. The technique of dynamic modeling with cellular automata shows great promise in modeling complex biological systems. Such systems can be broken down to subsystems of smaller scale to ease computational time and simulated independently so as to shed light on the processes on a larger scale.
The essential element in such applications is the extraction of useful topological-dynamic topodynamic patterns, which identify specific effects of topological structure on the dynamics of network processes while keeping all kinetic parameters constant. The beauty of the topological approach in studies of dynamics is in the generality of the patterns found, which are independent on the nature of the processes, and may be applied to any process of chemical transformation, as well as to any process of mass, energy or information transfer down the forward direction of the motifs.
The dynamics of the feed forward motifs observed in this study revealed important aspects of networks with such components. The topological hierarchy established in this study for four-node motifs predicts that the acceleration of the overall process in such motifs continue increasing with the decrease in the distance both along the shortest path and along all paths between the input and output nodes, whereas at the same distance the cellular automata and differential equation simulations produce in a similar manner a further distinction between the motifs dynamic efficacy.
The intriguing property of isodynamicity was identified showing motifs with the same number of nodes and different topology to have the same overall rate of input-to-output transformation. If shown to be present in larger biological networks, the observed isodynamic property could indicate a level of biological robustness at a topological level. Further topology-dynamics studies involving construction of networks from combinations of such structural blocks will aid in increasing our understanding of complex biological networks.
Apte performed the cellular automata simulations, J. Cain did the ODE modeling, D. Bonchev and S. Fong planned the study, and provided the topological D. All authors have read and approved the final manuscript.
The authors are indebted to Dr. National Center for Biotechnology Information , U. Journal List J Biol Eng v.
J Biol Eng. Published online Feb Author information Article notes Copyright and License information Disclaimer. Corresponding author. Advait A Apte: ude. Received Oct 1; Accepted Feb This article has been cited by other articles in PMC. Abstract Background Feed-forward motifs are important functional modules in biological and other complex networks. Results Feed-forward motif dynamics were studied using cellular automata and compared with differential equation modeling.
Conclusion It was shown for classes of structural motifs with feed-forward architecture that network topology affects the overall rate of a process in a quantitatively predictable manner. Background Modeling is a means of making predictions and testing our understanding.
Methods Cellular automata modeling of biochemical networks Cellular automata CA are modeling tools that represent dynamic systems discretely in space, time, and state.