DOCSLIB.ORG

The Role of Certain Post Classes in Boolean Network Models of Genetic Networks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Role of Certain Post Classes in Boolean Network Models of Genetic Networks The role of certain Post classes in Boolean network models of genetic networks Ilya Shmulevich*†, Harri La¨ hdesma¨ ki*‡, Edward R. Dougherty§, Jaakko Astola‡, and Wei Zhang* *Cancer Genomics Laboratory, University of Texas M. D. Anderson Cancer Center, Houston, TX 77030; ‡Institute of Signal Processing, Tampere University of Technology, 33720, Tampere, Finland; and §Department of Electrical Engineering, Texas A&M University, College Station, TX 77840 Communicated by Leo P. Kadanoff, University of Chicago, Chicago, IL, July 28, 2003 (received for review September 3, 2002) A topic of great interest and debate concerns the source of order most of its components being frozen. In this regime, the transfer and remarkable robustness observed in genetic regulatory net- of information between the components is impeded by large works. The study of the generic properties of Boolean networks walls of frozen components. In the chaotic regime, the system has proven to be useful for gaining insight into such phenomena. behaves in the opposite way, with a perturbation of one com- The main focus, as regards ordered behavior in networks, has been ponent propagating to many others in an avalanche-like manner, on canalizing functions, internal homogeneity or bias, and network with only a few isolated islands comprised of frozen components. connectivity. Here we examine the role that certain classes of Thus, networks in the chaotic regime are very sensitive to initial Boolean functions that are closed under composition play in the conditions and perturbations. The boundary between order and emergence of order in Boolean networks. The closure property chaos is called the complex regime or the critical phase, with the implies that any gene at any number of steps in the future is networks undergoing a kind of phase transition (11). It is in this guaranteed to be governed by a function from the same class. By regime that the networks are most evolvable. Kauffman (5) argues means of Derrida curves on random Boolean networks and perco- that life must exist on the edge of chaos such that the networks lation simulations on square lattices, we demonstrate that net- representing real genetic regulatory networks operate at or near works constructed from functions belonging to these classes have this critical phase. As Kauffman (12) puts it, ‘‘a living system must a tendency toward ordered behavior. Thus they are not overly first strike an internal compromise between malleability and sta- sensitive to initial conditions, and damage does not readily spread bility. To survive in a variable environment, it must be stable to be throughout the network. In addition, the considered classes are sure, but not so stable that it remains forever static.’’ significantly larger than the class of canalizing functions as the A Boolean network contains n elements (genes) {x1,...,xn}. ʦ ϭ connectivity increases. The functions in these classes exhibit the Each gene xi {0, 1} (i 1,...,n) is a binary variable, the value same kind of preference toward biased functions as do canalizing of which at time t ϩ 1 is completely determined by the values of functions, meaning that functions from this class are likely to be some other genes xj (i), xj (i), ..., xj ͑i͒ at time t by means of a 1 2 ki biased. Finally, functions from this class have a natural way of k 3 Boolean function fi:{0, 1} i {0, 1}. That is, there are ki genes ensuring robustness against noise and perturbations, thus repre- 3 assigned to gene xi and the mapping jk:{1, ...,n} {1,...,n}, senting plausible evolutionarily selected candidates for regulatory ϭ k 1,...,ki determines the ‘‘wiring’’ of gene xi. Thus, we can rules in genetic networks. ϩ ϭ write xi(t 1) fi(xj (i)(t), xj (i)(t),..., xj ͑i͒(t)). All genes are 1 2 ki assumed to update synchronously in accordance with the func- athematical and computational modeling is becoming tions assigned to them, and this process is then repeated. In a Mincreasingly important for understanding the complex random Boolean network, the functions fi are selected randomly, dynamical interactions in genetic regulatory systems. In light of as are the genes used as its inputs. recent development of high-throughput genomic technologies, Let us briefly consider several properties known to have modeling efforts have the potential to move from theory into an impact on the mode of operation of Boolean networks. The practice. The understanding of the underlying mechanisms used first two are the connectivity K and the function bias p. K ϭ by genetic networks not only sets the stage for a systematic view Ϫ1 ͚n n ϭ ki is the mean number of input variables used by the of physiology and pathophysiology but also carries tremendous i 1 random Boolean functions fi. We will assume in the sequel that practical potential in the context of model inference from real ϭ ϭ ki K for all i 1,...,n. The bias p of a random function fi measurement data. Because the implications of the logic of is the probability that it takes on the value 1. If p ϭ 0.5, then the genetic networks are difficult to deduce solely by means of function is said to be unbiased. By varying these parameters, the experimental techniques, computational and mathematical network can be made to undergo a phase transition. For modeling play a vital role (1). example, in the case of unbiased functions, the critical connec- Since their inception in 1969 by Kauffman, random Boolean tivity is K ϭ 2, meaning that for K Ͼ 2, we observe chaotic networks (2–5) have been one of the most intensively studied behavior. In general, for a given bias p, the critical connectivity models of discrete dynamical systems, enjoying a sustained is equal to K ϭ [2p(1 Ϫ p)]Ϫ1 (ref. 13). Alternatively, by inverting interest from both the biology and physics communities. Con- c this equation, a critical bias pc can be found for an arbitrary sidering their structural simplicity, these systems are capable of connectivity K. Strongly biased functions (when p is far away displaying a remarkably rich variety of complex behaviors and from 0.5) are said to have a high degree of internal homogeneity share many features of other dynamical system models. Analyt- (8) and are associated with increased order in Boolean networks. ical and numerical studies of the relationships between structural It has been known for quite some time that canalizing func- organization and dynamical behavior of Boolean networks have tions play a role in preventing chaotic behavior (2, 5, 14, 15). A yielded important insights into the overall behavior of large canalizing function is one in which at least one of the input genetic networks (5), evolutionary principles (6–8), and the variables is able to determine the value of the output of the development of chaos (9, 10). function regardless of the other variables. By increasing the Of particular interest are the relationships between local percentage of canalizing functions in a Boolean network, one can properties of the network and its global behavior. Various parameters specifying these local properties can be ‘‘tuned’’ such that the network is operating in one of several different regimes. †To whom correspondence should be addressed. E-mail: [email protected]. In the ordered regime, the system behaves in a simple way, with © 2003 by The National Academy of Sciences of the USA 10734–10739 ͉ PNAS ͉ September 16, 2003 ͉ vol. 100 ͉ no. 19 www.pnas.org͞cgi͞doi͞10.1073͞pnas.1534782100 Downloaded by guest on September 26, 2021 repeating this process until a satisfactory circuit is produced. A more reasonable and sustainable approach would be to develop a systematic way to design noise tolerance from the bottom up, for example by building in redundancy, exactly as von Neumann (20) envisioned in 1956. Thus, the issue is not whether evolution acted by means of random selection but what criteria or fitness functions it used to arrive at canalizing rules. As Sawhill and Kauffman (21) write, ‘‘canalization would be a natural way to design in robustness against noise and uncertainty.’’ It thus seems more plausible that evolution acted locally (at the level of regulatory rules) to ensure robustness and reliability, whereas global order came for free, exactly as Kauffman has been asserting. If canalizing functions indeed were found randomly and then selected, then it must have been quite a difficult task for evolution to find so few needles in such a large haystack. If a different class of functions, much more abundant in number than canalizing functions, could exhibit the same high fitness in terms of robustness and globally ordered behavior, it would have been much more likely for evolution to stumble on and select it. This possibility serves as one of the motivations of this article. Fig. 1. (a and c) Subplots, with K ϭ 4 and 5, respectively, show histograms of There is another issue, however. Canalizing functions, strictly the number of canalizing (solid line) and A2 ഫ a2 (dashed line) functions versus speaking, constitute a class of functions in the sense that a given ϭ the number of ones in their truth tables. (b and d) Subplots, with K 4 and 5, function either belongs or does not belong to this class. Biased respectively, show the probability that a randomly chosen Boolean function with bias p is a canalizing (solid line) or A2 ഫ a2 (dashed line) function. functions, on the other hand, do not constitute a class. That is, because the definition of bias is a probabilistic one, given a bias 0 Ͻ p Ͻ 1, any Boolean function can still be generated with a move closer toward the ordered regime and, depending on the positive probability.¶ Thus, a p-biased random network, meaning connectivity and the distribution of the number of canalizing one with functions that are all chosen randomly with a bias p, variables in the randomly chosen canalizing functions (16), cross could potentially contain any Boolean function.
Load more

Recommended publications
  • Markov Model Checking of Probabilistic Boolean Networks Representations of Genes
    Markov Model Checking of Probabilistic Boolean Networks Representations of Genes Marie Lluberes1, Jaime Seguel2 and Jaime Ramírez-Vick3 1, 2 Electrical and Computer Engineering Department, University of Puerto Rico, Mayagüez, Puerto Rico 3 General Engineering Department, University of Puerto Rico, Mayagüez, Puerto Rico or, on contrary, to prevent or to stop an undesirable Abstract - Our goal is to develop an algorithm for the behavior. This “guiding” of the network dynamics is automated study of the dynamics of Probabilistic Boolean referred to as intervention. The power to intervene with the Network (PBN) representation of genes. Model checking is network dynamics has a significant impact in diagnostics an automated method for the verification of properties on and drug design. systems. Continuous Stochastic Logic (CSL), an extension of Biological phenomena manifest in the continuous-time Computation Tree Logic (CTL), is a model-checking tool domain. But, in describing such phenomena we usually that can be used to specify measures for Continuous-time employ a binary language, for instance, expressed or not Markov Chains (CTMC). Thus, as PBNs can be analyzed in expressed; on or off; up or down regulated. Studies the context of Markov theory, the use of CSL as a method for conducted restricting genes expression to only two levels (0 model checking PBNs could be a powerful tool for the or 1) suggested that information retained by these when simulation of gene network dynamics. Particularly, we are binarized is meaningful to the extent that it is remains in a interested in the subject of intervention. This refers to the continuous domain [2].
    [Show full text]
  • Online Spectral Clustering on Network Streams Yi
    Online Spectral Clustering on Network Streams By Yi Jia Submitted to the graduate degree program in Electrical Engineering and Computer Science and the Graduate Faculty of the University of Kansas in partial fulfillment of the requirements for the degree of Doctor of Philosophy Jun Huan, Chairperson Swapan Chakrabarti Committee members Jerzy Grzymala-Busse Bo Luo Alfred Tat-Kei Ho Date defended: The Dissertation Committee for Yi Jia certifies that this is the approved version of the following dissertation : Online Spectral Clustering on Network Streams Jun Huan, Chairperson Date approved: ii Abstract Graph is an extremely useful representation of a wide variety of practical systems in data analysis. Recently, with the fast accumulation of stream data from various type of networks, significant research interests have arisen on spectral clustering for network streams (or evolving networks). Compared with the general spectral clustering problem, the data analysis of this new type of problems may have additional requirements, such as short processing time, scalability in distributed computing environments, and temporal variation tracking. However, to design a spectral clustering method to satisfy these requirement cer- tainly presents non-trivial efforts. There are three major challenges for the new algorithm design. The first challenge is online clustering computation. Most of the existing spectral methods on evolving networks are off-line methods, using standard eigensystem solvers such as the Lanczos method. It needs to recompute solutions from scratch at each time point. The second challenge is the paralleliza- tion of algorithms. To parallelize such algorithms is non-trivial since standard eigen solvers are iterative algorithms and the number of iterations can not be pre- determined.
    [Show full text]
  • Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence
    Hindawi Publishing Corporation EURASIP Journal on Bioinformatics and Systems Biology Volume 2007, Article ID 32454, 15 pages doi:10.1155/2007/32454 Research Article Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence Stephen Marshall,1 Le Yu,1 Yufei Xiao,2 and Edward R. Dougherty2, 3, 4 1 Department of Electronic and Electrical Engineering, Faculty of Engineering, University of Strathclyde, Glasgow, G1 1XW, UK 2 Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128, USA 3 Computational Biology Division, Translational Genomics Research Institute, Phoenix, AZ 85004, USA 4 Department of Pathology, University of Texas M. D. Anderson Cancer Center, Houston, TX 77030, USA Received 10 July 2006; Revised 29 January 2007; Accepted 26 February 2007 Recommended by Tatsuya Akutsu The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of proba- bilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation is that the characteristics of a single Boolean network without perturbation may be de- termined by its pairwise transitions. Because the network function is fixed and there are no perturbations, a given state will always be followed by a unique state at the succeeding time point. Thus, a transition counting matrix compiled over a data sequence will be sparse and contain only one entry per line. If the network also has perturbations, with small perturbation probability, then the transition counting matrix would have some insignificant nonzero entries replacing some (or all) of the zeros.
    [Show full text]
  • Boolean Network Control with Ideals
    Portland State University PDXScholar Mathematics and Statistics Faculty Fariborz Maseeh Department of Mathematics Publications and Presentations and Statistics 1-2021 Boolean Network Control with Ideals Ian H. Dinwoodie Portland State University, [email protected] Follow this and additional works at: https://pdxscholar.library.pdx.edu/mth_fac Part of the Physical Sciences and Mathematics Commons Let us know how access to this document benefits ou.y Citation Details Dinwoodie, Ian H., "Boolean Network Control with Ideals" (2021). Mathematics and Statistics Faculty Publications and Presentations. 302. https://pdxscholar.library.pdx.edu/mth_fac/302 This Working Paper is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Boolean Network Control with Ideals I H Dinwoodie Portland State University January 2021 Abstract A method is given for finding controls to transition an initial state x0 to a target set in deterministic or stochastic Boolean network control models. The algorithms use multivariate polynomial algebra. Examples illustrate the application. 1 Introduction A Boolean network is a dynamical system on d nodes (or coordinates) with binary node values, and with transition map F : {0, 1}d → {0, 1}d. The c coordinate maps F = (f1,...,fd) may use binary parameters u ∈ {0, 1} that can be adjusted or controlled at each step, so the image can be writ- ten F (x, u) and F takes {0, 1}d+c → {0, 1}d, usually written in logical nota- tion.
    [Show full text]
  • Random Boolean Networks As a Toy Model for the Brain
    UNIVERSITY OF GENEVA SCIENCE FACULTY VRIJE UNIVERSITEIT OF AMSTERDAM PHYSICS SECTION Random Boolean Networks as a toy model for the brain MASTER THESIS presented at the science faculty of the University of Geneva for obtaining the Master in Theoretical Physics by Chlo´eB´eguin Supervisor (VU): Pr. Greg J Stephens Co-Supervisor (UNIGE): Pr. J´er^ome Kasparian July 2017 Contents Introduction1 1 Biology, physics and the brain4 1.1 Biological description of the brain................4 1.2 Criticality in the brain......................8 1.2.1 Physics reminder..................... 10 1.2.2 Experimental evidences.................. 15 2 Models of neural networks 20 2.1 Classes of models......................... 21 2.1.1 Spiking models...................... 21 2.1.2 Rate-based models.................... 23 2.1.3 Attractor networks.................... 24 2.1.4 Links between the classes of models........... 25 2.2 Random Boolean Networks.................... 28 2.2.1 General definition..................... 28 2.2.2 Kauffman network.................... 30 2.2.3 Hopfield network..................... 31 2.2.4 Towards a RBN for the brain.............. 32 2.2.5 The model......................... 33 3 Characterisation of RBNs 34 3.1 Attractors............................. 34 3.2 Damage spreading........................ 36 3.3 Canonical specific heat...................... 37 4 Results 40 4.1 One population with Gaussian weights............. 40 4.2 Dale's principle and balance of inhibition - excitation..... 46 4.3 Lognormal distribution of the weights.............. 51 4.4 Discussion............................. 55 i 5 Conclusion 58 Bibliography 60 Acknowledgements 66 A Python Code 67 A.1 Dynamics............................. 67 A.2 Attractor search.......................... 69 A.3 Hamming Distance........................ 73 A.4 Canonical specific heat.....................
    [Show full text]
  • A Random Boolean Network Model and Deterministic Chaos
    A Random Boolean Network Model and Deterministic Chaos Mihaela T. Matache* Jack Heidel Department of Mathematics University of Nebraska at Omaha Omaha, NE 68182-0243, USA *[email protected] Abstract This paper considers a simple Boolean network with N nodes, each node’s state at time t being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali ( [5]) who consider the same number of parents for all nodes. We make use of the same Boolean rule as the authors of [5], provide a generalization of the formula for the probability of finding a node in state 1 at a time t and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period- doubling bifurcations which turn into reversed (period - halving) bifurcations for certain combinations of parameter values. 1. Introduction In recent years various studies ( [5], [16], [19], [20], [21], [22], [30], [31]) have shown that a large class of biological networks and cellular automata can be modelled as Boolean networks. These models, besides being easy to understand, are relatively easy to handle. The interest for Boolean networks and their appli- cations in biology and automata networks has actually started much earlier ( [8], [13], [14], [15], [25], [29], [32]), with publications such as the one of Kauffman ( [25]) whose work on the self-organization and adap- tation in complex systems has inspired many other research endeavors.
    [Show full text]
  • Reward-Driven Training of Random Boolean Network Reservoirs for Model-Free Environments
    Portland State University PDXScholar Dissertations and Theses Dissertations and Theses Winter 3-27-2013 Reward-driven Training of Random Boolean Network Reservoirs for Model-Free Environments Padmashri Gargesa Portland State University Follow this and additional works at: https://pdxscholar.library.pdx.edu/open_access_etds Part of the Dynamics and Dynamical Systems Commons, and the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y Recommended Citation Gargesa, Padmashri, "Reward-driven Training of Random Boolean Network Reservoirs for Model-Free Environments" (2013). Dissertations and Theses. Paper 669. https://doi.org/10.15760/etd.669 This Thesis is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Reward-driven Training of Random Boolean Network Reservoirs for Model-Free Environments by Padmashri Gargesa A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering Thesis Committee: Christof Teuscher, Chair Richard P. Tymerski Marek A. Perkowski Portland State University 2013 Abstract Reservoir Computing (RC) is an emerging machine learning paradigm where a fixed kernel, built from a randomly connected "reservoir" with sufficiently rich dynamics, is capable of expanding the problem space in a non-linear fashion to a higher dimensional feature space. These features can then be interpreted by a linear readout layer that is trained by a gradient descent method. In comparison to traditional neural networks, only the output layer needs to be trained, which leads to a significant computational advantage.
    [Show full text]
  • Solving Satisfiability Problem by Quantum Annealing
    UNIVERSITY OF CALIFORNIA Los Angeles Towards Quantum Computing: Solving Satisfiability Problem by Quantum Annealing A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical Engineering by Juexiao Su 2018 c Copyright by Juexiao Su 2018 ABSTRACT OF THE DISSERTATION Towards Quantum Computing: Solving Satisfiability Problem by Quantum Annealing by Juexiao Su Doctor of Philosophy in Electrical Engineering University of California, Los Angeles, 2018 Professor Lei He, Chair To date, conventional computers have never been able to efficiently handle certain tasks, where the number of computation steps is likely to blow up as the problem size increases. As an emerging technology and new computing paradigm, quantum computing has a great potential to tackle those hard tasks efficiently. Among all the existing quantum computation models, quantum annealing has drawn significant attention in recent years due to the real- ization of the commercialized quantum annealer, sparking research interests in developing applications to solve problems that are intractable for classical computers. However, designing and implementing algorithms that manage to harness the enormous computation power from quantum annealer remains a challenging task. Generally, it requires mapping of the given optimization problem into quadratic unconstrained binary optimiza- tion(QUBO) problem and embedding the subsequent QUBO onto the physical architecture of quantum annealer. Additionally, practical quantum annealers are susceptible to errors leading to low probability of the correct solution. In this study, we focus on solving Boolean satisfiability (SAT) problem using quantum annealer while addressing practical limitations. We have proposed a mapping technique that maps SAT problem to QUBO, and we have further devised a tool flow that embeds the QUBO onto the architecture of a quantum annealing device.
    [Show full text]
  • A Multi-Agent System for Environmental Monitoring Using Boolean Networks and Reinforcement Learning
    Journal of Cyber Security DOI:10.32604/jcs.2020.010086 Article A Multi-Agent System for Environmental Monitoring Using Boolean Networks and Reinforcement Learning Hanzhong Zheng1 and Dejie Shi2,* 1Department of Computer Science, The University of Pittsburgh, Pittsburgh, PA 15213, USA 2School of Computer and Information Engineering, Hunan University of Technology and Business, Changsha, 410205, China *Corresponding Author: Dejie Shi. Email: [email protected] Received: 10 February 2020; Accepted: 28 June 2020 Abstract: Distributed wireless sensor networks have been shown to be effective for environmental monitoring tasks, in which multiple sensors are deployed in a wide range of the environments to collect information or monitor a particular event, Wireless sensor networks, consisting of a large number of interacting sensors, have been successful in a variety of applications where they are able to share information using different transmission protocols through the communication network. However, the irregular and dynamic environment requires traditional wireless sensor networks to have frequent communications to exchange the most recent information, which can easily generate high communication cost through the collaborative data collection and data transmission. High frequency communication also has high probability of failure because of long distance data transmission. In this paper, we developed a novel approach to multi-sensor environment monitoring network using the idea of distributed system. Its communication network can overcome the difficulties of high communication cost and Single Point of Failure (SPOF) through the decentralized approach, which performs in-network computation. Our approach makes use of Boolean networks that allows for a non-complex method of corroboration and retains meaningful information regarding the dynamics of the communication network.
    [Show full text]
  • List of Glossary Terms
    List of Glossary Terms A Ageofaclusterorfuzzyrule 1054 A correlated equilibrium 3064 Agent 58, 76, 105, 1767, 2578, 3004 Amechanism 1837 Agent architecture 105 Anearestneighbor 790 Agent based models 2940 A social choice function 1837 Agent (or software agent) 2999 A stochastic game 3064 Agent-based computational models 2898 Astrategy 3064 Agent-based model 58 Abelian group 2780 Agent-based modeling 1767 Absolute temperature 940 Agent-based modeling (ABM) 39 Absorbing state 1080 Agent-based simulation 18, 88 Abstract game 675 Aggregation 862 Abstract game of network formation with respect to Aggregation operators 122 irreflexive dominance 2044 Aging 2564, 2611 Abstract game of network formation with respect to path Algebra 2925 dominance 2044 Algebraic models 2898 Accuracy 161, 827 Algorithm 2496 Accuracy (rate) 862 Algorithmic complexity of object x 132 Achievable mate 3235 Algorithmic self-assembly of DNA tiles 1894 Action profile 3023 Allowable set of partners 3234 Action set 3023 Almost equicontinuous CA 914, 3212 Action type 2656 Alphabet of a cellular automaton 1 Activation function 813 ˛-Level set and support 1240 Activator 622 Alternating independent-2-paths 2953 Active membranes 1851 Alternating k-stars 2953 Actors 2029 Alternating k-triangles 2953 Adaptation 39 Amorphous computer 147 Adaptive system 1619 Analog 3187 Additive cellular automata 1 Analog circuit 3260 Additively separable preferences 3235 Ancilla qubits 2478 Adiabatic switching 1998 Anisotropic elements 754 Adjacency matrix 1746, 3114 Annealed law 2564 Adjacent 2864,
    [Show full text]
  • Introduction to Natural Computation Lecture 18 Random Boolean
    Introduction to Natural Computation Lecture 18 Random Boolean Networks Alberto Moraglio 1 / 25 Random Boolean Networks Extension of Cellular Automata. Introduced by Stuart Kauffman (1969) as model for genetic regulatory networks. Uncover fundamental principles of living systems. Hypothesis: living organisms can be constructed from random elements. Simplification: Boolean states for all nodes in the network. 2 / 25 Definition Random Boolean Network (RBN) A random Boolean network consists of N nodes, each with a state 0 or 1 K edges for each node to K different randomly selected nodes a lookup table for each node that determines the next state, according to states of its K neighbours. Last few lectures: fixed graph, random process This time: random graph, deterministic process Note: randomness used to create network, afterwards network remains fixed! 3 / 25 Relation to Cellular Automata http://www.metafysica.nl/boolean.html 4 / 25 Lookup Tables Recall rules from Cellular Automata: RBN: each node has its own random lookup table: inputs at time t state at time t + 1 000 1 001 0 010 0 011 1 100 1 101 0 110 1 111 1 5 / 25 Example of a Simple RBN Gershenson C. Introduction to Random Boolean Networks. ArXiv Nonlinear Sciences e-prints. 2004; 6 / 25 States State space has size 2N . After at most 2N + 1 steps a state will be repeated. 7 / 25 States State space has size 2N . After at most 2N + 1 steps a state will be repeated. Attractors As the system is deterministic, it will get stuck in an attractor. one state → point attractor / steady state two or more states → cycle attractor / state cycle The set of states that flow towards an attractor is called attractor basin.
    [Show full text]
  • UCLA Samueli Engineering Annoucemente 2019-20
    ANNOUNCEMENT 2019-20 HENRY SAMUELI SCHOOL OF ENGINEERING AND APPLIED SCIENCE UNIVERSITY OF CALIFORNIA, LOS ANGELES OCTOBER 1, 2019 Contents A Message from the Dean . .3 Exceptional Student Admissions Program . 16 Henry Samueli School of Engineering and Applied Science. 4 Official Publications . 16 Grades . 16 Officers of Administration . 4 Nondiscrimination . 16 The Campus . 4 Harassment . 17 The School . 4 Undergraduate Programs . 19 Endowed Chairs . 4 Admission . 19 The Engineering Profession . 5 Requirements for B.S. Degrees . 22 Calendars. 8 Policies and Regulations . 24 General Information . 9 Honors. 25 Facilities and Services . 9 Graduate Programs . 26 Library Facilities . 9 Master of Science Degrees. 26 Services . 9 Continuing Education . 9 Admission . 27 Career Services . 10 Departments and Programs of the School. 28 Health Services . 10 Bioengineering. 28 Services for Students with Disabilities. 10 Chemical and Biomolecular Engineering. 40 International Student Services . 10 Civil and Environmental Engineering . 50 Fees and Financial Support. 11 Computer Science . 62 Fees and Expenses . 11 Living Accommodations . 11 Electrical and Computer Engineering . 83 Financial Aid . 11 Materials Science and Engineering. 101 Special Programs, Activities, and Awards . 13 Mechanical and Aerospace Engineering . 109 Center for Excellence in Engineering and Diversity (CEED) . 13 Master of Science in Engineering Online Programs . 125 Student Organizations. 15 Schoolwide Programs and Courses. 127 Women in Engineering . 15 Externally Funded Research
    [Show full text]