Luận án tiến sĩ: Modeling and Symbolic Analysis of Biological Protein Signaling Networks Using Hybrid Automata

This thesis proposes a hybrid automata framework for modeling such processes; hybrid automata theory being a hierarchical mathematical system that uses tial equations to model continuous

MODELING AND SYMBOLIC ANALYSIS OF BIOLOGICAL PROTEIN SIGNALING NETWORKS USING HYBRID

AUTOMATA

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF

AERONAUTICS AND ASTRONAUTICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Ronojoy GhoshDecember 2005

UMI Number: 3197434

Copyright 2006 byGhosh, Ronojoy

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ii

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

of Doctor of Philosophy

CL,

(Claire Tomlin) Principal Adviser

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

/ (Stephen Rock) ⁄

I certify that I have read this dissertation and that, in my opinion, it

I certify that I have read this dissertation and that, in my opinion, it

is fully adequate in scope and quality as a dissertation for the degree

Recent advances in quantitative biology have created a tremendous opportunity toapply dynamical systems modeling to biological phenomena, and to validate thesemodels using experimental data Using simulation and analysis, there is immensescope to discover non-intuitive design principles behind biological processes, and

successfully predict the effects of changing key variables Systems biology, defined asthe integration of mathematical analysis with experimental biology, has the potential

to revolutionize the way biology is done

Cellular protein signaling networks exhibit complex combinations of both discrete

and continuous behaviors The dynamics that govern the spatial and temporal

in-crease or dein-crease of protein concentrations inside cells are continuous differentialequations, while the activation or deactivation of these continuous dynamics are trig-

gered by discrete switches that involve regulating species concentrations reachinggiven thresholds

This thesis proposes a hybrid automata framework for modeling such processes;

hybrid automata theory being a hierarchical mathematical system that uses tial equations to model continuous dynamics, and discrete event-driven switches tomodel the governing equations in different modes of operation In particular, the the-sis proposes hybrid models of two interesting intercellular signaling pathways activeduring embryonic development: the lateral inhibitory Delta-Notch pathway respon-sible for pattern formation in the embryonic skin of Xenopus laevis, and the Planar

differen-Cell Polarity (PCP) signaling pathway in Drosophila melanogaster wings Thesemodels are validated against experimentally observed steady state protein concen-

tration patterns

iv

A fundamental objective of this work is to analytically compute constraints on

the kinetic parameters of the model, for particular biologically observed or interestingsteady states to exist The constraints are computed symbolically, i.e without having

to numerically instantiate the parameters This is a great advantage in the context

of biological processes, where exact numerical parameters cannot often be identified

from experimental data, but a range of values, or relative values for the parameterscan be obtained The particular structure of the hybrid automata models developed

in this work make symbolic constraint generation computationally tractable

Another key objective is the computation of initial conditions, or initial protein

concentrations, that converge to a particular steady state The initial conditions can

be interpreted as initial biases in the distribution of signaling species that lead to abiologically interesting steady state This is posed as a backward reachable set com-

putation problem An abstraction procedure is presented that converts the hybrid

automaton into a discrete transition system using symbolic solutions to the tial equations and Lie derivatives to compute transitions between discrete states Thebackward reachability problem is then computed on the discrete abstraction, which

differen-makes the analysis tractable for large state spaces The reachability computation isimplemented using MATLAB and the quantifier elimination tool QEPCAD and is

demonstrated for multiple cell Delta-Notch signaling networks with up to eighteen

continuous variables

Since the computed reachable sets are large, it is difficult to directly interpret

them in a biologically meaningful way To solve this problem, a query algorithm isdeveloped and presented that can be used to test whether a particular protein distri-bution is guaranteed to converge to a steady state of interest The use of the queryalgorithm is demonstrated for the Delta-Notch hybrid model The thesis concludes

with a description of the implementation of the analysis tools on a publicly availablesystems biology software platform known as Bio-SPICE A further example, lactose

metabolism inside a cell, is described as an illustration of the methods developed in

this work; and reachable sets are computed for this model using the tools integrated

with Bio-SPICE

I would like to thank my principal adviser, Professor Claire Tomlin, for her guidance

and support during my graduate career at Stanford University Her insight, teaching

skills, mentorship, and unwavering enthusiasm made this work possible I would alsolike to thank Professor David Dill for introducing me to formal verification, whichenabled me to approach control theoretic analysis from a different perspective, andfor his insightful comments on my research I gratefully acknowledge the research

collaboration with Professor Jeffrey Axelrod, who gave me an opportunity to observe

the wonderful world of experimental biology, and for his useful comments that helped

refine this thesis I would like to thank Professor Stephen Rock, for his valuable

comments on my dissertation research and for encouraging me to write this thesis in

a style accessible to both engineers and biologists

It is a pleasure to acknowledge my research collaborations with Dr Ashish

Ti-wari and Dr Patrick Lincoln, at SRI International The experience I gained working

with them on implementing symbolic abstraction and verification algorithms proved

invaluable when I was attempting to design and implement my own analysis

algo-rithms I would like to thank Dr Adam Halasz and Professor Vijay Kumar at the

University of Pennsylvania, for our research collaborations on the lactose metabolism

model My sincere thanks go to Professors Harley McAdams and Lucy Shapiro, for

sustaining my research interest in systems biology during the crucial early years of

my graduate career The discussions we had on mathematical modeling of geneticnetworks provided valuable insight into the biologist’s view of the problem I wouldalso like to thank Professor Hasan Suhail, my undergraduate adviser at the Indian

vi

Institute of Technology, Kharagpur, for initiating my interest in computational

biol-ogy.

In addition, I would like to thank Keith Amonlirdviman, Gokhan Inalhan, dre Bayen, Jung Soon Jang, Inseok Hwang, Meeko Oishi, Rodney Teo, Ian Mitchell,Hamsa Balakrishnan, Robin Raffard, Gabe Hoffmann, Steven Waslander, KaushikRoy, Peter Brende, Sriram Shankaran, Jianghai Hu, and Dusan Stipanovic for theircollaborations, informal conversations, and for making the hybrid systems laboratory

Alexan-an exciting Alexan-and stimulating place to work I would like to gratefully acknowledge thefinancial support of DARPA, and thank Dr Eric Eisenstadt and Dr Sri Kumar fortheir encouragement and interest in my research I would also like to thank Stanford

University for the School of Engineering Fellowship and the Interstate Electronics

Corporation Fellowship I would like to thank my friends, Daniel Levner in ular, for their encouragement and for adding color to life outside of work Lastly,

partic-I would especially like to thank my wife, Sheila, and my parents for their constantsupport and encouragement without which my graduate career would not have been

possible

vii

2 Protein Signaling

2.1 Developmental Signaling 0000000.2.2 Lateral Inhibition Through Delta-Notch Signaling

2.3 Planar Cell Polarity Signaling cv

2.4 Motivation for Hybrid Model 2 02 0008.4

2.5 Switching Functions 1 0 ee ee

3 Hybrid Automata

3.1 Hybrid Automata and Transition Systems

4 Delta-Notch Signaling

4.1 Delta-Notch Lateral Inhibition 000

4.1.1 Previous Work: Mathematical Models

4.2 Hybrid Automaton with Piecewise Linear Switch

4.21 Model Development 2 0 0.000 eee eee

4,2.2 Equilibrium Analysis and Constraint Generation

2020

4.2.3 Simulation Results for a Planar Array of£Cells

4.3 Hybrid Automaton with Signum Switch

4.3.1 Model Development 00.0004 eee 4.3.2 Equilibrium Analysis and Constraint Generation

4.3.3 Simulation Results for 1 and 2 Dimensional Networks of Cells 4.3.4 Comparison with Nonlinear ODE Model

Reachability Computation 5.1 Reachability: Mathematical Definitions

5.2 Lie Derivative 2 ee 5.3 Abstraction Procedure 1 0 ee 5.4 Reachable Set Computation Results

9.4.1 One Cell Delta-Notch Automaton

5.4.2 Two Cell Delta-Notch Automaton

5.4.3 Four Cell Delta-Notch Automaton

5.5 Query-Based Interpretation 0 0.000 eee eee 5.5.1 Structure of Computed Reachable Sets

5.5.2 Example: Four Cell Delta-Notch Automaton

5.5.3 Query Based Interpretation Algorithm

5.5.4 Query Results for Four Cell Delta-Notch Automaton

5.6 Proposed Biological Experiments 0000 eae Planar Cell Polarity Signaling 6.1 Model Properties 2 0 Q Q ng và và va 6.2 Hybrid Automaton Model Development

6.3 Equilibrium Analysis and Parameter Constraints

6.4 Simulation Results and Experimental Validation

Further Applications 7.1 Integration with Bio-SPICE Systems Biology Software Platform 7.2 Lactose Metabolism within a Bacterial Cel

7.21 Hybrid Model Development and Simplification

ix

55 59 56 58 69 70 71 74 76 77 78 81 82 87

7.2.2 Reachability Analysis Results

8 Future Work

A Definition of Lactose Metabolism Hybrid Model

Bibliography

Equilibria of the single cell automaton Hone cell, PWA - - 0 ee 33

Existence conditions for equilibrium points of Hạns se PWA- - - + - 33Unsatisfiable constraint list for possible equilibrium-containing modes

of Hiwo-cell, PW A- SS 36Existence conditions for equilibrium points of Honecell 2 0 0 - 45

Existence conditions for equilibrium points of Hiwoccy (the

composi-tion of two single-cell hybrid automata) .0.4 49

Steady state protein concentrations in four cell Delta-Notch network 78

Equilibria of the single cell automaton Hpgp Of the 256 equilibria,only the important ones are listed 2 0.0 0.20 000 96Existence conditions for equilibria of Hpcp .204 97

Parameter values for lactose metabolism hybrid automaton 114Equilibria of reduced order lactose metabolism hybrid automaton Aljgeg.115

xi

pre-of the wing below shows the hexagonal shape pre-of the cells and the hairspointing in a similar direction 2 0 eee ee 16Planar cell polarity signaling network between two adjacent cells inDrosophila pupal wing 2 1v nà và k va 17Piecewise linear switching function .0 19Sigmoidal switching function © 2 0.0.0.0 0000000 19

Continuous state space with geometrical representations of polynomialmodal invariants of a two dimensional hybrid automaton 24

(a) Hexagonal close-packed layout scheme for cells in two dimensionalarrays (b) Influence diagram for Delta-Notch protein signaling network 27

Transition diagram for a single cell hybrid automaton with piecewiselinear switch, © ung và g v v và VN va 31Hybrid automaton for a 3 x 3 array, modeling a nine cell network 32Equilibrium mode map for single cell automaton with piecewise linearswitching function © kg kg kg kg kg k va 34

Effect on equilibria of two cell automaton Hs se¡ pwA With changes

in switch slope m Note the disappearance of the two equilibria at

(1,0) and (0,1) when mm < au TH xặ:cừýáa 38

xii

Steady state protein concentration distribution in a planar array of

cells for nonlinear model with sigmoid switch, 41Steady state protein concentrations for hybrid model with shallow lin-

ear switch 1 a a ẶẼẶẼ.Ă 42Steady state protein concentrations for nonlinear model with shallow

Phase portrait for a single cell hybrid automaton 47Pruned transition diagrams for Delta-Notch hybrid automata 50Simulation results showing the steady state of each cell Red indicates

a differentiated cell and white indicates an undifferentiated cell 52

Phase plane projections for two cell system showing equilibria Labels

đi and dz are the Delta protein concentrations in cell 1 and 2 respectively 54

(a) Phase portrait of a hybrid automaton showing system dynamics

and discrete state partitions (b) States partitioned into boundary

(switching surface) and interior 0 0 00.02 eee 60(a) The vector field in each discrete state is used to compute Lie deriva-

tives that determine discrete state transitions (b) Transitions

com-puted between the discrete states, by abstracting the vector field Note

that there may be several transitions out of one discrete state (for ample, state q) E Sẽ Ma 61(a) Subdivision of state g, with multiple transitions, into states with

ex-one transition each The dividing polynomial is an exact solution of

the differential equations governing continuous flow in q, (b) Iterative

application of the refinement procedure to predecessor states of gq 64

Generation of sub-partitioning surfaces for iterative refinement of

par-titions This diagram shows the projection of the sub-partitioningsurface of state gịo of a two cell Delta-Notch hybrid automaton 66

xili

(a) An example of reachability computation using the abstracted

tran-sition system, the gray shaded area is completely reachable from the

final state (b) Schematic state transition diagram showing

approxi-mate backward reachable sets from final states 68Exact discrete abstraction for a single cell Delta-Notch automaton 70Projections showing computed backward reachable set from the equi-

libria for the two cell Delta-Notch automaton The cyan set representsthe reachable set for equilibrium 1, and the green one for equilibrium 2 73

(a) Layout of four cell Delta-Notch network showing the variables

as-sociated with each cell b) Biologically consistent steady states of thefour cell network, a shaded cell represents a high steady state concen-tration of Delta protein, and an unshaded cell has high Notch protein

Biologically consistent steady states of the four cell network with odic boundary conditions A shaded cell represents a high steady stateconcentration of Delta protein and low level of Notch protein, and anunshaded cell has low Delta protein and high Notch protein at steady

peri-Influence diagram showing initial and predicted steady state proteinconcentrations in a two cell network .0 008 89

Influence diagram showing initial and predicted steady state protein

concentrations in a four cell network .20202020202 0004 90

Compartmentalized Drosophila wing epithelial cell 93Dsh protein localization: (a) GFP-tagged Dsh protein localizes to thedistal boundary of the cell, indicated by the yellow arrows, and (b) dis-

tinctive localization pattern in a wild-type fly wing Figures courtesy

of Professor Jeffrey Âxelrod ee 100

Simulation results showing Dsh protein localization to the distal cellmembrane, at steady state in a wild-type wing 102

XIV

Simulation results showing incorrect Dsh protein localization, at steadystate, when threshold parameters do not satisfy constraints .Simulation results showing incorrect Fz protein localization, at steadystate, when threshold parameters do not satisfy constraints .Simulation results showing incorrect Pk protein localization, at steady

state, when threshold parameters do not satisfy constraints

Fz localization in Fz mutant wing: (a) direction of hairs reversedimmediately to the right of the Fz mutant patch (figure from Vinsonand Adler, Nature, 329, 549-551, 1987), and (b) simulation results

showing similar reversal of Fz localization

Screen snapshot showing Bio-SPICE dashboard with reachability

anal-ysis toolboxes 6 Q Q Q HQ gà gà Nà kiaSchematic diagram of lactose metabolism process inside a cell

State space diagram of lactose metabolism hybrid automaton showing

switching surfaces and computed partition

XV

Chapter 1

Introduction

Modeling, simulation and analysis of biological processes is a fast-expanding and

increasingly fruitful area of research Mathematical tools from a wide range of

en-gineering disciplines have been applied to biological systems at various levels of straction; from individual protein molecular dynamics to organ modeling and other

ab-physiological processes, to population models Mathematical modeling enables the

systematic organization and interpretation of experimental data to help understandthe design principles behind the process This is achieved by identifying the parame-ters of the system, validating the model against experimental observations, and thengenerating predictions that involve non-intuitive behavior of the system, which can

be tested experimentally to give new insight into the process under study In therecent past, there has been increasing interest in modeling one particular type ofbiological process: that which is associated with signaling and regulatory networksinvolving proteins and genes within individual cells This is due to rapid advances

in the two disciplines of molecular biology and mathematical modeling, which volve very different areas of scientific expertise, but which are now driving research

in-in computational systems biology [63], in-in complement

In the field of molecular biology, recent developments have enabled researchers

to construct experiments that generate large amounts of high quality data, reliablyand at relatively low expense These breakthrough technologies include genetic ma-nipulation techniques that can control the expression profile of single genes during

CHAPTER I INTRODUCTION 2

embryonic development [96], as well as post-transcriptional control using externallyintroduced chemical signals like RNAi [36] On the sensing and measurement side,

microarray technology has currently evolved to a state where gene expression

pro-files can be sensed for thousands of different genes simultaneously, at reasonablecost [27] In experiments where in vivo concentrations of proteins have to be vi-sualized and measured, advances in tagging proteins with fluorescent markers havegiven scientists a large palette of fluorophores to work with [71, 87] More excit-

ing, ongoing research in fluorescence spectroscopy, such as fluorescence recovery after

photobleaching (FRAP) [97], fluorescence correlation spectroscopy (FCS) [111], andfluorescence resonance energy transfer (FRET) [61], promises to provide key tools to

obtain quantitative data about protein dynamics and protein-protein binding, inside

living cells [28, 35, 107, 109]

On the computational side, biomathematical modeling and analysis have fited from the exponential increase in computing power over the last two decades

bene-A wide range of modeling frameworks and analysis techniques, developed primarily

for engineering and physical processes, have now been applied to cell biology [14,

51, 56, 62, 95, 108] These include formal verification [88], pathway logic [38], Petri

nets [91, 92], and cellular automata [113], from computer science; flux balance

anal-ysis [37], and stochastic kinetic modeling [10, 52, 80], from chemical engineering; as

well as several linear, nonlinear and switched ordinary and partial differential

equa-tion based dynamic models [7, 67, 76, 112], and multistable logic circuit analysis [100],

from control theory and engineering Although a large portion of the research efforthas been concentrated on model development and simulation; the control theoretic

contributions have included analysis of phenomena such as feedback (both negative

and positive) [42, 70], multistability, bifurcations, hysteresis, oscillations [8], and bustness [98] Phenomena such as heat shock response in E coli [39, 40], signaling

ro-in the mitogen-activated protero-in kro-inase (MAPK) pathway [15], quorum sensro-ing ro-in

bio-luminescent bacteria [2], and sporulation in B subtilis [34], have been modeled

as feedback networks using differential equations for continuous dynamics

One particular mathematical framework synthesizes logic-driven discrete events

and continuous dynamics described by differential equations to create a powerful

CHAPTER 1 INTRODUCTION 3

modeling tool: hybrid automata theory Formally, a hybrid automaton is a dynamicalsystem with temporal evolution of continuous state variables governed by differentialequations whose parameters change due to discrete input or event driven discretestate transitions First devised to model the interactions of a digital computer pro-

gram (or controller) with a continuous environment, hybrid automata have been usedwidely to model engineered systems, such as automated highway systems [58], airtraffic management systems [104], manufacturing systems [18], robots [17], commu-nication networks [55], automotive engine and transmissions [23], and active tractioncontrol [24], among others

Cellular regulatory and signaling networks exhibit complex combinations of bothdiscrete and continuous behaviors; the dynamics that govern the spatial and temporalincrease or decrease of protein concentration or activity inside a single cell are contin-

uous differential equations, while the activation or deactivation of these continuous

dynamics are triggered by switches which encode protein concentrations reachinggiven thresholds Therefore, hybrid automata theory presents an ideal framework to

model and analyze these processes [81] Hybrid models have an important advantage

over continuous nonlinear models that have traditionally been used: it is possible to

derive some analytical results about the structure and temporal evolution of hybrid

automata, which is not feasible for most nonlinear systems The author was one

of the first to apply hybrid modeling to molecular biological processes, intercellular

signaling and lateral inhibition in Xenopus laevis [46], and demonstrate its potential.Other examples of biological applications of hybrid modeling include signal trans-

duction and genetic regulatory networks in # coli and B subtilis [34], luminescence and quorum-sensing in V fischeri [2], lactose intake and metabolism in E coli [22],

and sporulation in B subtilis [73]

A key goal of mathematical modeling is to predict behavior of a biological system,

through analysis or simulation, which can be used to construct laboratory

experi-ments to help further understand, or identify, the system For example, given aninitial set of protein concentrations for a particular cellular regulatory process, themodel predicts what the system does after a given time or at steady state If thepredicted condition of the system has not been observed before in the laboratory,

CHAPTER 1 INTRODUCTION 4

experiments can be designed to test the prediction This is useful as it focuses the

experiments, which are usually complex and expensive to perform, on a specific targetobservation that is potentially interesting and helpful in understanding the system

better Similarly, it is also interesting to compute all possible combinations of initial

protein concentrations that lead the system to a particularly interesting tion Experiments can be designed to test some of the non-intuitive initial protein

configura-levels that can lead to a previously observed steady state, or configuration Thisgives insight into the non-canonical behavior of the process, as it identifies hithertounknown conditions that lead to the biologically significant configuration Analyti-

cally, this is posed as a reachability computation for the system, i.e determination of

the set of state variables that can be reached from a given initial set of states (known

as forward reachability), or the set of state variables that can lead to a given final

set of states (known as backward reachability) Reachability analysis, therefore, can

yield a large number of useful predictions from a mathematical model of a biologicalsystem The predictive property of reachable sets, in both biological and engineeringapplications, has inspired a great deal of research in constructing reliable reachabilityalgorithms

In the context of hybrid automata, reachability algorithms follow one of two

tracks: i) model checking and verification tools for discrete transition systems from

computer science are extended to hybrid automata This is done by first ing the temporal dynamics of the hybrid model, that is systematically converting

abstract-the hybrid automaton into a discrete transition system without continuous dynamics(i.e., without differential equations) while preserving the transition structure Reach-ability is then computed on the abstracted discrete transition system ii) Controller

design and stability analysis tools, such as Lyapunov theory, for continuous systemsare directly applied to hybrid automata to compute reachable sets of states Theabstraction track is particularly advantageous for systems with a large number ofcontinuous variables and discrete states, because current reachability algorithms for

discrete transition systems are much more efficient than those for continuous and

hybrid systems and can handle a much larger number of states Also, once the

reachable set has been computed for the abstracted system, efficient model checking

CHAPTER 1 INTRODUCTION 5

algorithms [30, 110] can be used to test whether the reachable sets of the discrete

transition model satisfy a given set of specifications and properties

The process of abstraction can introduce approximations in the computed able sets Depending on the abstraction algorithm, the reachable sets of the ab-

reach-stracted discrete transition system can either be over-approximate, i.e they are

larger than the exact reachable sets of the hybrid automaton and contain states thatare not reachable from the given set of states; or they can be under-approximate,i.e they are smaller than the exact reachable sets and exclude some states that

are reachable from the given set of states The two approximations guarantee two

different properties The over-approximation guarantees that it contains all the

reach-able states, even though it can contain some spurious reachreach-able states The

under-approximation guarantees that all the states it contains are real reachable states,even though it may exclude some real reachable states Depending on the nature ofthe questions being answered through the reachability computation, one or the other

type of approximation is more appropriate If the problem is the identification ofall the initial conditions that lead to a given set of states that define a condition toavoid, a cancerous cell fate for example; then the over-approximate reachable set has

to be computed This is because it is critically important not to exclude any initial

condition that may lead to the undesirable final state In this example, not

comput-ing the over-approximation would leave open the possibility that some seemcomput-ingly-safe

initial condition could lead to the cell becoming cancerous On the other hand, ifthe problem is the identification of initial conditions that lead to a given set of statesthat define a desired objective, such as regulating a bacteria to ingest and metabo-lize toxic waste; then the under-approximate reachable set should be computed In

this case, guaranteed achievement of the desired final condition is most important

Even though some reachable initial conditions are left out, choosing any of the initialstates in the under-approximate reachable set guarantees that the objective will be

CHAPTER 1 INTRODUCTION 6

The algebraic expressions that define the discrete states of the hybrid automaton as

well as the differential equations that govern its continuous variables can have bolic constants as coefficients This is very useful in developing biological models

sym-because the exact numerical values of biochemical parameters are difficult to

mea-sure The reachability computation performed on the model returns reachable setsthat are expressed in terms of the symbolic coefficients of the model, instead of beingnumerically instantiated After algebraic manipulation, the predicted results fromthe reachable sets can be expressed as relative values or as ranges For example,the predicted initial condition for a desired phenotype D can be expressed as “initialconcentration of protein A has to be greater than that of protein B and less than

that of protein C for phenotype D to occur” This is much easier to probe for, or

control, experimentally than an exact numerical initial concentration for the proteins

A, B and C

Computing reachable sets for hybrid automata is a complex task due to the

dif-ficulty of representing and propagating sets in high dimensional continuous spaces

For most hybrid automata that do not have the simplest linear continuous dynamics

(timers or two dimensional rectangular differential inclusions defined as 4# < b,z €

#*?), reachability computation is undecidable For timed automata, a verification

algorithm was developed in [5], and an algorithm for simultaneous reachability putation and minimization has been designed by [115] More recently, there has been

com-a focus on techniques which use com-approximcom-ations of vcom-arious types to mcom-ake the problem

of computing reachable sets tractable; these include approximating the continuousdynamics using differential inclusions [53, 94] Some methods attempt to approx-imate the structure of the reachable set using polyhedral representations [13, 29]

or ellipsoidal approximations [25, 65] Another approach uses numerical solutions

of levels sets of Hamilton-Jacobi equations to compute reachable sets and optimalcontrol strategy [21, 83, 84, 102] An approach utilizing optimal control techniques

has been developed by [66], which can analyze high dimensional constrained linear

and piecewise affine systems Other reachability algorithms involve the computation

of barrier certificates [93], and bisimulation and collapsing [9] Recently, qualitative

CHAPTER 1 INTRODUCTION 7

simulation models [20, 33, 64] have been proposed to abstract continuous phase

por-traits of hybrid automata to simpler transition graphs, on which reachability analysis

can be performed Predicate abstraction [3, 4, 48] and quantifier elimination [69, 101]

have been proposed for computing discrete abstractions of hybrid automata Most

of these methods suffer from one or both of two disadvantages: (a) The

complex-ity of the computations on the hybrid automaton restricts its dimensionalcomplex-ity, and

more importantly, (b) symbolic computations are not possible Quantifier

elimina-tion techniques have been used by the author to compute over-approximaelimina-tions of the

symbolic backward reachable sets for protein signaling automata in [44]

The author’s research focuses on a biological mechanism known as intercellularprotein signaling Found in all multicellular organisms from an early embryo stage,intercellular signaling is a feedback network which interrelates the fate of a singlecell and its neighbors In particular, this thesis presents models and analysis of twosignaling pathways active during development:

1 The Delta-Notch signaling mechanism, responsible for pattern formation in

many different biological systems, such as the emergence of ciliated cells in

Xenopus embryonic skin [77] The Delta-Notch pathway is an important model

system for studying localized cell-cell interactions that lead to distinctive globalpatterns By studying this pathway it may be possible to discover why aparticular pattern is formed, and also the means to choose and control thetype of pattern The author has studied initial protein concentrations that

lead to different steady state patterns, as well as non-canonical behavior of

the signaling network under certain conditions Two experiments have been

proposed to test these results and validate the non-canonical behavior

2 The planar cell polarity (PCP) signaling mechanism that controls the

posi-tion of trichome, or hair, growth in Drosophila melanogaster pupal wing cell

arrays [105] This is another important model system for linking cell-level

sig-naling pathways with tissue-level patterning and development Similar

path-ways have been implicated in defects in the human inner ear stereociliary (hair)bundle orientation [86], which decreases hearing ability For this system, the

CHAPTER 1 INTRODUCTION 8

objective is to show that the local signaling pathway model is sufficient to ulate the orientation and position of trichome in normal, or wild-type wings;

reg-as well reg-as robust enough to explain pattern disruptions in mutant wings

Both systems have been modeled [43, 46] using the mathematical framework of

hybrid automaton theory The key attributes of this research effort is that: (a)important properties of the systems are analytically derived This is an improvement

on large scale simulation, because the analytical properties are guaranteed to hold for

absolutely all conditions within a well-defined set, and (b) both modeling and analysis

is symbolic, i.e none of the parameters such as protein production/activation, decay

constants, or switching thresholds are numerically instantiated Rather, by doingsymbolic analysis, predictions are generated that involve ratios of symbolic kinetic

parameters (for example, the relative rates of production of two different proteins),

resulting in a model with valid parameter ranges given in the form of constraints This

is particularly important in biological systems, where the exact values of switching

thresholds and chemical reaction rates might be unknown, but a range of possiblevalues, usually expressed in terms of other symbolic constants, can be inferred

The author has analytically derived constraints on the system kinetic parameters

necessary for biologically feasible steady states to exist [46], in multi-stable systems

The constraints are expressed in terms of ratios of these parameters and switching

thresholds An abstraction procedure has also been developed [45] to compute

back-ward reachable sets of states, which are expressed in terms of the system’s symbolicparameters These backward reachable sets, when computed for the steady states ofthe system, represent sets of initial continuous variable values that are guaranteed

to converge to one particular steady state or the other Biologically, this implies

that one can identify sets of initial protein concentrations from which a biologicallyinteresting steady state can be achieved This result has been used to propose two

experiments to test the predictive property of the reachability analysis The novelabstraction procedure iteratively partitions the state space of a piecewise affine hy-brid automaton model to produce an abstracted discrete transition system It uses asystematic way of computing transitions and exact symbolic solutions of the contin-

uous differential equations to iteratively refine the partitions An under-approximate

polyno-resulting reachable set is under-approximate, which implies that all initial conditions

in the set are guaranteed to reach the steady state

1.1 Overview

The main contributions of this work can be summarized as follows:

1 It proposes a hybrid automata framework for modeling protein regulatory cesses In particular, the thesis proposes hybrid models of two interesting inter-cellular signaling pathways active during embryonic development: the lateralinhibitory Delta-Notch pathway responsible for pattern formation in the em-

pro-bryonic skin of Xenopus laevis, and the Planar Cell Polarity (PCP) signaling

pathway in Drosophila melanogaster wings ‘These models are validated against

experimentally observed steady state protein concentration patterns

2 Constraints on the kinetic parameters of the model are computed analytically,for particular biologically observed or interesting steady states to exist Theconstraints are computed symbolically, i.e without having to numerically in-stantiate the parameters This is a great advantage in the context of biologicalprocesses, where exact numerical parameters cannot often be identified fromexperimental data, but a range of values, or relative values for the parameters

may be obtainable

3 An abstraction procedure is presented that converts the hybrid automaton into

a discrete transition system using symbolic solutions to the differential

equa-tions and Lie derivatives to compute transiequa-tions between discrete states Thebackward reachability problem is then computed on the discrete abstraction,which makes the analysis tractable for large state spaces The backward reach-able set gives initial protein concentrations that converge to a particular steady

pre-algorithm is demonstrated for the Delta-Notch hybrid model.

5 The analysis tools developed have been implemented on a publicly available

systems biology software platform known as Bio-SPICE Using the Bio-SPICE

toolset, a further example, lactose metabolism inside a cell, is analyzed andreachable sets are computed for this model

This thesis is divided into eight chapters The core of this work is bracketed

by Chapter 1, which is the introduction, and Chapter 8, which summarizes possiblefuture research directions Chapter 2 explains the basic biological processes that

are modeled, as well as the motivation behind using hybrid automata as a modeling

framework In Chapter 3, the formal definitions of the hybrid automata and discrete

transition systems that make up the modeling framework, as well as mathematicalconcepts referred to in the reachability analysis, are given This is followed by adetailed description of the model design, analysis, and simulation results for theDelta-Notch protein signaling pathway, in Chapter 4

Chapter 5 details the abstraction procedure developed by the author, for

reach-ability analysis It also contains reachreach-ability results for several Delta-Notch hybridmodels and their biological implications This chapter also includes the query al-gorithm and its results, which provide biologically interesting deductions from the

computed reachable sets The design and analysis of the Planar Cell Polarity (PCP)

protein signaling network is given in Chapter 6, as well as simulation results thatreplicate biologically observed behavior In Chapter 7, Bio-SPICE, the open-sourceplatform for computational systems biology, is introduced The analysis tools de-veloped by the author were implemented in Bio-SPICE, and a description of thetools are given Finally, a further application of reachability computation for hybrid-automata based molecular biological models, is also described in Chapter 7: the

CHAPTER 1 INTRODUCTION 11

lactose metabolism cycle

1.2 Glossary of Biological Terms

This section defines the biological terms that have been commonly used in this thesis,

according to standard usage as given in [1, 68]

Cooperativity phenomenon displayed by molecules that have multiple binding tachment) sites Binding of one ligand alters the affinity of the other site(s)

(at-Delta transmembrane protein commonly associated with intercellular signaling inconjunction with receptors like Notch, implicated in pattern formation andneurogenesis

Dimer a compound formed by the union of two identical units of a simpler pound

com-Disheveled cytoplasmic protein associated with signal transduction in pathwaysregulating tissue and segment polarity

Epithelium one of the simplest types of tissues A sheet of cells, one or severallayers thick, organized above a membrane, and often specialized for mechanicalprotection or active transport Examples include skin, and the lining of lungs,gut and blood vessels

Flamingo also known as Starry night, is a surface receptor protein involved in the

establishment of tissue polarity

Frizzled transmembrane protein designated as a receptor involved in pattern mation, tissue and segment polarity.

for-Genotype all or part of the genetic constitution of an individual or group

Larva the immature stage, between the egg and the pupa, of an insect

Phenotype the visible properties of an organism that are produced by the

interac-tion of the genotype and the environment

Polymer a chemical compound or mixture of compounds formed by the union of

repeating structural units

Prickle cytoplasmic protein functional in tissue polarity regulation, in conjunction

with Frizzled protein

Proteolysis the degradation of proteins with formation of simpler and soluble

prod-ucts.

Pupa the stage between the larva and the adult in an insect

Puparium a protective case formed by the hardening of the next to the last larvalskin in which the pupa is formed

Receptor a membrane-bound or membrane-enclosed molecule that binds to, or sponds to something more mobile (the ligand), with high specificity

re-Chapter 2

Protein Signaling

The basic concepts of protein signaling during embryonic development, in the context

of cell differentiation, are introduced in this chapter This is followed by a description

of the biological processes studied in this work: lateral inhibition through Delta-Notch

signaling, and Planar Cell Polarity signaling A summary of previous mathematicalmodels developed for lateral inhibition, and the basic motivation behind using hybridautomata as an appropriate modeling framework, are also mentioned here Thechapter ends with a discussion of suitable switching functions for modeling proteinproduction regulation

2.1 Developmental Signaling

Cellular differentiation in embryonic tissue is a complex control process regulated by

a set of developmental genes, most of which are conserved in form and function across

a wide spectrum of organisms Classic model organisms like the fruit fly Drosophilamelanogaster, the nematode Caenorhabditis elegans, the South African claw-toed frog

Xenopus laevis and the zebrafish Danio rerio have been extensively studied to tify the key signaling pathways behind differentiation The concentration levels and

iden-activity of various proteins in a mature cell decide its phenotype Genes, therefore,

control cell fate by regulating the type and amount of proteins produced in a cell [11]

13

CHAPTER 2 PROTEIN SIGNALING 14

Proteins in turn affect gene activity by turning on or off gene expression thereby fecting the production of proteins themselves This forms a complex network of

af-gene and protein inhibitors and promoters linked through cascades of positive and

negative feedback [42] Hence differential gene activity is considered the key to celldifferentiation [114] and protein concentrations in a cell are a good measure of gene

activity and environmental input

One ubiquitous type of differentiation mechanism is intercellular signaling Found

in almost all multicellular organisms from an early embryo stage, intercellular naling interrelates the fate of a single cell and its neighbors in a population of ho-mogeneous cells The spectrum of signals taken together typically form feedbackmechanisms, but most smaller scale signaling systems are less feedback dependent

sig-Among the various signaling channels, the Delta-Notch protein pathway in particular

has gained wide acceptance as the arbiter of cell fate for an incredibly varied range

of organisms [12]

2.2 Lateral Inhibition Through Delta-Notch

Sig-naling

Delta and Notch are both transmembrane proteins that are active only when cells

are in direct contact, in a densely packed epidermal layer for example [72] Delta is a

ligand that binds and activates its receptor Notch in neighboring cells The activation

of Notch in a cell has a very rapid effect on the expression of a variety of other geneswhich lead to a particular cell fate being chosen Hence Notch signaling directlycontrols switching in genetic networks and cascades The activation of Notch in a

cell affects the production of Notch ligands (i.e Delta) both in itself and indirectly in

its neighbors, thus forming a feedback control loop In the case of lateral inhibition,high Notch levels suppress ligand production in the cell and thus a cell producingmore ligands forces its neighboring cells to produce less

The Delta-Notch signaling mechanism has been found to cause pattern formation

in many different biological systems Examples include the emergence of ciliated cells

CHAPTER 2 PROTEIN SIGNALING 15

in Xenopus embryonic skin [77], sensory cell differentiation in the zebrafish ear [49],

chick feather array [32], neurogenesis, wing vein morphogenesis, and the eye R3/R4

photoreceptor differentiation and planar polarity, all in Drosophila [41, 47, 59, 74, 78]

An example of the distinctive salt-and-pepper pattern formed due to lateral inhibition

is the Xenopus epidermal layer where a regular set of ciliated cells form within a

matrix of smooth epidermal cells as seen in Figure 2.1 Apart from pattern formation,

a Delta-Notch mechanism has been used to explain lineage decisions and boundary

formation [26, 60], as well as stem cell function and formation of skin appendages [72]

Figure 2.1: Xenopus embryo labeled by a-tubulin, a marker for ciliated cell precursorsseen as black dots

2.3 Planar Cell Polarity Signaling in Drosophila

In adult Drosophila, each epithelial cell on the wing produces a single hair, or

tri-chome The hairs grow from the distal (toward the wing tip) side of each cell and

all point in the same direction, toward the wing tip, as shown in Figure 2.2 Thisphenomenon is caused by spatially asymmetric distributions of certain proteins in

CHAPTER 2 PROTEIN SIGNALING 16

the plane of the epithelium The process by which the proteins controlling hair larization localize to different areas within each cell during the development of the fly

po-is called planar cell polarity (PCP) signaling The wing epithelial cells aggregate in

a hexagonal close-packed array (Figure 2.2, courtesy of Professor Jeffrey Axelrod.)

It is assumed that cell-to-cell contact is required for PCP signaling

Figure 2.2: Drosophila adult wing epithelium The figure of a magnified portion ofthe wing below shows the hexagonal shape of the cells and the hairs pointing in asimilar direction

Using mutant clones, which lack the ability to produce one or more of the coresignaling proteins, it has been possible to identify the sequence of the control cascade

for intercellular PCP signaling [16, 105] Note that this signaling network only acts

at the cell membrane and thus requires direct contact between neighboring cells to

be effective The regulatory cascade is drawn schematically in Figure 2.3 Frizzled

CHAPTER 2 PROTEIN SIGNALING 17

(Fz) protein promotes Disheveled (Dsh) recruitment and co-localization to the cellmembrane Dsh then promotes the stabilization of Fz at that cell membrane, possi-

bly by the formation of a Dsh-Fz complex Fz then acts through intermediaries to

promote the localization of Prickle (Pk) in the adjacent membrane of the neighboring

cell Pk represses the recruitment of Dsh to the cell membrane, thus completing theloop Experimentally, it has been observed that, in steady state, Dsh and Fz proteinslocalize to the distal edge and Pk to the proximal edge of all cells in the array

Figure 2.3: Planar cell polarity signaling network between two adjacent cells inDrosophila pupal wing

2.4 Motivation for Hybrid Model

A wide range of cell regulatory and signaling mechanisms seem to be ideal candidatesfor hybrid systems models The physical reasons behind this include: gene expressions

are represented by the existence (or absence) of certain proteins; protein

concentra-tion dynamics are described by constant exponential growth and decay rates coupledwith discrete switches; protein production is switched on or off depending on theexpression of other genes, i.e presence or absence of other proteins in sufficient con-centrations; complexity is introduced by the massive interconnections in the discrete

switching circuit and logic (it is not uncommon to find complicated repressive andpromoter feedback channels forming genetic circuits, e.g [79]) These observations

CHAPTER 2 PROTEIN SIGNALING 18

suggest that a piecewise affine hybrid model would be a very good choice for modelingthese systems Using simple continuous dynamics and lumping the complexity intothe discrete inputs gives us the capability to: analyze the model mathematically andprove reachability and convergence for a wide set of initial conditions, extract impor-

tant parameters and predict their effects on the system evolution without simulation,and suggest biological experiments to validate the model as well as refine it

2.5 Switching Functions

Genes, which control protein production, are switched on or off by low level

cooper-ative binding of proteins to DNA strands This results in a fairly steep sigmoid gene

expression switch as a function of protein concentration The Hill equation [19], given

by equation (2.1), is an empirical function used to describe the binding of ligands to

proteins:

k

where a and & are empirical constants and k is known as the Hill coefficient Previous

models, such as those developed by [31] and [77] have used nonlinear sigmoid functions

given as f(u—h) = 0.5(1+4+ Tớ): drawn in Figure 2.5, to model the gene

expression switch While this works well in simulation it makes analysis difficult,

apart from linearization solutions around an equilibrium [31] The model proposed

by [85] is particularly relevant because it incorporates a sharp sigmoid switching

function with switching thresholds For modeling the protein production switch,within the framework of piecewise affine hybrid automata, a piecewise linear switch

(Figure 2.4) can be defined as follows:

m(u—h)+4 when h— sL <usht+s

when > h + 5+

where h is the switching threshold and m is the slope of the switch The piecewise

linear switch can be considered as the best possible compromise between biological

CHAPTER 2 PROTEIN SIGNALING 19

accuracy and mathematical tractability It eliminates biologically infeasible

phe-nomenon (like Zeno states) while retaining the advantage of having an analyzable

piecewise affine system Another important consideration behind choosing a wise linear switch is the fact that the slope of the biological switch is related to theprotein dynamics and therefore controls the steady state of the system This led the

piece-author to explore the behavior of the equilibria of the hybrid automaton with respect

to the slope of the piecewise linear switch and the subsequent analysis produced a set

of constraints on the slope, for a biologically viable system These constraints mayhave some biological significance as it is known that the slope of the Hill equation is

linked to the number of binding sites available

Chapter 3

Hybrid Automata

This chapter formally defines a hybrid automaton, and a restricted class of hybridautomata that is used in the Delta-Notch model development This is followed by thedefinition of a discrete transition system that represents an abstraction of the hybrid

automaton A more general discussion related to abstractions of hybrid automata

and their decidability can be found in [6, 54]

3.1 Hybrid Automata and Transition Systems

The mathematical definition of a general hybrid automaton, developed and refined

by [75, 103], is given by:

Definition 1 A hybrid automaton: H = (Q,X,%, V, Init, f, Inv, R), is defined such

that

1 Q={0,da, dQm} ts the set of discrete states, or modes;

2 X ER" is the set of continuous state variables;

Gs & is the set of discrete inputs;

V is the set of continuous inputs;

—

5 Init = Qo x Xo is the set of initial conditions;

20

CHAPTER 3 HYBRID AUTOMATA 21

6.2 € X : # = f(q,2,V) ts the continuous vector flow associated with each

discrete state;

7 Inv(q) C #?, assigns to each discrete state an invariant set that defines the

state This is also known as the modal invariant,

& R:QxX xD — 2°** is the transition map.

Traces of the hybrid system H consists of continuous evolutions according to the

differential equations # = f(¢,z,V) keeping the discrete state constant and discrete

jumps from one discrete state to another which may involve a reset of the continuous

state variables Note that the differential equations are perfectly general and may

be nonlinear Also, the continuous state variables z(t) can exit the invariant Inv(q)

under continuous evolution, in which case a discrete transition is forced to another

discrete state without a reset of the continuous variables occurring

Restrictions have to be introduced in the hybrid automaton, developed in [103], to

tailor it for modeling the biological systems of interest and also to make it amenable

to abstraction and analysis using the procedure presented in Chapter 5 Therefore,

a restricted class of hybrid automata can now be defined:

Definition 2 A piecewise affine hybrid automaton, H = (Q,X,TM, Init, f, Inv, R),

is defined such that

1 Q = {q, G, -; Gm} is the set of discrete states, or modes;

2 X CR” is the set of continuous state variables;

3 Y= {01,00, ,0m} is the set of discrete inputs;

4 Init = Qo x Xo is the set of initial conditions;

5 ƒ(q,#) = Agr + bạ is the continuous vector field associated with each discretestate, where Ag € R"*” is a diagonal matriz, and bạ € 3È";

6 Inv(q) = (Nữ < 0)) A(Aj (2; = 0))AAy0z > 9)) ACA ŠS 9)) (Am (Pm 2

0)), where p¿ € Pula), p; € Peq(q) pe € Pot(q) pi € Pre(@),Pm € Pye(q), ts the

CHAPTER 3 HYBRID AUTOMATA 22

invariant defining each discrete state py : X — 3\ is a polynomial, and Po(q)

represents a list of polynomial expressions;

7 R:Qx XxX — 29% ¡s the transition map.

For this class of hybrid automata, the state transition matrix A, is diagonal with

real eigenvalues However, the elements of A, are free to be symbolic, ie theeigenvalues, À¡, A2, An, of Ag need not be numerically instantiated The elements

of vector bg are also free to be symbolic Constraints may be imposed on these

symbolic constants to restrict the behavior of the model The polynomials defining

the invariant of each mode, can be separated into five classes: Py(q), Peg(q), Pot(@),

Pie(q), and P,-(q), according to their signs in the state For example, all polynomials

Dị € Py(q) are negative, or less than (/t) zero, in state g, and similar definitions

hold for the other classes Peg(q), etc Pi(q), Peg(q),- , Poe(q) are mutually disjoint,

and Vạ, Pi(q) U Peg(q) U Pot(g) U Pie(q) U Pye(q) is invariant This implies that the

polynomials defining each state are identical, their sign alone varies from state tostate These classes are used to determine adjacency, i.e whether two states aregeometrical neighbors in state space, in several different steps of the abstraction

procedure presented in Section 5.3

It should be noted that, as the abstraction procedure progresses, additional nomials are added to the modal invariants to partition the modes This may give

poly-rise to redundancies in the polynomials defining the modal invariants The dant constraints can be removed from the invariant using a decision procedure such

redun-as QEPCAD at every partitioning step However, this is unnecessary because the

adjacency check performed during transition computation will ensure that there are

no transitions between non-adjacent modes, thus taking care of the redundant straints in the invariant In the transition map, transitions caused by the continuousflow of the automata crossing switching boundaries defined by the state invariant arecalled forced transitions When a network of automata is built by composing several

con-of them together, their discrete inputs, ©, are coupled to the internal state variables

of other automata in the network, as will be seen in the multiple cell Delta-Notch work model in Chapter 4 In that case, the entire network behaves as an autonomous

net-CHAPTER 3 HYBRID AUTOMATA 23

hybrid automaton If the assumption of zero boundary conditions, i.e no influence

from outside the network, is made, then the state transitions in that automaton are

OA x1 +as#¿ + bs > 0 Each of the sets Pr(qi), Peg(q1), - Poe(q) are lists that can

now be populated by polynomial expressions according to their sign in the invariant of

qi Therefore, it can be seen that (gi) = {21 +a1%2+b1, 21 +03%2+b3, 71 +0422 +b4}

and Pye(q1) = {£1 + da#za + bo, 1 + as2¿ + bs}, and the others are empty, P.g(qi) =Đ„(m) = Pie(gi) = 0 Similarly, for state go, the lists of polynomials are P(q2) =

{x1 +aiza+bi, #1 +a4r2+bs}, Poe(qe) = {v1 +aa+a +ba, 21 +3%2 ba, 21 +a5r2+b5},

and P›z(q›) = P„(q›) = Pie(go) = @ As previously mentioned, these lists are used to

determine whether two discrete states are geometrically next to each other in ous state space In the example, the polynomial expression 7; +a3%_ +63 has different

continu-signs in the two states, it is an element of P(qi), but is also a member of P,.(q2)

This implies that the sign change occurs at the boundary 7; + as#a + b3 = 0, which

is part of g2 and that the two discrete states g, and qs are geometrically contiguous

or adjacent, as can be seen in Figure 3.1 This test can be performed automaticallyand efficiently for high dimensional hybrid automata to check adjacency

Definition 3 A finite discrete transition system, T = (Q,3,—, Qo, Qr), is defined

such that

1 Q= {i,da ,dn} is a set of states;

2 XL = {01,02, ,0n} is a set of events;

8 -CQxzx Q is a transition relation;

CHAPTER 3 HYBRID AUTOMATA 24

Figure 3.1: Continuous state space with geometrical representations of polynomial

modal invariants of a two dimensional hybrid automaton.

4 Qo GQ is the set of initial states;

5 Qr CQ is the set of final states.

The transition system T can be thought of as a graph with directed edges denoting

transitions between nodes that are representations of the states g € Q The transition

system is finite if the cardinality of Q is finite, and it is deadlock free if for every

state g € Q, there exists a state q’ € Q and an event sigma € & such that q 4“,

q Additionally, the transition system is live if for each state g € Q, transition

g = q’ is eventually taken A dual representation of a finite transition system is

an adjacency matrix A € {0,1}"*”, where 7 € 1,2, ,n represents a discrete state.

In the adjacency matrix, a,; € A: ø = 1 means that a transition gq, — q; exists,

and a;; = 0 means no transition exists from g¡ to g; Note that the event ø, which

triggers a transition, is not relevant in the adjacency matrix notation and has been dropped from the transition relation The final states, g € Qr, of the transition system are states that have no transitions out of them, i.e in the adjacency matrix,