A Broad Dynamical Model for Pattern Formation by Lateral Inhibition

0.19 MB·English

by  Murat Arcak

A Broad Dynamical Model for Pattern Formation by Lateral
Inhibition
Murat Arcak∗
January 9, 2012
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0
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n
a 1 Introduction
J

6
Spatial patterns ofgene expression arecentral to thedevelopment ofmulti-cellular organisms. Most math-

]
S ematical studies of pattern formation investigate diffusion-driven instability, which is a mechanism that
D amplifies spatial inhomogeneities in a class of reaction-diffusion systems (see, e.g., [1]). However, many
. patterning events in multi-cellular organisms rely on cell-to-cell contact signaling, such as the Notch path-
h
t way [2], and do not involve diffusible proteins for intercellular communication. A particularly interesting
a
m phenomenon inthisformofcommunication islateral inhibition wherebyacellthatadoptsaparticular fate
[ inhibits its immediate neighbors from doing likewise [3], thus leading to ‘fine-grained’ patterns. There is

1 increasing interest inunderstanding theNotchsignaling circuitry inmammaliancellsthatleadstosuchlat-
v
eral inhibition [4, 5]. Recent studies showed thatalateral inhibition pathway also functions inE.Coli, and
7
7 enablesthebacteria toinhibitthegrowthofotherE.Colistrains indirectcontact[6].
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Dynamical models areofgreatinterest forunderstanding the circuit topologies involved inlateral inhi-
1
. bition and for predicting the associated patterns. Several simplified models have been employed for Notch
1
0 signallingpathwaysin[3]and[5]. Theobjectiveofthispaperistopresentanabstractdynamicalmodelthat
2
captures the essential features of lateral inhibition and to demonstrate with dynamical systems techniques
1
: thatthesefeaturesindeedleadtopatterning. AlthoughthismodelisnotmeantspecificallyforNotchsignal-
v
i ing,itencompasses asspecialcasesthelateralinhibition modelin[3]aswellasaslightly modifiedversion
X
oftheonein[5].
r
a Ourmodeltreatstheevolutionofconcentrations ineachcellasaninput-outputsystem,wheretheinputs
represent theinfluenceofadjacentcellsandtheoutputscorrespond totheconcentrations ofthespecies that
interact with adjacent cells. The input-output models for the cells are then interconnected according to an
undirected graph where the nodes represent the cells, and the presence of a link between twonodes means
that the corresponding cells are in contact. The main assumption on the input-output model is that each
constantinputyieldsauniqueandgloballyasymptoticallystablesteady-state,andthatthevalueoftheoutput
at this steady-state is a decreasing function of the input. This decreasing property captures the inhibition
of the cell function by its neighbors. The model allows multiple inputs and outputs, and is restricted by a
monotonicity assumption, following the definition of monotonicity for dynamical systems with inputs and
outputs[7].
∗Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. Email: ar-
[email protected].
1
Using this model, wefirstgive aninstability condition for the homogeneous steady-state, applicable to
arbitrary contact graphs. We then focus our attention on bipartite graphs, and demonstrate the emergence
of a “checkerboard” pattern, exhibiting alternating high and low values of concentrations in adjacent cells.
Next, we establish a strong monotonicity property of the interconnected model for bipartite graphs, which
implies that almost every bounded solution (except for a measure-zero set of initial conditions) converges
to a steady-state [8, 9]. A graph is bipartite if and only if it contains no odd-length cycles, and Cartesian
products of bipartite graphs are also bipartite [10]. Thus, the results of this section are applicable, among
others, to grid graphs (one dimensional path graphs and their Cartesian products in higher dimensions)
whichareappropriate forrepresenting arraysofcells.
2 Lateral Inhibition Model and Preliminaries
Welet G be anundirected, connected graph where the nodes represent the cells, and the presence of a link
betweentwonodesmeansthatthecorrespondingcellsareincontact. Inpreparationforthedynamicalmodel
studiedbelow,weletN denotethenumberofcellsanddefinethematrix P∈RN×N:
d−1 ifnodesiand jareadjacent,
p = i (1)
ij  0 otherwise,

whered denotesthedegreeofnodei. Itfollowsthat Pisanonnegative row-stochastic matrix,thatis:
i
P1=1 (2)
where 1 denotes the vector of ones. The matrix P is identical to the probability transition matrix for a
random walk on the graph G. Theproperties summarized below therefore follow from standard results for
randomwalks(see,e.g.,[11]):
Lemma 1. P possesses real eigenvalues λ ≤···≤ λ all of which lie in the interval [−1,1], and corre-
N 1
sponding real, linearly independent eigenvectors v, i = 1,···,N. In particular, λ = 1, and v = 1 is a
i 1 1
corresponding eigenvector. IfGisbipartite,thenλ =−1,andaneigenvectorv issuchthattheentriesare
N N
either1or−1,andtwoentries corresponding toadjacent nodeshaveopposite signs.
Leti=1,···,N denotethecells,andconsider thedynamicalmodel:
x˙i= f(xi,ui) yi=h(xi) (3)
where xi∈X ⊂Rnisavectordescribingthestateofreagentconcentrations incelli,ui∈U ⊂Rm describes
the ‘input’ from adjacent cells, and yi ∈Y ⊂Rm describes the ‘output’ that serves as an input to adjacent
cells. Inparticular,
U =(P⊗I )Y (4)
m
wherePisasdefinedin(1),U :=[u1T···uNT]T andY :=[y1T···yNT]T. Iffollowsfrom(1)thattheinputui
istheaverage oftheoutputs yk overallneighbors kofcelli. Thus, wehenceforth taketheinput andoutput
spacestobeidentical: U =Y .
Weassumethat f(·,·)andh(·)arecontinuously differentiable andfurthersatisfythefollowingproperty:
Assumption 1. For each constant input u∗, system (3) has a globally asymptotically stable steady-state
x∗:=S(u∗)withtheadditional property that:
∂f(x,u)
det ,0. (5)
∂x (cid:12)(cid:12)(x,u)=(x∗,u∗)!
(cid:12)
(cid:12)
(cid:12)
2
ThemapS :U →X and,therefore, themapT :U →U defined by:
T(·):=h(S(·)), (6)
arecontinuously differentiable.
Following the terminology in [7], we will refer to S(·) as the input-state characteristic, and to T(·) as
theinput-output characteristic. Ournextassumption isthat(3)isamonotone system inthesense of[7],as
definedbelow. Accordingtotheclassicaldefinition forsystemswithoutinputsandoutputs [9],amonotone
system is one that preserves a partial ordering of the initial conditions as the solutions evolve. The partial
orderingisdefinedwithrespecttoapositivitycone K intheEucledeanspacethatisclosed,convex,pointed
(K∩(−K)={0}), and has nonempty interior. Givensuch acone, x(cid:22) xˆ means xˆ−x∈K, x≺ xˆ means x(cid:22) xˆ
and x, xˆ, and x≪ xˆ means that xˆ−x is in the interior of K. The system x˙ = f(x) is then defined to be
monotone iftwosolutions x(t)and xˆ(t)starting withtheorder x(0)(cid:22) xˆ(0)maintain x(t)(cid:22) xˆ(t)forall1 t≥0.
Themorerestrictivenotionofstrongmonotonicitystipulatesthat x(0)≺xˆ(0)implies x(t)≪xˆ(t)forallt>0.
Themonotonicity conceptwasextended tosystemswithinputsandoutputs in[7]:
Definition 1. Given positivity cones KU,KY,KX for the input, output, and state spaces, the system x˙ =
f(x,u), y=h(x) is said to be monotone if x(0)(cid:22) xˆ(0) and u(t)(cid:22)uˆ(t) for all t ≥0 imply that the resulting
solutions satisfy x(t)(cid:22) xˆ(t)forallt≥0,andtheoutputmapissuchthat x(cid:22) xˆimpliesh(x)(cid:22)h(xˆ).
Assumption2. Thesystem (3)ismonotonewithrespecttoKU =Rm , KY =−KU,and KX =K,where K is
≥0
somepositivity coneinRn.
Asobserved in[7,RemarkV.2], monotonicity implies thattheinput-state andinput-output characteris-
ticsarenondecreasingwithrespecttothesameordering;thatis,u(cid:22)uˆwithrespecttoKU impliesS(u)(cid:22)S(uˆ)
withrespecttoKX andT(u)(cid:22)T(uˆ)withrespecttoKY. SinceKY =−KU inAssumption2,weconcludethat
T(·)isnonincreasing withrespect tothestandard orderinduced by KU =Rm . Thisnonincreasing property
≥0
means that, if two cells are in contact, an increase in the output value of one has the opposite effect on the
other, which is why (3)-(4) is referred to as a “lateral inhibition” model. We note from the nonincreasing
property ofT(·)that:
∂T(u)
T′(u):= (7)
∂u
isanonpositive matrixinRm×m,anddenoteitsspectral radiusas:
ρ(T′(u)). (8)
Weconcludethissectionbyquotinglemmasthatwillbeusedinthesequel. Lemmas2and3arefrom[12]:
Lemma 2. Given the system x˙ = f(x,u), y=h(x) with continuously differentiable f(·,·) and h(·), the lin-
earization x˙=Ax+Bu, y=Cx about a point (x∗,u∗) satisfying f(x∗,u∗)=0 is also monotone with respect
tothesamepositivity cones.
Lemma3. Thelinearsystem x˙=Ax+Bu,y=Cxismonotoneifandonlyif:
1) x∈KX implies Ax∈KX,
2)u∈KU implies Bu∈KX,
3) x∈KX impliesCx∈KY.
1Here,“forallt”isunderstoodas“foralltimestthatbelongtothecommondomainofexistenceofthetwosolutions.”
3
Thefollowinglemma,provenin[12]forsingle-input, single-output systemsandextendedin[13]tothe
multivariablecase,determinesstabilityofapositivefeedbacksystembasedonthe‘dcgain’oftheopen-loop
system:
Lemma 4. Suppose the linear system x˙= Ax+Bu, y=Cx is monotone with respect to cones KU,KY,KX
suchthat KU =KY andAisHurwitz. If−(I+CA−1B)isHurwitz,thensoisA+BC. If−(I+CA−1B)hasan
eigenvalue withapositiverealpart,thensodoes A+BC.
Inthespecialcaseofsingle-input, single-output systems,thestabilitycondition aboveamountstochecking
whether the dc gain −CA−1B is greater or smaller than one. In the multi-input, multi-output case, this
condition is equivalent to inspecting whether the spectral radius of the dc gain matrix is greater or smaller
thanone.
Thefollowingtestfrom[7,14]isusefulforcertifying monotonicity withrespecttoorthantcones:
Lemma 5. Consider the system x˙ = f(x,u), y = h(x), x ∈ X ⊂ Rn, u ∈ U ⊂ Rm, y ∈ Y ⊂ Rp, where
the interiors of X and U are convex, and f(·,·) and h(·) are continuously differentiable. If there exist
ǫ ,···,ǫ ,δ ,···,δ ,µ ,···,µ ∈{0,1}suchthat:
1 n 1 m 1 p
∂f
(−1)ǫj+ǫk j(x,u)≥0 ∀x∈X,∀u∈U,∀j,k (9)
∂x
k
∂f
(−1)ǫj+δk j(x,u)≥0 ∀x∈X,∀u∈U,∀j,k (10)
∂u
k
∂h
(−1)ǫj+µk k(x,u)≥0 ∀x∈X,∀j,k, (11)
∂x
j
then the system is monotone with respect to the positivity cones KU ={u∈Rm | (−1)δjuj ≥0}, KX ={x∈
Rn |(−1)ǫjxj≥0},KY ={y∈Rp |(−1)µjyj≥0}.
3 Instability of the Homogeneous Steady-State
Notethatsystem(3)-(4)admitsspatiallyhomogeneous solutionsoftheform xi(t)=x(t),i=1,···,N,where
x(t)satisfies:
x˙ = f(x,h(x)). (12)
Inparticular, ifthemapT(·)hasafixedpoint:
u∗=T(u∗), (13)
then(12)admitsthesteady-state:
x∗=S(u∗). (14)
Forsingle-input,single-outputsystemswithU =R ,thenonincreasingpropertyofthemapT :R →R
≥0 ≥0 ≥0
indeedguarantees auniquefixedpointu∗ in(13).
The “lumped model” (12) describes the dynamics of the Nn-dimensional system (3) reduced to the n-
dimensional invariant subspace wherethesolutions arespatially homogeneous. Thus,thesteady-state x∗ of
thelumpedmodeldefinesthehomogeneoussteady-state xi=x∗,i=1,···,N,forthefullsystem(3)-(4). Asa
startingpointfortheanalysisofpatternformation,wenowgiveaninstabilitycriterionforthehomogeneous
steady-state:
4
Theorem1. Considerthesystem(3)-(4)andsupposeAssumptions1and2hold. Letλ denotethesmallest
N
eigenvalue ofPasinLemma1,andletu∗,x∗ beasin(13),(14). If:
λ ρ T′(u∗) <−1, (15)
N
(cid:0) (cid:1)
thenthehomogeneous steady-state xi=x∗,i=1,···,N,isunstable.
Proof: LetX:=[x1T···xNT]T,andnotethatthelinearization of(3)-(4)aboutthehomogeneous steady-state
[x∗T,···,x∗T]T givestheJacobian matrix:
I ⊗A+P⊗(BC) (16)
N
where:
∂f(x,u) ∂f(x,u) ∂h(x)
A:= , B:= , C:= . (17)
∂x (cid:12)(cid:12)(x,u)=(x∗,u∗) ∂u (cid:12)(cid:12)(x,u)=(x∗,u∗) ∂x (cid:12)(cid:12)x=x∗
(cid:12) (cid:12) (cid:12)
WerecallfromLemma1that (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
λ
1
V−1PV =Λ:= ... , (18)
whereV =[v ···v ],andapplythefollowingsimilaritytransformλNation to(16):
1 N
(V−1⊗I )[I ⊗A+P⊗(BC)](V⊗I )=I ⊗A+Λ⊗(BC). (19)
n N n N
Thismatrixisblock-diagonal, withthekthdiagonal blockgivenby:
A+λ BC. (20)
k
Claim: If
λ ρ T′(u∗) <−1, (21)
k
(cid:0) (cid:1)
then(20)hasapositiveeigenvalue.
The theorem follows from this claim because, if (15) holds, then (20) has a positive eigenvalue for
k = N, which implies instability. To prove the claim, we note from Assumption 2 and Lemma 2 that the
linearsystem x˙=Ax+Bu,y=CxismonotonewithrespecttoKU =Rm ,KY =−KU,andKX =K. Wewrite
≥0
A+λ BC=A+BC whereC :=λ Candnotethat(21)impliesλ <0. Thus,thesystem x˙=Ax+Bu,y=C x
k k k k k k
is monotone with KU =KY. In addition, Assumptions 1 and 2 imply that A is Hurwitz, as can be deduced
from [12,Lemma6.5]. Thus,itfollowsfrom thesecond statement ofLemma4thatif−(I+C A−1B)hasa
k
positiveeigenvalue, thensodoes(20). Theremainingtaskisthustoprovethat
−(I+C A−1B)=−I−λ CA−1B (22)
k k
hasapositiveeigenvalue. Tothisend,wefirstshowthat
T′(u∗)=−CA−1B. (23)
Since
f(S(u),u)≡0, (24)
differentiation gives:
∂f(x,u) ∂S(u) ∂f(x,u)
+ =0. (25)
∂x (cid:12)(cid:12)x=S(u) ∂u ∂u (cid:12)(cid:12)x=S(u)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
Next,itfollowsfromthedefinition(6)that
∂h(x) ∂S(u)
T′(u)= . (26)
∂x (cid:12)(cid:12)x=S(u) ∂u
(cid:12)
(cid:12)
Combining(25)and(26),andsubstituting (17),weve(cid:12)rify(23). Substituting (23),wethenrewrite(22)as
−I+λ T′(u∗), (27)
k
andconclude thatitindeed hasapositive eigenvalue, because λ <0impliesthatλ T′(u∗)isanonnegative
k k
matrix and (21) implies that its spectral radius exceeds one. Since the spectral radius is an eigenvalue for
nonnegative matrices(see,e.g.,[15]),theconclusion follows. (cid:3)
The eigenvectors v of P used in the similarity transformation (19) may be interpreted as the spatial
k
modes of the system. Thus, the stability properties of the matrix (20) for each k determines whether the
corresponding mode decays orgrowsintime. Since thespectral radius isnonnegative andλ ,k=1,···,N,
k
areindecreasingorder,whenevertheinstabilitycriterion(21)holdsforaparticularmodek,italsoholdsfor
highervaluesofk. Becauselargerwavenumberskimplyhigherspatialfrequencycontentinv ,weconclude
k
thattheinstability condition abovesetsthestagefortheformationofhigh-frequency spatialpatterns.
4 Patterning in Bipartite Graphs
4.1 Emergence ofCheckerboard Patterns
Forbipartitegraphs, whereλ =−1asstatedinLemma1,theinstability condition inTheorem1is:
N
ρ(T′(u∗))>1. (28)
Thiscondition indicates thegrowthofthehighest spatial-frequency modev whichexhibits opposite signs
N
foradjacentnodes. Thus,concentrations inadjacentnodesmoveinoppositedirectionsinthevicinityofthe
homogeneous steady-state. Wenowshowthat,ifthemap
T2(·):=T(T(·)) (29)
hastwofixedpointsu ,u otherthanu∗,satisfying:
1 2
u =T(u ), u =T(u ), (30)
1 2 2 1
then the system (3)-(4) has an inhomogeneous steady-state with two sets of concentrations, each assigned
to one of two adjacent cells. We will refer to this steady-state as a “checkerboard” pattern, since adjacent
cells adopt distinct states. Although this term may be associated with cells arranged as agrid graph in two
dimensional space, wewilluseitbroadlyforanyspatial arrangement thatformsabipartitegraph.
Proposition1. LetGbeabipartitegraphandletthesetsI⊂{1,···,N}andI′={1,···,N}−Ibesuchthat
notwonodesinthesamesetareadjacent. Ifthereexistu ∈U andu ∈U,u ,u ,satisfying (30),then
1 2 1 2
xi=S(u ), i∈I, xi=S(u ), i∈I′, (31)
1 2
and
xi=S(u ), i∈I, xi=S(u ), i∈I′, (32)
2 1
aresteady-states forsystem(3)-(4).
6
Proof: Toshowthat(31)isasteady-state, wenotethat,ifi∈I,thenyi=T(u )and,ifi∈I′,thenyi=T(u ).
1 2
From (4), the input ui to a node in I is T(u ) because all neighbors of this node belong to I′. Likewise,
2
the input ui to a node in I′ is T(u ) because all neighbors of this node belong to I. Since T(u )=u and
1 2 1
T(u )=u ,weconclude that(31)isindeedasteady-state, andidentical arguments applyto(32). (cid:3)
1 2
Theorem2. Considerthesystem(3)-(4)andsupposeAssumptions1and2,andthehypotheses ofProposi-
tion1hold. If,inaddition,
ρ(T′(u )T′(u ))<1, (33)
1 2
thenthesteady-states (31)and(32)areasymptotically stable.
Beforegivingtheproof,wenotethat(30)corresponds toaperiod-twoorbitofthediscrete-timesystem:
u(t+1)=T(u(t)), (34)
and (33)implies theasymptotic stability ofthis orbit. Likewise, (28)indicates instability ofthefixedpoint
u∗forthisdiscrete-timesystem. Thus,aninterestingdualityexistsbetween(34)andthespatially-distributed
system (3)-(4) defined on a bipartite graph: A bifurcation from a stable fixed point to a stable period-two
orbitin(34)correspondstotheemergenceofstablecheckerboardpatternsfromahomogeneoussteady-state
in(3)-(4).
Inthesingle-input, single-output casewithU =R ,whereT :R →R isanonincreasing function
≥0 ≥0 ≥0
byAssumption2,condition(28)indeedimpliestheexistenceofaperiod-twoorbit(30). Toseethis,assume
to the contrary that u∗ is the unique fixed point of T2(·). Since T(·) is continuous and nonincreasing, this
uniquenesspropertywouldimplythatu∗isaglobalattractorforallsolutionsofthedifferenceequation(34)
starting in R [16, Lemma1.6.5]. This, however, contradicts (28), which implies instability of u∗ for this
≥0
scalardifferenceequation.
The argument above does not suggest the uniqueness of the pair (u ,u ), and multiple pairs satisfying
1 2
(30)mayexist. However,weclaimthatatleastonepairsatisfies:
dT2(u) dT2(u)
= =T′(u )T′(u )<1, (35)
du (cid:12) du (cid:12) 1 2
(cid:12)(cid:12)u=u1 (cid:12)(cid:12)u=u2
(cid:12) (cid:12)
(cid:12) (cid:12)
whichisthescalarequivalent of(33),(cid:12)sinceT′(u )T′(cid:12)(u )isnonnegative. Toseethis,notefrom(28)that:
1 2
dT2(u)
=T′(u∗)T′(u∗)>1 (36)
du (cid:12)
(cid:12)(cid:12)u=u∗
(cid:12)
(cid:12)
andsuppose,incontrastto(35),thatthederiva(cid:12)tiveofT2(·)isgreaterthanorequaltooneateachofitsfixed
points. ThisimpliesthatT2(u)≥uforallu≥u∗,because T2(u)−uhasnonnegativeslopeatzero-crossings
and, thus, remains nonnegative for u ≥u∗. The inequality T2(u) ≥ u implies unbounded growth of T2(·)
whichisacontradiction becauseT(·)iscontinuous andnonincreasing, thus,bounded.
ProofofTheorem2: LetNI and NI′ :=N−NI denotethecardinalities ofthesetsIandI′,andindexthe
cellssuchthati=1,···,N belongtoI,andi=N +1,···,N belongtoI′. ThenthematrixPhastheform:
I I
0 P
P= 12 (37)
 P 0 
21
 
7
where P12∈RNI×NI′,P21∈RNI′×NI. LetX:=[x1T···xNT]T,andnotethatthelinearization of(3)-(4)about
(31)givestheJacobian matrix:
I ⊗A P ⊗(B C )
NI 1 12 1 2 (38)
 P ⊗(B C ) I ⊗A 
where  21 2 1 NI′ 2 
∂f(x,u) ∂f(x,u) ∂h(x)
A := , B := , C := , j=1,2. (39)
j j j
∂x (cid:12)(cid:12)(x,u)=(S(uj),uj) ∂u (cid:12)(cid:12)(x,u)=(S(uj),uj) ∂x (cid:12)(cid:12)x=S(uj)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
From the definition(cid:12)(1), the matrix DP, where D (cid:12)is a diagonal matrix of the n(cid:12)ode degrees, is symmetric.
Since D−1/2(DP)D−1/2=D1/2PD−1/2 isalsosymmetric,wewrite:
0 R
D1/2PD−1/2= (40)
 RT 0 
 
whereR∈RNI×NI′ isappropriately defined. Then,weapplythefollowingsimilaritytransformation to(38):
I ⊗A P ⊗(B C ) I ⊗A R⊗(B C )
(D1/2⊗I ) NI 1 12 1 2 (D−1/2⊗I )= NI 1 1 2 . (41)
n  P ⊗(B C ) I ⊗A  n  RT⊗(B C ) I ⊗A 
 21 2 1 NI′ 2   2 1 NI′ 1 
Thestructure of(40)issuchthatitcandiagonalized withanorthonormal matrixoftheform:
Q Q Q 0
Q= 1 1 3 (42)
 Q −Q 0 Q 
2 2 4
 
whichresultsin:
Λ
+
0 R  −Λ+ 
where Λ is a diagonal matrix ofthReTstr0ictlyQp=osQitive eigenvalues0of0P, the columns of Q and Q s(p4a3n)
+ 3 4
the null spaces of RT and R, respectively, and the dimensions of the zero diagonal blocks in (43) are con-
sistent with the dimensions of these null spaces (which we denote as n and n , respectively). From the
3 4
orthonormality of Q,wegettheidentities:
1
QTQ =QTQ = I (44)
1 1 2 2 2 r
QTQ =I QTQ =I (45)
4 4 n4 3 3 n3
QTQ =0 QTQ =0, (46)
1 3 2 4
whereristhedimension ofΛ . Likewise,equation(43)implies:
+
RQ =Q Λ RTQ =Q Λ (47)
2 1 + 1 2 +
RQ =0 RTQ =0. (48)
4 3
WenowreturntotheJacobian matrix(41)andfurtherapplythefollowingsimilarity transformation:
2QT ⊗I 0
1 n
 QT300⊗In 2QQT4T20⊗⊗IInn RTIN⊗I(⊗BA2C11) RI⊗NI(′B⊗1CA12) Q10⊗In Q20⊗In Q30⊗In Q40⊗In (49)
8
where the leftmost matrix is the inverse of the rightmost matrix from (44)-(46). Likewise, using (44)-(48),
itisnotdifficulttoshowthattheproduct (49)equals:
I ⊗A Λ ⊗(B C )
r 1 + 1 2
 Λ+⊗(B2C1) Ir⊗A2 
 In3⊗A1 In4⊗A2 . (50)
SinceAssumptions1and2implythatA andA areHurwitz[12,Lemma6.5],stabilityof(50)isdetermined
1 2
bytheupperleftblockswhich,uponasimilaritytransformationwithanappropriatepermutationmatrix,are
block-diagonalized intorblocksoftheform:
A λB C
1 i 1 2
(51)
 λB C A 
i 2 1 2
 
i=1,···,r.
Wewillnowshowthat(51)isHurwitzforanyλ ∈[−1,1]. Sincealleigenvalues ofPlieinthisinterval
i
byLemma1,thiswillconcludetheproof. Wedonotprovideaseparateprooffortheasymptoticstabilityof
(32), as identical arguments apply when the indices 1 and 2 are swapped in (51). If λ =0, (51)is Hurwitz
i
because A and A areHurwitz. Ifλ ,0,thenweapplythesimilaritytransformation:
1 2 i
I 0 A λ B C I 0 A λ2B C
1 i 1 2 = 1 i 1 2 (52)
 0 λ−1I  λ B C A  0 λI   B C A 
 i  i 2 1 2  i   2 1 2 
andrewritetheresultas:
A+BC (53)
where
A λ2B C 0
A:= 1 i 1 2 , B= , C=[C 0]. (54)
1
 0 A   B 
2 2
Weclaimthatthelinearsystemdefinedbythetriplet(C,A ,B)ismonotonewithrespecttoKU =KY =Rm ,
≥0
and KX =−K×K where K isasinAssumption2. Toseethis,firstnotefromLemma2that(C ,A ,B )and
1 1 1
(C ,A ,B ) are monotone with respect to the cones specified in Assumption 2. By Lemma 3, this means
2 2 2
that:
x∈K ⇒ A x∈K, u∈Rm ⇒ B u∈K, x∈K ⇒ C x∈Rm , j=1,2. (55)
j ≥0 j j ≤0
Wenowshowthattheconditions ofLemma3holdfor(C,A,B)withKU =KY =Rm ,KX =−K×K:
≥0
1)Suppose x=[xTxT]T ∈−K×K,thatis x ∈−K, x ∈K. Then,
1 2 1 2
A x +λ2B C x
Ax= 1 1 i 1 2 2 ∈−K×K (56)
 A x 
2 2
 
because, from(55), A x ∈−K,A x ∈K,C x ∈Rm and,hence, B C x ∈−K.
1 1 2 2 2 2 ≤0 1 2 2
2)Wewanttoshowthatu∈Rm impliesBu∈−K×K. FromthedefinitionofBin(54),Bu∈−K×K means
≥0
B u∈K. Itfollowsfromthesecondimplication in(55)thatu∈Rm indeedimplies B u∈K.
2 ≥0 2
3)Toprovemonotonicity withKY =Rm ,weneedtoshowthat x ∈−K and x ∈K implyC[xTxT]T ∈Rm .
≥0 1 2 1 2 ≥0
Thisisindeedtrue,sinceC[xTxT]T =C x and,from(55), x ∈−K impliesC x ∈Rm .
1 2 1 1 1 1 1 ≥0
9
Having verified the conditions of Lemma 3, we conclude that (C,A,B) is monotone with respect to
KU =KY =Rm . In addition, the matrix A in (54) is Hurwitz, as A and A are Hurwitz. Thus, it follows
≥0 1 2
fromthefirststatement inLemma4that,if−(I+CA−1B)isHurwitz,thensois(53). Notethat
A−1 −λ2A−1B C A−1 0
CA−1B=[C 0] 1 i 1 1 2 2 =−λ2C A−1B C A−1B (57)
1  0 A−1  B  i 1 1 1 2 2 2
 2  2 
and,fromaderivation similartotheonefor(23),T′(u )=−C A−1B , j=1,2.Thus,(57)gives:
j j j j
−(I+CA−1B)=−I+λ2T′(u )T′(u ), (58)
i 1 2
and(33)andλ ∈[−1,1]implythat−(I+CA−1B)isindeed Hurwitz. FromLemma4,this meansthat (53)
i
and,thus,(51)isHurwitzi=1,···,r,concluding theproof. (cid:3)
4.2 Generic Convergence to Steady-States
Thusfarwehave studied local asymptotic stability properties ofthe steady-states. Strongly monotone sys-
tems(as defined intheparagraph above Definition 1)have been shown topossess a“generic convergence”
property [8, 9] which means that almost every bounded solution (except for a measure-zero set of initial
conditions) converges to the set of steady-states. Below wefirst prove monotonicity of (3)-(4) in Theorem
3 and, next establish strong monotonicity in Theorem 4, thereby concluding generic convergence for this
system.
Theorem3. IfGisbipartite andAssumption2holds,thenthesystem(3)-(4)ismonotone.
Proof: LetI⊂{1,···,N}andI′={1,···,N}−IbedefinedasinProposition 1,andsuppose thatin(3),the
cells are indexed such that i=1,···,N belong to I, and i= N +1,···,N belong to I′ as in the proof of
I I
Theorem 2, where NI is the cardinality of set I. Let XI :=[x1T···xNIT]T, XI′ :=[xNI+1T···xNT]T, and
defineUI,UI′,YI,YI′ similarly. Then,theinterconnection condition (4)becomes:
UI = (P ⊗I )YI′ (59)
12 m
UI′ = (P ⊗I )YI (60)
21 m
where P and P areasin(37). Ablockdiagram illustrating thisinterconnection isdepicted inFigure1.
12 21
x1
U:=UI x2 YI UI′ YI′ Y
P ⊗I P ⊗I
... 21 ... 12
xNI xN
Figure1: Ablockdiagramforthesystem(3)-(4)whenthecontactgraphisbipartiteandthecorrespondingintercon-
nectionmatrixPisdecomposedasin(37).
To prove the monotonicity of this feedback system, we establish the monotonicity of the feedforward
systemwithinputU:=UI andoutputY:=(P ⊗I )YI′:
12 m
10