Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional

Approach

Sohei Ito

1,2

, Kenji Osari

, Shigeki Hagihara

and Naoki Yonezaki

5,6

Department of Fisheries Distribution and Management, National Fisheries University, 2-7-1 Nagata-Honmachi,

Shimonoseki, Yamaguchi, Japan

School of Business Administration in Karvin´a, Silesian University in Opava, Univerzitn´ı n´am. 76,

733 40 Karvin´a, Czech Republic

Yahoo Japan Corporation, 9-7-1 Akasaka, Minato-ku, Tokyo 107-6211, Japan

Department of Computer Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan

Graduate School of Information E nvironment, Tokyo Denki University, 2-1200, Muzai Gakuendai,

Inzai-shi, Chiba 270-1382, Japan

The Open University of Japan, 2-11 Wakaba, Mihama-ku, Chiba City,

Chiba 261-8586, Japan

Keywords:

Gene Regulatory Network, Systems Biology, Homeostasis, Temporal Logic, Realisability.

Abstract:

Homeostasis is an important property of life. Thanks to this property, living organisms keep their cellular

conditions within an acceptable range to function normally. To understand mechanisms of homeostasis and

analyse it, the systems biology approach is indispensable. For this purpose, we proposed a qualitative approach

to model gene regulatory networks with logical formulae and formulate the homeostasis in terms of a kind

of logical property – called realisability of l inear temporal logic. This concise formulation of homeostasis

naturally yields the method f or analysing homeostasis of gene networks using realisability checkers. However,

the realisability problem is wel l-known for its high computational complexity – double-exponential in the size

of a formula – and the applicability of this approach will be limited to small gene networks, since the size

of formula increases as the network does. To overcome this limitation, we leverage a compositional method

to check realisability in which a formula is divided into a few sub-formulae. The difﬁculty in compositional

approach is that we do not know how we obtain a good division. To tackle this issue, we introduce a new

clustering algorithm based on a characteristic function on formulae, which calculates the size of formulae and

the variation of propositions. The experimental results show that our method gives a good division to beneﬁt

from the compositional method.

1 INTRODUCTION

Since the end of the 20th century, the advances of ex-

perimental and high-throughpu t technologies in mo-

lecular biology have enabled us to produce huge

amount of biological data. Compared to such advan-

cement, the understanding of biolo gical systems and

how th ey work seems less understood than expected.

As genes an d proteins are components of the system,

knowledge about each component does not give us

how the system is working as entirety. Therefore , a

new research ﬁeld - systems biology - has emerged.

Systems biology aims to understan d organisms as

systems. Systems biology covers many topics such

as structural identiﬁcation of biological systems, con-

struction of mathematical models which ﬁt to obser-

ved phenom ena, and development of modelling and

analysing tools (Funahashi et al., 2003; Naldi et al.,

2009; Helikar et al., 2012).

Usual approach in systems biology is to model

a system with ordinary differential equations. Since

there is uncertainty in kinetic pa rameters, we ﬁt them

to observed data to obtain a plausible model. Ho-

wever, the more comp onents we have in the system,

the more equations w e have thus parameter ﬁtting

or computing analytical solution of the equations be-

come almost infeasible. To overcome this difﬁculty,

computational models have been used in systems bi-

ology (Fisher and Henzin ger, 2007), which abstract

real numer ic al behaviours into some discrete state se-

quences. There are several approaches to such com -

putational models depending on underlying forma-

Ito S., Osari K., Hagihara S. and Yonezaki N.

Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional Approach.

DOI: 10.5220/0006093600170028

In Proceedings of the 10th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2017), pages 17-28

ISBN: 978-989-758-214-1

lisms; Boolean networks (Thomas, 1991), Petri nets

(Heiner et al., 2008 ), timed-automata (Batt et al.,

2007) and process algebras (Ciocchetta and H illston,

2009). The limitation of such computational models

is that they re quire concrete mo le cular me chanisms,

regulatory logics and sometimes a kinetic parameters.

Since biological systems are incomple te in most ca-

ses, the applicability of such computational models

are limited. In this regard, the constra int-based mo-

delling is promising in which systems are described

as a set of constraints (Palsson, 2000). We can mo del

biological systems with incomp lete information. The

more information we have on the target system, the

more constraints we have in a model.

The constraint-based modelling paradigm has a

good connection with lo gical spec iﬁcation of rea c tive

systems in whic h the system behaviour is described as

a set of logical constraints (Mori and Yonezaki, 1993;

Vanitha et al., 2000; Osari et al., 2014). Based on

this close connection, we proposed a qualitative met-

hod for modelling and analysing gene networks (Ito

et al., 2010; Ito e t al., 2015b) using linear temporal

logic (LTL). In this approach, we specify possible be-

haviours of networks by LTL as a set of logical con-

straints an d analyse properties of networks (also w rit-

ten in LTL) by checking satisﬁability of the formulae.

This idea is based on the veriﬁcation of reactive sy-

stem speciﬁcations. Using this correspondence, we

later formulated the notion of homeostasis by reali-

sability of reactive system speciﬁcations (Pnueli and

Rosner, 1989; Abadi et al., 198 9) and demonstrated

how homeostasis in gene networks can be analysed

(Ito et al., 2014a).

The problem in our approach was the high-

complexity of checking LTL realisability which is

doubly-exponential in the size of a formula. As we

reported in (Ito et al., 2015a), c hecking hom e ostasis

of gene ne tworks with even a few nodes takes several

minutes. Since the sizes of formulae characterising

possible behaviours of a network is proportional to

the size of the network, we need to devise a me thod

to mitigate the computational difﬁculty in th e an alysis

of homeostasis.

To facilitate realisability checking, a compositio-

nal approach has been a promising method to analyse

large for mulae (Filiot et al., 2011) in which an LTL

formu la is d ivided into several sub-formulae that are

analysed separately. The results are the n merged to

obtain the ﬁnal outcome of the veriﬁcation. The non-

trivial p roblem in compositiona l analysis is how to di-

vide a formula, which is critical to the perf ormance of

analysis.

The aim of this paper is to provide a method for

ﬁnding good divisions as a front-end of a compo-

sitional analysis of homeostasis. Our approach to

this problem is to develop a new clustering algorithm

which clusters clauses (a piece of formula) based on

a ‘score’ of clusters. The ‘score’ of clusters indicates

how the clustering is good for compositional analysis.

We implement our algorithm and show experimental

results of analysing homeostasis of several gene net-

works. For larger gene networks, the gain from com-

positional approach is quite much: with compositi-

onal approach with our clustering front-en d, we can

verify in a few minutes, or even in a few seconds, net-

works that cannot be veriﬁed with monolithic appro-

ach in 1 hour.

The rest of the paper is o rganised a s follows.

Section 2 introduces LTL and the notion of realisa-

bility of reactive system speciﬁcations. Section 3 re-

views how we model gene networks in LTL and for-

mulates homeostasis by realisability. Section 4 intro-

duces the compositional analy sis of realisability. We

show how we divide a network speciﬁcations to com-

positionally ana lyse homeostasis. Section 5 shows

and discusses the experime ntal results of compositi-

onal analysis. In section 6, we discuss the relations-

hip between our m ethod and other similar approaches.

Section 7 offers conclusion and discusses some future

directions.

2 PRELIMINARY

2.1 Linear Temporal Logic

If A is a ﬁnite set, A

denotes the set of all inﬁnite

sequences on A. The i-th element of σ ∈ A

is deno-

ted by σ[i]. Let AP be a set of propositions. A time

structure is a sequence σ ∈ (2

)

where 2

is the

powerset of A P. The formulae of LTL are deﬁned as

follows.

• p ∈ AP is a formula.

• If φ and ψ are for mulae, then ¬φ,φ∧ ψ,φ∨ ψ and

φUψ are also formulae.

We introduce the usual abbreviations: ⊥ ≡ p ∧ ¬p

for some p ∈ AP, ⊤ ≡ ¬⊥, φ → ψ ≡ ¬φ ∨ ψ, φ ↔

ψ ≡ (φ → ψ) ∧ (ψ → φ), Fφ ≡ ⊤Uφ, Gφ ≡ ¬F¬φ,

and φW ψ ≡ (φU ψ) ∨ Gφ. We assume that ∧, ∨ and

U binds more strongly than → and unary connectives

binds more strongly th an binary ones.

Let σ be a time structure and φ be a formula. The

semantics of LTL is deﬁned by the relation σ |= φ re-

presenting φ is true in σ. The satisfaction relation |=

is deﬁned as follows.

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

σ |= p iff p ∈ σ[0] for p ∈ AP

σ |= ¬φ iff σ 6|= φ

σ |= φ ∧ ψ iff σ |= φ and σ |= ψ

σ |= φ ∨ ψ iff σ |= φ or σ |= ψ

σ |= φUψ iff (∃i ≥ 0)(σ

|= ψ and

∀ j(0 ≤ j < i)σ

|= φ)

where σ

= σ[i]σ[i + 1].. . , i.e. the i-th su fﬁx of σ.

An LTL formula φ is satisﬁable if there exists a time

structure σ such that σ |= φ. We a lso say that σ is a

model of φ.

2.2 Reactive Systems and Realisability

A reactive system is deﬁned as a triple hX ,Y,ri, whe re

X is a set of events caused by the environment, Y is a

set of events caused by the system and r : (2

)

→ 2

is a reaction function. Th e expression (2

)

denotes

the set of a ll ﬁnite sequences on subsets of X. A re-

action function determines how the system reacts to

environmental (or external) input sequences. Reactive

system is a natural formalisation o f systems which ap-

propriately respond to requests from the environment.

Systems controlling vending machines, elevators, air

trafﬁc and nuclear power pla nts are examples of re-

active systems. Gene networks which respond to in-

puts or stimulation from the environme nt such as glu-

cose increase, change of temperature or blood pres-

sure can also be considered as reactive systems.

A speciﬁcation of a reactive system stipulates how

it responds to inputs from the environment. For exam-

ple, for a controller of an elevator system, a speciﬁca-

tion will be e.g. ‘if the open button is pushed, the door

opens’ or ‘if a call button of a certain ﬂoor is pushed,

the lift will come to the ﬂoor’.

Realisability is an important property of reactive

system speciﬁcations (Pnueli and Rosner, 1989;

Abadi et al., 1989). A realisable speciﬁcation guaran-

tees that there exists a r eactive system wh ich respo nds

to any environmental inpu t of any timing without vi-

olating the speciﬁcation.

To check realisability of a reactive system spe-

ciﬁcation, it should be described in a language with

formal and rigorous semantics. LTL is known to be

one of many other formal lang uages suitable for this

purpose and several realisability checkers of LTL are

available (Jobstmann et al., 2007; Filiot et al., 2 009;

Bloem et al., 2010).

Now we deﬁne the notion of realisability of LTL

speciﬁcations. Let AP be a set of atomic propositions

which is partitioned into X and Y . X corr esponds to

external events and Y to internal events. We denote

a time structure σ on AP as hx

ihx

i. .. where

⊆ X, y

⊆ Y and σ[i] = x

∪ y

. Let φ be an LTL

speciﬁcation. We say hX,Y,φi is realisable if there

x y

Figure 1: A gene network in which x and y are ge-

nes. Plus-edges represent activation relationship and

minus-edges represent inhibition relationship. Ge ne x

receives positive input from environment.

exists a reactive system RS = hX ,Y, ri such that

∀ ˜x.behave

( ˜x) |= φ,

where ˜x ∈ (2

)

and behave

( ˜x) is the inﬁnite beha-

viour determined by RS, that is,

behave

( ˜x) = hx

ihx

i. .. ,

where ˜x = x

.. . and y

= r(x

.. .x

Intuitively a speciﬁcation φ is realisable if there

exists a system which controls its internal events in

reaction to any sequence of external events, so that its

behaviour satisﬁes the speciﬁcation φ.

3 QUALITATIVE ANALYSIS OF

HOMEOSTASIS IN GENE

NETWORKS

In this section we review how we model the gene net-

works and analyse homeostasis of them by LTL.

3.1 Modelling Possible Behaviours of

Gene Networks in LTL

Qualitative pr inciples for characterising behaviours of

a gene network are as follows:

• A gene is ON when its activators are expressed

beyond so me thresholds.

• A gene is OFF when its inhibitors are expressed

beyond so me thresholds.

• If a gene is ON, its expression level increases.

• If a gene is OFF, its expression level decreases.

To form a lly describe these principles in LTL, we in-

troduce propositions for a given network which repre-

sent the state (or c onﬁgu ration) of a network, such as

whether genes are ON and whether gene s are expres-

sed beyond c ertain thresholds.

We show how we describe behaviours of networks

using an example network depicted in Fig. 1, where

x and y repre sent gene s, p lus-edges mean activation

and the minus-edge inhibition. Since gene x activates

gene y, there exists a threshold expression level of x

Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional Approach

above which gene x activates gene y. We write x

for

this threshold level. Similarly, gene y has the thres-

hold y

to inhibit x. Moreover, gen e x receives the

positive input from environment. We wr ite e

for the

threshold of the input to activate x.

For th is network we introduce th e following pro-

positions.

• on

, on

: whether gene x and y are ON r especti-

vely.

• x

, y

: whether gene x and y are expressed beyond

the threshold x

and y

respectively

• in

: whether the input to x is ON.

• e

: whethe r the positive input from the environ-

ment to x is beyond the threshold e

Using these propositions, we specify the above

qualitative principle s in LTL. For example, the fact

‘gene y is positively regulated by gene x’ is descr ibed

G(x

↔ on

)

in LTL. Intuitively this formula says gene y is ON if,

and only if, gene x is expressed beyond the threshold

(i.e. proposition x

is true) due to positive effect of

gene x toward gene y. As for gene x, it is negatively

regulated by gene y and has positive inp ut from the en-

vironm ent. A co ndition for activation and inactivation

of such multi-regulated genes depends on a function

which merges the multiple effects. We assume that

gene x is ON if gene y is not expressed beyon d the

threshold y

and, in addition, the input from the envi-

ronment to gene x is beyo nd the threshold e

; that is,

the negative e ffect of gene y is not opera ting and the

positive effect of the input is ope rating. Then this can

be described as

G(e

∧ ¬y

→ on

This formula says that if the input level is beyond the

threshold e

and gene y is not exp ressed beyond the

threshold y

(i.e. proposition y

is false; ¬y

is true),

then gene x is ON.

If gene x is ON, the exp ression level of gene x is

growing. In other words, it will reach the threshold x

unless gene x becomes OFF prematurely. This fact is

described as

G(on

→ F(x

∨ ¬on

)).

If g e ne x is ON and expressed beyond the threshold

, it keeps the level until gene x is OFF since this is

the largest threshold of gene x. This can be described

G(on

∧ x

→ x

W ¬on

Note that the threshold levels are denoted in roman

whereas the propositions corresponding to them are denoted

in italic.

This formula means ‘if gene x is ON and the current

expression level of gene x is above x

, gene x keeps

its level as long as gene x is ON’.

If gene x is OFF, its transcription produ c t decrea-

ses due to degradation. We have the symmetric for-

mulae to the case of gene x being ON:

G(¬on

→ F(¬x

∨ on

)),

G(¬on

∧ ¬x

→ ¬x

W on

We have similar c la uses for expression of gene y.

We do not th oroughly explain how w e d escribe

constraints which model the possible behaviours of

gene network s. Interested readers may wish to con-

sult (Ito et al., 2 015b) for detail.

3.2 Analysing Homeostasis by

Realisability Checking

Biological homeostasis is informally deﬁned as the

tendency of a system to maintain its internal stabi-

lity or f unction against any situation or stimulus. This

property is closely related to realisability since it says

that there is a system which responds to any environ-

mental inputs of any timing while keeping its speci-

ﬁed internal conditions. Using this correspondence

we for mulate homeostasis by realisability.

In the previous section, we showed that a gene net-

work can be modelled by an LTL formula which is the

conjunc tion of clau ses. Precisely speaking, the speci-

ﬁcation of a given network is a triple hE, I, ϕi where

• E: the set of external propositions (inputs),

• I: the set of internal propositions (genes and thres-

holds),

• ϕ: the formula which characterises the possible

behaviours of a given network.

We also write a biological property or function of

the network, which we are to check whether the given

network maintains it in response to any external input

sequence. The examples of such property will be ‘the

expression level of a certain gene is within a to le rable

range’, ‘once a gene becomes ON, its expression will

be suppressed afterwards’, et cetera. Such properties

can be easily described in LTL.

In realistic situation, we sometimes put an as-

sumption (or constraints) on the environment which

restricts the in put sequ ence. For example, we may

consider the case that an input is oscillating, an input

is always coming, or a certa in input com es only after

another comes, and so on. Sometime s it is useful to

consider homeostasis of a system under such environ-

mental assumptions. We also describe suc h assumpti-

ons in LTL.

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

Now we deﬁne homeostasis of a gene network

with respect to a p roperty under an environmental as-

sumption. The following deﬁnition is an extended

version of our previous deﬁnition of homeostasis (Ito

et al., 201 4a) in that we include environmental as-

sumption ξ.

Deﬁnition 1. A property ψ is homeostatic with re-

spect to a behaviour speciﬁcation hE, I, ϕi u nder the

environmental assumption ξ if hE, I, ξ → ϕ∧ ψi is re-

alisable.

Note that a homeostasis of a gene network is de-

ﬁned with a certain bio logical property. A network

which is homeostatic with r espect to a certain pro-

perty has a strategy to respond for any input sequence

which satisﬁes ξ, witho ut violatin g neither behaviour

constraint ϕ nor the property ψ. We can set ξ as true

(i.e. empty clause), which amounts to put no assump-

tion on environment.

Example 1. The network depicted in Fig. 1 is ho-

meostatic with respect to a p roperty ‘whenever gene x

becomes ON, the expression of gene x will be suppres-

sed afterwards’. This property is described in LTL as

follows:

G(on

→ F(¬on

∧ ¬x

)).

Let ϕ be a behaviou ral speciﬁcation of the network

(partly shown in the previous section). This is ve-

riﬁed by checking realisability of the speciﬁcation

h{in

}, {on

,on

}, ϕ ∧ G(on

→ F(¬on

∧

¬x

))i.

This property is also homeostatic if we put an en-

vironm e ntal assumption ‘the input to x is always ON’

which is described as:

Gin

That is to say, the speciﬁcation

h{in

}, {on

,on

}, Gin

→ ϕ ∧ ψi is re -

alisable (where ψ represents the above property).

Since we de ﬁned homeostasis by realisability, we

can use realisability checkers to conduct such ana-

lysis. A n obstacle with this framework is the h igh-

complexity of LTL realisability problem which is

2EXPTIME- c omplete in the size o f a formula (Pnu-

eli and Rosner, 1989). Since the size of a network

speciﬁcation is proportional to the size of a network,

the analysis of a larger network will be intractable in

general. The next section discusses the compositional

analysis method to mitigate this difﬁculty.

4 COMPOSITIONAL ANALYSIS

OF HOMEOSTASIS

Recent advances in realisability checking technique

enable us to handle large speciﬁcations. Since our

framework to analyse homeostasis uses realisability

checking, we can re ap the beneﬁt from the advan-

ces. Here we leverage the compositional algorithm

in realisability checking (Filiot et al., 2011). In our

setting, the com positional analysis of homeostasis of

gene networks is stated as fo llows:

1. For a given speciﬁcation hE, I, ξ →

1≤i≤n

compute a ‘good’ clustering of the formula

,. .. , c

}, where each c

ℓ

consists of ϕ

s, e.g.

network speciﬁcations and bio logical properties.

2. We check the realisability of each sub-

speciﬁcation

hE, I, ξ →

ϕ∈c

ℓ

ϕi(ℓ ∈ {1,. .. ,k}).

3. We merge the r esult of individual results.

Step 2 and 3 can be automatically done by the tool

Acacia+ (Boh y et al., 2 012). The problem is how

we obtain a ‘good ’ clustering of a given speciﬁcation.

Before proceedin g, we ﬁrst review the algorithm of

Acacia+ to solve realisability compositionally.

4.1 Compositional Approach to

Realisability Problems

The algorithm implemented in Acacia+ to solve rea-

lisability of LTL formula ϕ is as follows:

1. Translate ϕ to a universal co-B¨uchi a utomaton A

2. Translate A

to a safety game G(A

3. Compute the winning strategy of the game G(A

It is reported that th e ﬁrst step, i.e. translating an

LTL formula to a n equivalent automaton, is the bott-

leneck of the algorithm. Known algorithms to trans-

late LTL formulae into equivalent automata run in ex-

ponen tial time with respect to the sizes of formulae.

Especially for large LTL formulae the ﬁrst step do es

not ﬁnish. To overcome this p roblem, the compositi-

onal approach is proposed. Large LTL formulae are

often written as ϕ = ϕ

∧ ϕ

∧ ··· ∧ ϕ

. Thus in com-

positional appro ach, each sub-formula ϕ

is transla-

ted into autom aton A

and thus translated into a local

game G(A

), then the winning strategy is computed

for each G(A

). Thanks to the nice property of sa-

fety game s, the winning strategy fo r G(A

), the stra-

tegy for the original g ame, can be computed from the

winning strategies for each local game G(A

In general, the speciﬁcation is of the form ξ → ϕ

where ξ is an environmental assumption. Therefore ξ

Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional Approach

is distributed to each ϕ

, that is, we solve the realisa-

bility of each ξ → ϕ

separately.

The m e rit of this compositiona l approach is that

we c an reduce the time of translating formulae into

equivalent auto mata, since eac h ϕ

is much smal-

ler than the original formula ϕ. Note that, however,

we have extra cost of merging the winning strategies

for local games, compared to non-compositional (i.e.

monolith ic ) approach. For compositional approach to

be u seful, the reduction in automata translation must

prevail the additional cost of merging the local win-

ning strategies. Thus the performance of compositio-

nal approac h depend s on how we divide ϕ into several

4.2 Applying to Our Problem

Since behaviour sp e ciﬁcations for g ene ne tworks are

written as conjunctions of clauses (see section 3.1),

we can apply the compositional app roach introduced

in the previous section. Our speciﬁcation in gene-

ral consists of many clauses, thus it is not efﬁcient

to solve every single clauses sepa rately as the over-

head of integrating the winning strategies of the local

games increases according to the num ber of local ga-

mes.

Thus the possible approach is to distribute the

clauses in to several clusters c

,. .. , c

, i.e. ϕ =

(

) ∧ (

) ∧ ·· · ∧ (

) where

φ∈c

φ.

The problem is how to distribute them. What is a good

clustering?

For example, suppose that we have 20 clauses and

distributing them into 2 clusters. It seems better to put

10 clauses into ea ch cluster than to put 1 clau se into

one cluster and 19 clauses into the other. In such une-

ven clustering, the 19 clauses cluster will be a bottle-

neck and the efﬁciency may not improve as a whole.

Thus a goo d clustering distributes clauses into clus-

ters as equally as possible. However, the number of

clauses may not be a good criterion, since the sizes

of form ulae are not the same; in an extreme situation,

the size of a ce rtain formula might be equal to the size

of the rest. Thus we distribute clauses into clusters

whose sizes are as equal as possible.

To facilitate winning strategy computation of each

local game, it is desirable to obtain small automata.

The size of automata A

is determined by the number

of sub-formulae in ϕ. To compute the number of sub-

formu lae for each clause is not realistic because it is

exponential to the size of the formula. The plausible

criterion is the number of variation of propositions –

the more variation of propositions we have, the more

the number of sub-formulae. Thus we also take the

variation of propositions in a cluster into co nsidera-

tion, in addition to the size of a cluster.

Now we formalise the above idea. Let

C = {c

,. .. , c

} be a clustering o f a speciﬁcation

hE, I, ξ →

1≤i≤n

i, that is, each cluster c

ℓ

consists

of ϕ

s. Note that the environmental constraint ξ is co-

pied to each cluster. Let N(χ) and V (χ) respectively

denote the number and th e variations of propositions

in a formula χ. We introduce the evaluation function

F of clustering C as follows:

F (C) =

∑

1≤i≤k

E(ξ →

ϕ∈c

ϕ),

where E (χ) = N(χ)

+V (χ)

The fun ction E rep resents the evaluation of a single

formu la. Due to the squared terms in the function

E, we can easily see that a clustering consists of two

clusters of the sizes 5 and 5 is better than that of 9 an d

1. A good clusterin g is the one which minimises the

value of this function.

The number of possible clustering of n formulae

into k clusters is a S(n, k), i.e. the Stirling number

of the second kind. Thus the na¨ıve algorithm to ﬁnd

the optimal cluster is not realistic in general. Thus we

introdu ce a suboptimal algorithm whose complexity

is O(kn

) which we show in Algorithm 1.

Algorithm 1: Suboptimal clustering algorithm.

Input: {ϕ

,. .. , ϕ

} (speciﬁcation), k (the number of

the clusters)

Output: C = {c

,. .. , c

}

← {ϕ

,. .. , ϕ

}

for i = 2 to k do

←

end for

C ← {c

,. .. , c

}

min ← F (C)

repeat

for all ϕ ∈ c

change ← false

for i = 2 to k do

′

← {c

\{ϕ}, .. ., c

∪ {ϕ}, .. ., c

}

f ← F (C

′

)

if f < min then

C ← C

′

min ← f

change ← true

end if

end for

until change = false

This algorithm starts with the cluster c

with

{ϕ

,. .. , ϕ

} and the others with empty sets. For each

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

step we move one ϕ in c

to another clu ster c

and

evaluate th e new clustering by function F . If the

evaluation decreases for some i, we adopt the mini-

mal one among such clusterings. We repeat this pr o-

cess until moving ϕ from c

to any other clusters no

longer decreases the evaluation. Since the outermost

loop is executed at most n times, and each step de-

crements the number of elements of c

, this a lgorithm

runs in time O(kn

). We implemented the algorithm

in OCaml and conﬁrme d that this suboptimal algo-

rithm computes a clu stering whose evaluation is as

good as the optimal clustering.

5 EXPERIMENTAL RESULTS

AND DISCUSSION

This section discusses the expe rimental results of ho-

meostasis analysis pe rformed on several networks.

Since comp ositional analysis is available for only re-

alisable speciﬁcations, all speciﬁcations used in this

experiment are realisable.

To solve the realisability of an LTL spe ciﬁcation

ϕ, we ﬁr st construc t an ω-automa ton which is equi-

valent to ϕ, then construct a game on the automaton

and compute a winning strategy of the system which

can respon d any environm ental req uests. Co mposi-

tional analysis decreases the cost o f constructin g ω-

automata from LTL formulae since the form ula is di-

vided into several small sub-formulae. However, we

have extra cost of merging all winning strategies com-

puted for each sub-formulae. Therefore , increasing

the number of clusters ma y not ne c essarily reduce the

cost of analysis as a whole. However, to predict the

optimal number of clustering a priori is difﬁcult. We,

hence, demonstrate our clustering meth od with diffe-

rent number of clusters to see the impact of the num-

ber of clusters in our compositional method.

x y

Figure 2: A bistab le switch ( Ito et al., 2014a).

x y

- -

Figure 3: A bistable switch with a dditional inputs (Ito

et al., 2014a).

The result of experiments are shown in table 1.

These experiments are car ried out on a computer with

Intel Core i7-3820 3.60GHz CPU and 32GB memory.

The realisability checker used is Acacia + (Bohy et a l.,

2012). The network ‘bistable switch’ is the network

that consists of two genes x and y where x activates y,

y activates x and x receives negative input from envi-

ronment (Fig. 2). The network ‘bistable switch2’ is

the same as ‘bistable switch’ except both gene x and

y receive negative inputs from environment (Fig. 3).

The networks ‘anti-stress response (a )(b)(c)’ are th e

stress response networks studied in (Zhang and An-

dersen, 2007) (Fig. 4). The networks ‘3 genes’ to ‘6

genes’ are de picted in Fig s. 5 to 8. The homeosta-

tic properties we analysed are tendency to ease stress

(described as G(stress → F¬stress)) for anti-stress re-

sponse networks and stability of a certain gene ex-

pression (described as Gon) for others. A s environ-

mental assumptions, we put every input is oscillating;

it is written as G((inp ut → F¬input) ∧ (¬input →

Finput)).

As we can see from the result, we beneﬁt from

compositional analyses compared to the monolithic

approa c h (k = 1). Th e improvements in analyses of

speciﬁcations from ‘3 genes’ to ‘6 genes’ are remar-

kable.

Concernin g the number of clusters, as we stated

at the beginning of this section, incre asing the num -

ber of clusters does not necessarily reduce the entire

veriﬁcation time. We can see it from ‘anti-stress re-

sponse(c)’ wher e the cases for k > 4 are worse than

the monolithic approach. Even for larger speciﬁcati-

ons, the best number of clusters seems to be 2 or 3.

The computational time of our clustering algo-

rithm is less than 0.01 regardless of the number of

clusters for a ll networks excep t for ‘6genes’ at k =

6 (0.11 seconds). This means that we can ignore

the cost of computing clusters to use compositional

analysis even for small networks such as ‘bistable

switch’. Note that table 1 includes the clustering ti-

mes.

Now we compare our m ethod to ran dom cluste-

ring. For each speciﬁcation, we generate 50 random

clusterings. We ﬁrst randomly choose the size (i.e. the

number of clauses) of each cluster, then we randomly

distribute clauses to each cluster accordin g to the size.

We show the results o f random clustering in table

2. When calculating average time of veriﬁcation, the

clusterings which cannot be veriﬁed within 1 hour are

treated as 1 hour. The mark ⋆ in the table shows that

there were suc h clusterings. Thus the true average is

greater than the shown value.

For larger spec iﬁca tions such as ‘bistable

switch2’, ‘3genes’, ‘4genes’ and ‘5genes’, we can

clearly see that our method outper forms random clus-

tering. Readers might notice that for the network of

Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional Approach

(a)

(b) (c)

Figure 4: Anti-stress networks.

Table 1: Experimental results. Columns ‘E’ and ‘I’ respectively show the numb e rs of external propositions and

internal propositions. Column ‘S’ shows th e size of formula. (i.e. numbe r of connectives and propositions).

The columns ‘Time(s)’ show the total time of the veriﬁcation for various number of clusters (k). ‘k = 1’ means

non-compositional ana lysis. For ‘k > 2 the times include computation of cluster ing.

Time(s)

Network E I S k = 1 k = 2 k = 3 k = 4 k = 5 k = 6

bistable switch 1 6 232 0.40 0.07 0.08 0.10 0.10 0.07

bistable switch2 2 8 353 206.81 0.81 0.41 0.42 0.51 0.53

anti-stress response (a) 1 9 289 0.48 0.31 0.35 0.41 0.47 0.51

anti-stress response (b) 1 7 237 0.22 0.12 0.13 0.15 0.16 0.18

anti-stress response (c) 1 9 288 0.39 0.31 0.38 0.41 0.47 0.51

3 genes 3 10 418 1617.87 5.03 5.43 6.08 5.82 6.30

4 genes 2 12 444 > 3600 4.07 4.48 5. 16 5.69 6.21

5 genes 3 15 547 > 3600 173.31 123.80 192.17 135.77 194.75

6 genes 3 18 646 > 3600 2676.37 2792.61 2877.03 2942.62 3087.91

Table 2: Results of random clustering. It shows average veriﬁcation times of 50 randomly clustered specs. The

ﬁgures show the time (in seconds) to check homeostasis. The stars on the ﬁgures show that some trials failed due

to time out of 1 hour. Such clusterings are tr eated as ﬁnished in 3600 seconds for the sake of expedience.

Network k = 2 k = 3 k = 4 k = 5 k = 6

bistable switch 0.11 0.11 0.11 0.11 0.12

bistable switch2 12.20 4.39 2.24 1.62 1.23

anti-stress response (a) 0.36 0.38 0.43 0.44 0.47

anti-stress response (b) 0.13 0.14 0.15 0.16 0.17

anti-stress response (c) 0.35 0.37 0.40 0.43 0.46

3 genes 81.80 29.44 50.88 10.44 10.94

4 genes 687.47

⋆

441.53

⋆

166.43 45.99 130.66

5 genes 1746.24

⋆

845.6

⋆

772.39

⋆

841.6

⋆

842.74

⋆

6 genes 3220.6

⋆

2953.01

⋆

2767.81

⋆

2830.68

⋆

2641.21

⋆

‘6gene s’, the random clustering seems to ou tperform

our method for some k. However, this is because

we treated timed-out samples as 3600 seconds when

averaging the results. Though we do not k now the

true average, it will be worse than the resu lts of ou r

method.

For smaller speciﬁcations, such a s ‘bistable

switch’ and ‘anti-stress response’ networks, our met-

hod is no t clearly better than r andom clustering. The

reason is that the formulae are not so large that the

automata construction is not a heavy task in the veri-

ﬁcation process. For such speciﬁcations our strategy

– to decrease the time of automata construction – may

not be suitable.

Readers might wonder why ‘4genes’ for k = 5 is

considerably smaller than others in r a ndom cluste-

ring. Plausible expla nation is that the random sam-

pling happened to give biased clusterings. What is

important here is that even for such biased random

sampling, our method is much better.

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

Figure 5: A ne twork consisting of 3 genes.

Figure 6: A ne twork consisting of 4 genes.

Figure 7: A ne twork consisting of 5 genes.

Figure 8: A ne twork consisting of 6 genes.

We also show the standard deviations of the

random clustering in table 3. As we can see from the

table, the standard deviations are hig h for larger spe-

ciﬁcations. This means that our evaluation function is

actually a statistically meaningful cr iterion, because

random clustering ranges from better to worse. Our

clustering algorithm tactfully chooses better ones.

We show another proof of the beneﬁt of our al-

gorithm: relative standin gs of our results (the values

shown in Table 1) within the run-times of r andom

clusterings. In othe r words, the percentile r anks of

our results in the random clusterings. Tab le 4 shows

them. As we can see, for k = 2 or 3 our algorithm

gives good scores.

However, this table also shows that for k ≥ 4 our

evaluation function might not b e a good criterion.

This is because if we increase the number of clus-

ters, the size of for mulae in each cluster tends to be

small, so that the automata construction time decr e-

ases and becomes less im portant in the veriﬁcation

process. Thus the larger k tends to impair the bene-

ﬁt of our method. Rather, we on ly rea p the overhead

of me rging the winning strategies of the local games.

This suggests another tactics for clustering: we focus

on decreasing the computation time of winn ing stra-

tegy. We leave this issue for future work.

Before closing this section, we make a brief re-

mark on a treatmen t of unrealisable speciﬁcations. As

we men tioned at the beginning of this section, com-

positional appro ach is o nly applicab le for realisable

speciﬁcation (Filiot et al., 2011; Bohy et al., 2012). In

the current implementation of Acacia+, a user choo-

ses options whether he checks realisability or u nreali-

sability (or both in parallel). Thus when we are to ve-

rify a speciﬁcation for which we do not know whether

it is realisable or not, ﬁrst we try realisability checking

compositionally. If the veriﬁer cannot prove realisa-

bility, we try unrealisability checking monolithically.

However, since Acacia+ uses heuristic in which a cer-

tain bound is set when searching winning strategies,

sometimes neither realisability nor unrealisability is

proved. If this is the case, we set a larger bound and

repeat the process un til we have a deﬁnite result.

We summar ise this section. For larger sp eciﬁcati-

ons, our method works well because the bottleneck in

the veriﬁcation process is the automata construction.

If the n umber of clusters is small (2 or 3), ou r method

is especially effective. If we increase the number of

clusters, the bottleneck seems to shift from automata

construction to winning strategy compu ta tion. To in-

vestigate a method to reduc e the time for winning

strategy co mputation is an interesting future work.

6 RELATED WORK

Sensitivity or robustness of biological systems is close

but different from homeostasis. It analyses whe ther a

property holds for a similar degree/probability when a

parameter is mod iﬁed, w hile homeostasis me ans that

a given property holds for a certain set of input se-

quences. (Rizk et al., 2009) de ﬁnes r obustness of a

biological system by measuring the deviation of the

‘satisfaction degree’ of an LTL formula (Fages and

Rizk, 2008) c a used by perturbations on parameters.

They can treat the initial value of an environmental

input as a parame ter, thus homeostasis is partially co-

vered. (Donz´e et al., 2011) is a similar approach to

(Rizk et al., 2009) for robustness analysis. They ana-

lyse continuous trajectories by means of signal tem-

Efﬁcient Analysis of Homeostasis of Gene Networks with Compositional Approach

Table 3: Standard d eviation of the results of random clustering .

Network k = 2 k = 3 k = 4 k = 5 k = 6

bistable switch 0.05 0.03 0.02 0.01 0.01

bistable switch2 23.00 12.1 4.51 3.74 2.30

anti-stress response (a) 0.09 0.08 0.13 0.04 0.04

anti-stress response (b) 0.02 0.02 0.01 0.01 0.01

anti-stress response (c) 0.06 0.09 0.03 0.04 0.05

3 genes 183.01 138.24 273.49 13.04 22 .63

4 genes 1239.13

⋆

1021.42

⋆

618.16 218.94 377.97

5 genes 1593.59

⋆

1318.82

⋆

1183.29

⋆

1295.33

⋆

1270.92

⋆

6 genes 700.55

⋆

763.89

⋆

835.77

⋆

819.86

⋆

989.15

⋆

Table 4: Relative standing of the veriﬁcation time with our method within the run-times of random clusterings.

The value ‘best’ means that our method was the best.

Network k = 2 k = 3 k = 4 k = 5 k = 6

bistable switch 10% 14% 39% 43% best

bistable switch2 38% 13% 15% 41% 41%

anti-stress response (a) 27% 17% 64% 88% 88%

anti-stress response (b) 31% 45% 70% 7 0% 92%

anti-stress response (c) 23% 86% 6 8% 92% 92%

3 genes 14% 26% 4 7% 27% 27%

4 genes 17% 8% 34% 38% 43%

5 genes 19% 12% 4 3% 15% 35%

6 genes 14% 26% 4 1% 44% 50%

poral logic (STL) and its satisfaction-degree. In their

approa c hes, it is unclear how to model arbitrary inpu t

scenario and assess the effect of it. SReach (Wang

et al., 2015) is a too l to analyse stochastic hybrid sys-

tems, which can be used to m odel biological systems.

SReach focuses the probabilistic reachability, that is,

a property whether a given system reaches safe (or

unsafe) states within a given probability range. SRe-

ach can conduct sensitivity analysis: Are the re sults

of reachability analysis the same for different p ossible

values of a certain system parameter? Since SReach

only considers probabilities on state transitions, it is

not clear how to model perturbations on inputs.

(Zhang and Andersen, 2007) analyses homeosta-

sis of anti-stress gene regulatory networks by means

of control theory. They consider stead y-state response

curves of anti-stress genes to a dose of stress. Their

control- theoretical method is tailored to anti-stress

networks and feasible to predict the effect of chan-

ging local r e sponse coefﬁcients, but does not provide

a gene ric fr amework applicable to any network.

In contrast to above approaches, our approach is

purely qualitative. We do not ne e d any kinetic pa-

rameters, which is usually difﬁcult to obtain. There

have been proposed several qualitative approa ches to

model and analyse biological systems based on se-

veral different fo rmalisms - BIOCHAM (Fages et al. ,

2004) based on rewriting system, SMBioNet (Ber-

not et al., 2004 ) and Qualitative networks (Schau b

et al., 2007) based on extended Boolean networks,

and GNA (de Jong et al., 2003) based on piecewise

linear differential equation. Compared to these for-

malisms, our LTL-based model of gene network s are

ﬂexible and easy to construct since the basic principle

to model the behaviours are quite simple. Th e change

of condition or assumption (e.g. bias of multivariate

regulation) is im mediate. Although the above tools

are useful for checking wheth er a biological property

can be true in network behaviours, it is unknown how

to deﬁne and analyse homeostasis. As to the size of

analysable networks, we cannot directly compare th e

scalability since the properties ana lysed and behavi-

ours modelled in these tools are different and the cost

of analysis also depends on the complexity of the net-

work structure as well as the size. Fur thermore, the

examples demonstrated are very few and we do not

have canonical benchmarks in this ﬁeld. For informa-

tion, we refer the size of networks analysed in these

BIOINFORMATICS 2017 - 8th International Conference on Bioinformatics Models, Methods and Algorithms

tools; SMBioNet analy ses a 3 n odes network, GNA

a 10 nodes network and Qualitative networks a 2160

nodes network (in 10 hours). Qualitative network can

handle very large networks compared to others inclu-

ding our framework, but we should keep in mind that

Qualitative networks only consider pro positional pro-

perties on steady states of the network behaviour, i.e .

a temporal property such as ‘when a gene is activated

it will be suppressed later’ cannot be checked. Furt-

hermore, the behaviour of a Qualitative n etwork is de-

terministic. In contrast we consider all possible net-

work behaviours and their temporal properties with

responses to environmental inputs.

As for compositional analysis of gene networks,

we proposed to divide a network into several sub-

networks and verify them individually (Ito et al.,

2013; Ito et al., 2015b). T his means we manually di-

vide an LTL formula in to several sub-formulae, con-

sidering the network structure. We have not discussed

any algorithm to divide a network and the correspon -

ding fo rmula. In this paper, our method is based on

a syntactic structure o f a formula, not on a network

structure. Thu s it is easy to cluster a formula algo-

rithmically.

7 CONCLUSION

In this paper we proposed compositional analysis of

homeostasis of gene networks. We introduced a new

clustering algorithm based on char a cteristic function

on LTL formulae to divide a network sp eciﬁcation.

The experimental results show that our method dras-

tically imp roves the performance in analysing larger

network spec iﬁcations. Our clustering algorithm is

not limited to analyse homeostasis of gene n etworks

but for r ealisability checking of any r eactive system

speciﬁcation.

All examples in our experiments are car ried out

to see the impact of the sizes of networks. Th us the

homeostatic properties are simple and described as re-

latively small formulae compared to network speciﬁ-

cations. We would like to assess the impact of homeo-

static properties to be more conﬁdent in our approach.

For furthe r improvement, we are interested in im-

porting Ito et al.’s approximate method (Ito et al.,

2014b) to check homeostasis of gene networks. Anot-

her important future direction is to analy se real net-

works w hich may now be feasible thanks to this work,

and elucidate regulation mechanisms contributing bi-

ological homeostasis.

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