Temporal Logic based Framework to Model and Analyse Gene Networks

with Alternative Splicing

Sohei Ito

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

Shimonoseki, Yamaguchi, Japan

Keywords:

Gene Regulatory Network, Systems Biology, Alternative Splicing, Temporal Logic.

Abstract:

Toward system-level understanding of biological systems, we need a formalism to model and analyse them.

Due to incompleteness of knowledge about quantitative parameters and molecular mechanisms, qualitative

methods have been useful alternatives. We have been working on temporal logic-based approach for qual-

itative modelling and analysis of gene regulatory networks. Although our framework is well-established to

model several aspects of gene regulation, we still lack treatment of alternative splicing, which contributes to

proteomic diversity of eukaryotic organisms. In this paper we extend our logic-based qualitative framework

to be able to capture alternative splicing, which is crucial to model the gene regulatory networks in eukaryotic

organisms. We study mechanisms of alternative splicing and propose how we model each mechanism, then

demonstrate the modelling method by analysing the regulatory network of sex determination in Drosophila

and verify that the network ensures sex determination.

1 INTRODUCTION

To understand complex activities of the cell, mathe-

matical and computational approach is indispensable.

For precise mathematical modelling, we need huge

amount of quantitative information. Such quantita-

tive information available, however, is unfortunately

limited and not sufﬁcient despite of recent advances

in biology. Instead, a lot of qualitative information

about biological systems has been accumulated such

as schematic network representations of gene-gene

interactions, protein-protein interactions, signalling

pathways, and so on. Thus a qualitative method for

modelling and analysing biological processes based

on qualitative information is desired.

In this context, several computational formalisms

in biological modelling have been proposed: Boolean

network (Thomas, 1991), Petri net (Heiner et al.,

2008), timed automata (Batt et al., 2007) and process

algebra (Ciocchetta and Hillston, 2009), though all

of them are not necessarily qualitative. In these for-

malisms, the possible behaviours of a system can be

characterised by the traces of the model. Such com-

putational formalisms need concrete information on

molecular mechanisms and regulatory logics to con-

struct a model. Since biological information is in-

herently incomplete, it is pointed out that constraint-

based modelling is well-suited in biological mod-

elling (Palsson, 2000) in which we give several con-

straints reﬂecting incomplete knowledge on the sys-

tem to limit possible behaviours (solution space) of

biological systems.

In accordance with the motivation of constraint-

based modelling, we have been working on a logic-

based qualitative approach to model and analyse be-

haviours of gene regulatory networks (Ito et al., 2010;

Ito et al., 2013b; Ito et al., 2013a; Ito et al., 2014; Ito

et al., 2015) which uses linear temporal logic (LTL)

as the modelling language. In contrast to the orig-

inal constraint-based modelling paradigm which in-

tends to limit the quantitativepossible behaviours, our

approach aims to characterise qualitative possible be-

haviours using qualitative information of gene-gene

interactions which is represented as gene regulatory

networks. Since we only use qualitative information,

the reasoning is also limited to qualitative properties.

However, we can still analyse important properties of

gene networks such as oscillation, stability and reach-

ability, as it is pointed out that the overall behaviour

is relatively insensitive to the exact numerical values

of the kinetic constants (Palsson, 2000).

One of the difﬁculties in modelling and analysing

gene networks is the alternative splicing in eukary-

otic organisms. Alternative splicing of a precursor

mRNA (pre-mRNA) gives rise to multiple transcrip-

Ito, S.

Temporal Logic based Framework to Model and Analyse Gene Networks with Alternative Splicing.

DOI: 10.5220/0005655001510158

In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2016) - Volume 3: BIOINFORMATICS, pages 151-158

ISBN: 978-989-758-170-0

151

tion products from one gene. The selection of alter-

native splicing at an appropriate timing is critical for

cell differentiationand sex determination. In quantita-

tive approach, the splicing process can be modelled as

thermodynamical reactions (Louis et al., 2003; Wen,

2013). However, it is unclear how alternative splicing

is modelled in qualitative approach. The aim of this

paper is to establish a method for modelling alterna-

tive splicing in our LTL-based framework. In this pa-

per we study mechanisms of alternative splicing and

show how each mechanism can be modelled in LTL.

We demonstrate our formal framework by modelling

and analysing the network of sex determination in

Drosophila (Camara et al., 2008; Salz and Erickson,

2010).

The rest of this paper is organised as follows. In

section 2 we review our LTL-based framework for

modelling and analysing gene networks using LTL as

a baseline of this work. In section 3 we study mecha-

nisms of alternative splicing and present how we for-

mally model them in our framework. In section 4 we

demonstrate our formal framework in analysing the

network of sex determination in Drosophila. The ﬁ-

nal section offers conclusion and future directions.

2 QUALITATIVE FRAMEWORK

FOR MODELLING AND

ANALYSING GENE

NETWORKS USING LTL

A gene network is represented as a directed graph

whose nodes and edges (labelled by +/-) represent

genes and regulation relation (activation/inhibition)

among them, respectively. In Fig. 1 we show an ex-

ample of a gene regulatory network which consists of

three genes. A behaviour of a gene network is rep-

resented as a time series of expression proﬁles of the

genes in the network. Fig. 2 shows an example time

series of the network depicted in Fig. 1. In this be-

haviour the value x

is the threshold level of gene x

to activate gene y, y

the threshold level of gene y to

activate gene z and z

the threshold level of gene z

to inhibit gene y. At the beginning, no genes are ex-

pressed. At time t

, gene x begins to be expressed

and its expression level begins to grow. At time t

gene x crosses the threshold x

, thus gene y becomes

ON due to the positive effect from gene x. At time

gene x stops to be expressed and the level decreas-

ing. At time t

gene x falls below x

, thus gene y

becomes OFF. This way the network changes its state

over time.

This time series can be represented as a discrete

y z

Figure 1: An example gene regulatory network. Gene x

activates gene y, y activates z and z inhibits y. A plus edge

represents activation and a minus edge represent inhibition.

state transition system (called linear time structure)

depicted in Fig. 3. It consists of states (represented as

circles) and transitions (represented as arrows). The

conﬁgurations of the network at each state are shown

by the propositions depicted below states. We have

the following propositions to describe the conﬁgura-

tions of the network:

• on

, on

: whether genes x, y and z are ON or

OFF, respectively.

• x

, y

, z

: whether the expression level of gene x,

y and z are beyond the threshold x

, y

and z

respectively

We easily see that state 0 represents the conﬁguration

of the network at the beginning, state 1 represents the

conﬁguration betweent

and t

, state 2 between t

and

, and so on.

In general, there are many behaviours which can

be produced by a single network depending on the ini-

tial conditions, input scenarios, response times, and so

on. Our purpose is to model (or characterise) the set

of possible behaviours for a given gene network. In

quantitative approach, ordinary differential equations

(ODEs) are widely used. In the current setting, we

do not handle a numerical time series but a symbolic

time series of a behaviour (a linear time structure).

To characterise and reason about such structures, lin-

ear temporal logic (LTL) is the suitable mathemati-

cal language. LTL can be seen as propositional logic

equipped with temporal operators such as G (Glob-

ally), F (Future), U (Until) and W (Weak until). Gφ

means φ is always true, Fφ means φ is eventually true,

φUψ means φ is true until ψ is true and φWψ means

φ is true until ψ is true or φ is indeﬁnitely true (in

this case ψ need not be true in any future). The for-

mal syntax and semantics are omitted due to the space

limitation.

We are to characterise the set of possible be-

haviours (linear time structures) for a given network.

This can be done by specifying an LTL formula φ

for a given network N such that the set of possible

behaviours of a network is characterised as {σ | σ |=

}, i.e. all linear time structures which satisfy the

behaviour speciﬁcation φ

. The problem of analysing

Threshold values are written in Roman while proposi-

tions are written in italics.

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152

base



୷

୸

୷



Figure 2: Time series of expression levels of gene a, b and c in the network Fig. 1.

݋݊

1 2 3 4 5 6 7 8 9 10

11 12

݋݊

Figure 3: Symbolic representation of the time series of Fig. 2.

network behaviours, e.g. checking whether there is a

behaviour which satisﬁes a certain property ψ (also

written in LTL), can be solved by ﬁnding σ such that

σ |= φ

∧ ψ, i.e. checking satisﬁability of the for-

mula φ

∧ ψ. Thus analysing a gene network is re-

duced to satisﬁability checking of LTL. Once we have

a formula φ

∧ ψ, the analysis can be automatically

done by LTL satisﬁability checkers. LTL satisﬁability

checkers construct a B¨uchi automaton of a given LTL

formula which precisely accepts the linear time struc-

tures in which the formula is true (Vardi and Wolper,

1994). Hence if the language accepted by the au-

tomaton is empty, the formula is not satisﬁable. The

non-emptiness problem of B¨uchi automata is solved

by checking the existence of a maximal strongly con-

nected component containing accepting states of the

automata.

Due to the space limitation, we do not show in

detail how to specify φ

which characterises possible

behaviours of a given network N. Interested readers

would like to consult our previous work (Ito et al.,

2015). The key idea is the following qualitative prin-

ciples of gene network behaviours:

• A gene is ON when its activators are expressed

beyond some thresholds.

• A gene is OFF when its inhibitors are expressed

beyond some thresholds.

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

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

By expressing these principles in LTL, we have a

characterisation of possible behaviours of the net-

work. We only show an example characterisation

of the possible behaviours of the network in Fig. 1.

For this network we introduce the set of propositions

{on

, on

, x

, y

, z

}. Using these propositions,

we have the following behaviour speciﬁcation.

G(x

∧ ¬z

→ on

) ∧ G(z

→ ¬on

) ∧ G(y

↔ on

)∧

G(on

→ F(x

∨ ¬on

))∧

G(on

∧ x

→ (x

W¬on

))∧

G(¬on

→ F(¬x

∨ on

))∧

G(¬on

∧ ¬x

→ (¬x

Won

)) ∧ . . . .

In this speciﬁcation we assumed that gene y is OFF if

gene z is inhibiting y nevertheless gene x is activating

y. For this network let us check the bistability of the

expression of gene y, which is written in LTL as:

(Gon

→ FGon

) ∧ (G¬on

→ FG¬on

)

We check the satisﬁability of the conjunction of

the above formulae. We used T

-builder (Aoshima,

2003) to check it and had the answer ‘Yes’. This

means that the network of Fig. 1 produces two op-

posite behaviours: a behaviour in which gene y is al-

ways ON after some time point and another behaviour

in which gene y is always OFF after some time point,

which is determined by whether gene x is ON.

Temporal Logic based Framework to Model and Analyse Gene Networks with Alternative Splicing

153

3 MODELLING ALTERNATIVE

SPLICING BY LTL

In the formal framework described in the previous

section, we did not take alternative splicing into con-

sideration. This section discusses how we model the

alternative splicing in our framework.

In most eukaryotic organisms, the process of gene

expression consists of three steps: (i) a DNA region

which encodes a gene is transcribed into a precursor

messenger RNA (pre-mRNA), (ii) introns (and some

exons) in a pre-mRNA are removed, and (iii) a pro-

cessed mRNA is transported outside of a nucleus and

translated into a protein. Alternative splicing happens

in the step (ii) which causes the diversity of processed

mRNA from a single pre-mRNA by removing some

exons selectively as well as introns (Fig. 4). Due to

alternative splicing several isoforms of a protein are

obtained from one gene.

A natural solution to handle alternative splicing is

that we regard each isoform as being produced by dif-

ferent (virtual) genes. However, this treatment causes

blow-up of the number of propositions and the size of

a behaviour speciﬁcation, which deteriorates the per-

formance of analysis.

In this section we study the molecular mech-

anisms of alternative splicing (David and Manley,

2008; Hertel, 2008; Kornblihtt, 2005; Matlin et al.,

2005) and propose how to model these mechanisms

in LTL without introducing extra genes.

Mechanism 1. One of the mechanisms of alterna-

tive splicing is the usage of multiple promoters. The

mouse α-amylase gene is known to have this mech-

anism. For the purpose of illustration, let us con-

sider a gene u which has two promoters X and Y (Fig.

5). Gene u has two splicing patterns depending on

promoters. The choice of promoters is made by a

transcription complex. To model this mechanism in

LTL, we introduce propositions TC

and TC

to rep-

resent whether the levels of transcription complexes

for promoter X and Y are sufﬁcient, respectively. We

write R

(u) and R

−

(u) for LTL terms representing

conditions for activation(+) and inhibition(-) of gene

u, respectively

. For example if gene v activates u

and gene w inhibits u, R

(u) will be v

∧ ¬w

and

−

(u) will be ¬v

∧ w

. Then conditions for activat-

ing/inhibiting gene u can be described as:

G(R

(u) ∧ TC

∧ ¬TC

→ on

∧ ¬on

), (1)

G(R

(u) ∧ TC

∧ ¬TC

→ on

∧ ¬on

), (2)

G(R

−

(u) → ¬on

∧ ¬on

), (3)

In general we have several conditions for activat-

ing/inhibiting gene u, but treatment is the same.

where the propositions on

and on

represent gene

u is expressed from promoter X and Y, respectively.

Formula (1) says that if gene u is activated and the

transcription complex for promoter X is sufﬁcient and

that for promoter Y is not sufﬁcient, gene u is ex-

pressed from promoter X, not from promoter Y. For-

mula (2) describes the case that gene u is expressed

from promoter Y. Formula (3) says that if gene u is

not activated, gene u is expressed neither from pro-

moter X nor from promoter Y. Note that we can use

↔ instead of → in the above formulae depending on

our assumptions for a system to be modelled. (The

same argument applies to the other mechanisms.)

The problem is that how we describe the case

when both TC

and TC

are true, i.e. R

(u) ∧ TC

∧

→?. The situation is almost the same as the case

when both activators and inhibitors are active for a

gene, which is discussed in our previous work. The

solution depends on the knowledge or assumption we

have on a given network or a problem. We may write

∧ on

which means both on

and on

are true,

or write on

∨ on

which means on

or on

are true

(both are true is allowed). If we do not add any clause,

the values of on

and on

are free in this case. An-

other choice is to assume that both TC

and TC

cannot be true simultaneously by adding the clause

G¬(TC

∧ TC

For each gene that is regulated by the translated

products of gene u from promoter X/Y, we intro-

duce propositions of the expression levels for u

and

such as u

, u

, . . . , u

, u

, . . . , and clauses for the

changes of the expression levels of them. These

clauses are the same as those we have for normal

genes.

Readers might wonder what is the difference from

the modelling manner in that we split gene u into the

(virtual) genes u

and u

. If we do so, we must dupli-

cate regulating terms R

(u) and R

−

(u) into R

), R

−

) and R

−

). For example, if we as-

sume that gene u has two regulators v and w, we need

to introduce the threshold levels of v and w for both

gene u

and u

: v

, v

, w

and w

. This blows up

the number of clauses for the changes of the expres-

sion levels of gene u and w. In the above speciﬁca-

tion, however, such blow-ups of the number of propo-

sitions and clauses are avoided.

Mechanism 2. The next mechanism of alternative

splicing is the presence/absence of splicing factors

(SFs). SFs bind to introns or exons and change the

splice sites of an transcribed pre-mRNA of a gene.

SFs can activate or inhibit a certain splice sites, but

the important fact is that splicing is determined by

whether SFs are binding or not. We assume that a

gene u produces two isoforms u

(when the SF is

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154

Exon 1 Exon 2 Exon 3 Exon 4

Intron 1 Intron 2 Intron 3

Exon 1 Exon 2 Exon 4 Exon 1 Exon 2 Exon 3

DNA

Figure 4: Alternative splicing produces different mRNAs from a single pre-mRNA.

Exon 1 Exon 2 Exon 3

Exon 1 Exon 3 Exon 2 Exon 3

Gene u

From promoter X From promoter Y

Figure 5: Gene u has two promoters X and Y.

binding) and u

(when the SF is not binding). We in-

troduce a proposition SF

which represents the level

of the SF exceeds the threshold SF

upon which the

SF affects on splicing. Then conditions for activat-

ing/inhibiting gene u can be described as:

G(R

(u) ∧ SF

→ on

∧ ¬on

), (4)

G(R

(u) ∧ ¬SF

→ on

∧ ¬on

), (5)

G(R

−

(u) → ¬on

∧ ¬on

). (6)

Formula (4) says that if gene u is activated and the

level of the SF is beyond the threshold SF

, gene u

produces the isoform u

. Formula (5) describes the

case when the level of the SF is not enough to bind

pre-mRNAs of gene u. In this case gene u produces

the isoform u

. Formula (6) says that if gene u is not

activated, u does not produce any isoform.

In general, multiple SFs (SF1, SF2, ...) involve

the splicing of a gene u which results in many iso-

forms u

, u

, . . . . By generalising the above for-

mulae we can easily model such complex splicing.

For each combination of effectiveSFs, we specify that

the corresponding isoform is expressed (ON).

4 DEMONSTRATION

In this section we apply our method for modelling al-

ternative splicing described in section 3 to analyse the

network of sex determination in Drosophila (Camara

et al., 2008; Salz and Erickson, 2010).

Genes involved in this sex determination process

are Sxl, tra, tra-2 and dsx. Sxl, tra and dsx have both

male-speciﬁc and female-speciﬁc splicing. More-

over, Sxl has two promoters – the early promoter

and the late promoter. Sxl is known to have two

female-speciﬁc splicing – one from the early pro-

moter and the other from the late promoter. Male-

speciﬁc splicing of Sxl occurs only from the late pro-

moter. Thus we have three isoforms from Sxl. We rep-

resent S

(from e

arly promoter), S

(female-speciﬁc

splicing from the late promoter) and S

ale-speciﬁc

splicing) for each isoform. We similarly write t

(female-speciﬁc) and t

(male-speciﬁc) for tra, and

(female-speciﬁc) and d

(male-speciﬁc) for dsx.

The network controlling sex determination in

Drosophila is illustrated in Fig. 6. First the isoform

is produced from Sxl by the early promoter. S

ac-

tivates female-speciﬁc splicing of Sxl itself and pro-

duces the isoform S

, which inhibits male-speciﬁc

splicing of Sxl and tra. As a result tra produces

female-speciﬁc isoform t

. This t

with tra-2 acti-

vates female-speciﬁc splicing of dsx.

To model this network in LTL we introduce the

following propositions.

• on

, on

: representing

whether the isoforms S

, S

, t

, d

and d

are expressed, respectively.

• S

, S

, t

, t2: these propositions correspond to

whether each isoform is expressed beyond the

threshold level for each activation/inhibition be-

tween genes. S

corresponds to S

−→ S

, S

−

−→ S

, S

to S

−

−→ t

, t

to t

−→ d

and t2

to tra-2

−→ d

(see Fig. 6). We consider S

and

as splicing factors for tra and dsx, respectively.

• TC

, TC

: representing whether the levels of tran-

scription complexes of Sxl for the early(E) and

late(L) promoters are sufﬁcient, respectively.

Here we show the essential part, i.e. how the splic-

ing is controlled, of behaviourspeciﬁcation of the net-

work.

Temporal Logic based Framework to Model and Analyse Gene Networks with Alternative Splicing

155

Sxl

tra

dsx

tra

Figure 6: The network controlling sex determination in Drosophila.

G(TC

∧ ¬TC

↔ on

∧ ¬on

)∧ (7)

G(S

∧ TC

∧ ¬TC

↔ on

∧ ¬on

)∧ (8)

G(¬S

∧ TC

∧ ¬TC

↔ on

∧ ¬on

)∧ (9)

G(TC

↔ (on

∨ on

))∧ (10)

G(S

→ ¬on

)∧ (11)

G(S

→ (¬on

∧ on

))∧ (12)

G(¬S

→ (on

∧ ¬on

))∧ (13)

G(t2∧t

→ on

∧ ¬on

)∧ (14)

G(t2∧ ¬t

→ ¬on

∧ on

)∧ (15)

G(¬t2 → ¬on

∧ ¬on

) ∧ . . . (16)

Formulae (7)-(9) are directly derived from the

mechanism 1 in the previous section. Since we do

not have an explicit regulator for Sxl in the network,

the regulating condition (R

(·) in section 3) is empty

in formula (7). Formula (10) reﬂects the assumption

that isoforms S

and S

need to be expressed from the

late promoter. Formula (11) stipulates the negative ef-

fect of S

to the expression of S

. Formulae (12)-(16)

are derived from the mechanism 2 where S

and t

splicing factors.

For this networkwe check the bistability – female-

speciﬁc stability and male-speciﬁc stability. The crit-

ical switch to determine this is the female-speciﬁc

transcription complex for early promoter of Sxl. If its

intracellular level is sufﬁcient at the initial time, the

cell eventually reaches female-speciﬁc stability, oth-

erwise the cell eventually reaches male-speciﬁc sta-

bility. This property is described in LTL as:

(TC

→ FG(on

∧ on

))∧

(¬TC

→ FG(on

∧ on

))

in which female(male)-speciﬁc stability is written as

that female(male)-speciﬁc splicing of the three genes

Sxl, tra and dsx are maintained. Using the LTL satisﬁ-

ability checker we have the result ’Yes’, which means

that this network surely satisﬁes the bistability.

We analysed the B¨uchi automaton constructed by

the LTL formula by the LTL satisﬁability checker

and investigate the witnesses of the satisﬁability

We used GOAL (Tsay et al., 2007) to analyse B¨uchi

Since there are many possible behaviours from all

possible initial state, we extracted a behaviour from

the initial state where gene Sxl is OFF (other genes

are arbitrary) and transcription complex of Sxl for

the early promoter is present/absent. We depict the

behaviours (linear structures) obtained from the au-

tomaton as witnesses in Fig. 7. The states are

represented as vectors of propositional values for

(TC

, TC

, on

) in Fig. 7,

where 1 represents true and 0 represents false. For

simplicity the other propositions are omitted, thus the

edge in the ﬁgure does not necessarily corresponds

the one atomic step of the original behaviour because

the states in which only the values of the omitted

propositions differ are identiﬁed.

In both Fig. 7(a) and (b), after some initial pertur-

bation, the network reaches the stable state in which

all the sex-speciﬁc splicing is maintained. Note that

this is just instances of the possible behaviours hap-

pened to be produced by the automaton as witnesses.

The behaviours include some interesting features: the

transcription for the late promoter of Sxl is some-

times cut off in the female-speciﬁc behaviour (a) and

the transcription factor for the early promoter of Sxl,

which is known to be a female-speciﬁc transcription

complex, is once produced in the male-speciﬁc be-

haviour (b), nevertheless, the network reaches the

ﬁnal desirable states (only sex-speciﬁc splicing are

maintained). This can be interpreted that the network

has homeostasis against the perturbation on the tran-

scription complexes for Sxl.

Compared to Fig. 7 (b), the interpretation of

(a) might be a bit difﬁcult: especially the step from

the state 00001000 to the last state 01010101 looks

somewhat mysterious. The key to understand this

behaviour is the previous two states (1111

1000 and

01011000) where gene Sxl is expressed in the female-

speciﬁc isoform (underlined). Therefore the isoform

is sufﬁciently stored while the supply of the transcrip-

tion complex for the late promoter is instantaneously

suspended (at the state 00001000). Thus the female-

speciﬁc splicing of Sxl is re-started once the supply

of the transcription complex for the late promoter is

resumed and maintained (at the last state 01010101).

automata.

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156

10000101

01010100

00000101

01101011

(a) (b)

11111000

01101010

11111010

01011000

00001000

00000100

01010101

01101010

Figure 7: Example behaviours of the sex determina-

tion network in Drosophila. For simplicity the vec-

tor only consists of the values of the propositions

(TC

, TC

, on

). (a) A female-

speciﬁc behaviour. Underlined are female-speciﬁc iso-

forms. (b) A male-speciﬁc behaviour. Underlined are male-

speciﬁc isoforms.

Next we check another property: this network is

not able to keep female- and male-speciﬁc splicing

simultaneously. This property is formally written as:

FG(on

∧ on

)

The result is ’No’, as we expected. There is no

such behaviour in the possible behaviours of the net-

work controlling sex determination. Both veriﬁca-

tions show that this network ensures the sex determi-

nation.

The sex determination of Drosophila is also mod-

elled and analysed in (Louis et al., 2003). In that

work, the authors model the behaviour of the expres-

sion of gene Sxl and analyse how the bistable female-

speciﬁc and male-speciﬁc differences arise. They use

ODE models for the transcription of Sxl from theearly

promoter and probabilistic models for that of the late

promoter. The two models are combined to employ

the overall analysis. Their model is elaborated and re-

quires deep knowledge in molecular mechanisms. In

addition, the mathematical inference of unavailable

kinetic parameters is required. For simplicity they

abstracted the downstream genes of Sxl such as tra

and dsx. We guess the reason of this simpliﬁcation

in the modelling and analysis as that the cascading

the quantitative model makes the entire model very

sensitive to the changes of parameter values in each

stage and to ‘correct’ inference of the uncertain pa-

rameters is more crucial to reproduce the sex determi-

nation. In contrast, our qualitative framework allows

us to model the network concisely with the same con-

clusion (bistability of the network). The downstream

genes of Sxl are also included in the analysis without

difﬁculty. We must note, however, that their quanti-

tative model enables the robustness analysis such that

how the sex determination works against the modula-

tion of gene doses. Due to qualitativeness of our for-

malism, such analysis is infeasible in our framework.

We, however, showed that both female-speciﬁc and

male-speciﬁc gene expressions cannot be maintained,

i.e. the possible behaviours of the network does not

contain such behaviours. To prove this with quanti-

tative model is very difﬁcult since we need to test all

combinations of possible quantitative parameters and

conﬁrm that such behaviours are never produced.

(Fear et al., 2015) modelled the sex determina-

tion regulatory network of Drosophila using struc-

tural equation models. The purpose of their work is

to infer statistically likely links between genes in the

known network or to ﬁnd new genes which can likely

be included in the network, rather than to investigate

how the sex determination is ensured by the network.

Although the aim of modelling the network is differ-

ent from ours, the prediction of plausible extension of

the known network is interesting aspect of systems bi-

ology. This line of research will be future work in our

formalism. Whereas their method for ﬁnding plausi-

ble extension is brute force: they enumerate all possi-

ble interactions and insertion of new genes in all pos-

sible locations in the graph, in our framework some

logical inference may be helpful to ﬁnd plausible ex-

tensions of the network instead of using brute force.

5 CONCLUSION

In this paper we presented a qualitative framework to

model and analyse gene networks using linear tem-

poral logic. We studied molecular mechanisms of al-

ternative splicing and showed how such mechanisms

are modelled in our framework. As a demonstra-

tion, we modelled the network of sex determination

in Drosophila and checked the sex-speciﬁc bistability

of the network.

Since this work is still at a theoretical stage, we

are investigating applications of our frameworkto real

biological problems. For this we are to develop (semi-

)automatic modelling method from gene regulation

information. Compared to quantitative approaches

which need manual parameter inference or tuning, a

(semi-)automatic model construction of a gene regu-

latory network ismore feasible in ourframework. The

start point will be to devise a machine readable uni-

form presentation of splicing networks. One promis-

ing approach is to extend SBML Qualitative Models

Package (Chaouiya et al., 2013).

Temporal Logic based Framework to Model and Analyse Gene Networks with Alternative Splicing

157

ACKNOWLEDGEMENT

This work was supported by JSPS KAKENHI Grant

Number 26730153.

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