BioSystems 55 2000 93 – 105
A logic for biological systems
Zhenhua Duan, Mike Holcombe , Alex Bell
Department of Computer Science, The Uni6ersity of Sheffield, Sheffield S
1 4
DP, UK
Abstract
This paper proposes a specification language, hybrid projection temporal logic of modelling, analyzing and verifying biological systems which can be considered, in general, to be hybrid systems consisting of a non-trivial
mixture of discrete and continuous components. The syntax and semantics of the logic are presented, and some examples of hybrid systems are modelled to illustrate the formalism. © 2000 Elsevier Science Ireland Ltd. All rights
reserved.
Keywords
:
Temporal logic; Dynamic systems; Biological systems; X-Machine; Automata; Hybrid systems www.elsevier.comlocatebiosystems
1. Introduction
Biological systems can be viewed as complex, parallel collections of communicating and cooper-
ating processing systems with a complex hierarchy Holcombe, 1991. Attempts to model parts of
these systems at various levels in the hierarchy have demonstrated that the processing that is
going on is essentially a combination of both discrete events, perhaps involving threshold ef-
fects and continuous, evolving, dynamic pro- cesses, for example chemical rate equations. Thus
these systems are examples of hybrid systems.
Hybrid systems are systems that consist of a non-trivial mixture of continuous activities and
discrete events. To model, analyze, verify, and control such a system, transition systems Manna
and Pnueli, 1993, hybrid automata Alur et al., 1993, hybrid graphs Nicollin et al., 1993, state
charts Harel, 1987; Keten and Pnueli, 1992, and integration graphs Keten et al., 1993, as well as
formalisms based on temporal logic, e.g. duration calculus Chaochen et al., 1993 systems based on
a two-phase step assumption. In these models we explicitly describe the system state by a combina-
tion of two types of variables, continuous ones that evolve without discontinuity during the life-
time of the system and discrete variables which model disjoint state changes and external event:
onsets and completions, for example.
The basic systems model we will use is a hybrid version of the X-machine Eilenberg, 1974; Hol-
combe and Ipate, 1998. Here we consider a sys- tem to possess a finite number of internal states.
Associated with the system are two sets of vari- ables, continuous variables which might represent
the concentration of a substance, temperature of some part of the system, real time value etc. and
Corresponding author. Tel.: + 44-114-2221802; fax: + 44- 114-2221810.
E-mail address
:
m.holcombedcs.shef.ac.uk M. Holcombe 0303-264700 - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved.
PII: S 0 3 0 3 - 2 6 4 7 9 9 0 0 0 8 7 - 8
discrete variables which will represent specific ab- stractions such as whether a particular substance
is present or not, whether a given molecule is in a particular mode and whether some attribute is
present or not etc. There are many examples which will be found of these variable types in
every biological system. The modeller must decide the le6el of detail and the scope of the model at
the outset and the states and variables will be determined in the light of this. Models may be of
molecular systems, cellular systems or tissueor- gan systems. Other phenomena which we need to
identify are the sort of environmental influences that the system reacts to, these will be represented
as discrete events. There will also be system ac- tions which describe discrete responses which the
environment of the system experiences as a result of the system’s behaviour.
The next ingredients is a sets of equations of- ten differential equations which are associated
with each internal state. These describe how the continuous variables change while time passes and
the system is in that state. There may be special conditions associated with a state, called lea6ing
conditions which will be conditions on one or more of the continuous variables that determine if
a transition to another state must occur. Finally we have a set of discrete transitions which operate
between states. These transitions are triggered ei- ther by a leaving condition becoming true or
because a specific external event input has oc- curred. In some cases there may be several possi-
ble transitions that can be taken once a leaving condition is true, which one happens, in a deter-
ministic system, will depend on which event oc- curs first. Clearly there has to be some careful
thought given to the relationships between leaving conditions, events and transitions to ensure that
the system description is consistent and complete in the sense that we don’t find a situation where a
leaving condition is satisfied and no valid transi- tion exists The system then moves instanta-
neously to another state where there is a new set of equations describing how the continuous vari-
ables will behave, naturally the continuous vari- ables behave in a continuous way after such a
state change. A state transition may manipulate the discrete variables and there may be some sort
of environmental action or output that occurs at the transition. To deal with the problem that
some state transitions may take a non-trivial amount of time to operate we introduce an inter-
mediate state and have instantaneous start and end transitions. The model operates in conjunc-
tion with a continuous clock which provides a reference point for all activity. This is a hybrid
X-machine, see Fig. 1.
Using it we can greatly simplify many mathe- matical models which try to capture the behaviour
of variables but do not take advantage of the inherent state structure of systems, see Fig. 2 Bell
and Holcombe, 1998.
The purpose of the logic is to provide a rigor- ous language for describing these machines and
the properties that they may posses. We can use a formal symbolic logic like this to either derive
Fig. 1. A simple hybrid X-machine diagram.
Fig. 2. Complex behaviour modelled using a hybrid X-ma- chine.
e
i
, a
i
, e
i
1
, a
i
1
,… 1
This model is similar to the interleaving model for concurrent computations.
Since discrete events take zero time, a discrete event e
i
j
takes place actually at the exact point in time when continuous activity a
i
j
starts to evolve. Thus, it is better to describe the sequence Eq. 1
of the computation run by the following sequence: {e
i
, a
i
}, {e
i
1
, a
i
1
},… 2
where {e
i
j
, a
i
j
} represents a step of the computa- tion which is the same as in sequence 1 but e
i
j
and a
i
j
take place at the same time. However, a
i
j
takes a time duration while e
i
j
occurs only at the beginning of the time duration for a
i
j
. This com- putational model is somewhat similar to the true
concurrency model for concurrent computations. Furthermore, the projection construct Duan et
al., 1994b, p
1
,…, p
m
prjq, can be thought of as a special parallel operator. Intuitively, it means that
q is executed in parallel with p
1
,…, p
m
over an interval obtained by taking the endpoints ren-
dezvous points of the intervals over which p
1
,…, p
m
are executed. The projection construct permits the processes p
1
,…, p
m
, q to be autonomous, each process having the right to specify the interval
over which it is executed. In particular, the se- quence of processes p
1
,…, p
m
and process q may terminate at different time points. Although the
communication between processes is still based on shared variables, the communication and synchro-
nization take place only at the rendezvous points global states. Otherwise they are executed inde-
pendently. See Fig. 3.
The two time scales of the projection construct may allow us to model hybrid systems. Naturally,
if processes p
1
,…, p
m
are treated as continuous activities over local time subintervals while pro-
cess q is thought of as a discrete process which proceeds in a synchronous way with processes
p
1
,…, p
m
but communicates with them only at the time instants when processes p
1
,…, p
m
start, then the continuous activities and discrete events can
be combined in a uniform way. A computation run of a projection construct modelling a hybrid
system is exactly as indicated in sequence 2. Therefore, we believe that the projection construct
Fig. 3. Projection construct.
consequential properties deduced from the ma- chine definition or to implement a logic program-
ming language which could be used as a simulator deriving scenarios of behaviour that the system
can undertake.
A computation run of a hybrid system is as- sumed to be a sequence of two-phase steps. The
first phase of a step corresponds to a continuous state evolution usually described by differential
equations, which may be describing, for example the evolution of specific enzyme driven process-
ing; in the second phase, the state is submitted to a discrete change taking zero time, this results in
the system being switched into a situation where a, possibly, different collection of differential
equations is modelling the way the continuous variables behave. More precisely, a computation
run of a hybrid system is an alternate sequence between continuous activities a
i
j
and discrete events e
i
j
for 0 5 j N:
is more suitable for modelling hybrid systems. It is not straightforward, however, to generalize the
projection construct from a time-free notation to a timed notation for hybrid systems. The general-
ization requires a radical shift in the semantics of the underlying logic.
This paper proposes a Logic, HPTL, for mod- elling, analyzing, verifying and understanding hy-
brid biological systems. The syntax and semantics are presented. It is intended to provide a fully
general formal logic to allow for the precise rea- soning about arbitrary biological systems. Since
these systems are going to be incredibly complex any formal reasoning has to be automated and a
formal logic is the prequisite for achieving this. We are not considering the process as being only
one of simulation. What we want to achieve is a mechanism whereby mathematical facts, which
will be valid in precisely described conditions, can be deduced from the description of the model.
Simulation can only provide a finite number of scenarios describing system behaviour and cannot
generate statements that are universally true or true in infinitely many situations, however simula-
tion is a valuable way of getting greater under- standing
of the
model and
should be
accommodated as well. This paper is organized as follows: Section 2
explores hybrid systems; Section 3 is devoted to formalizing the hybrid projection temporal logic;
in Section 4, examples of two simple biological systems are modelled with HPTL to illustrate the
formalisms; finally, conclusions are drawn in Sec- tion 5.
2. Hybrid systems
