SPICE, un logiciel de simulation pour physiologistes

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Spice : a circuit circulation program for physiologists

P. Cruiziat, R. Thomas

To cite this version:

P. Cruiziat, R. Thomas. Spice : a circuit circulation program for physiologists. Agronomie, EDP

Sciences, 1988, 8 (7), pp.613-623. �hal-02720463�

AGRONOMIE

SPICE -

a

circuit simulation

_program

for

physio-logists

P. CRUIZIAT _Randy THOMAS

LN.R.A., Laboratoire de Bioclimatologie, Domaine de Crouelle, F 63039 Clermont-Ferrand Cedex

( *

) INSERM, U 192, H6pital Necker, TT 6" _étage, 149, rue de Sèvres, F 75015 Paris Cedex 15.

SUMMARY This article _presents a _{program named SPICE for simulation in plant physiology.} It was _{originally developed} to

simulate electric and electronic circuits. For several years it has now been also used in various fields of human and animal _{physiology (classical physiology,} membrane _{biophysics, pharmacokinetics...). ).}

After a short description of SPICE’s main features and capabilities, 3 simple illustrative examples are _{given :} a

whole plant water flow model, a _coupling fluxes of water and solute across a membrane and a process obeying

the Michaelis-Menten law.

The advantage of this approach is that the modeler need know nothing of numerical analysis and can nonetheless

treat _complex, non _{linear-systems. Cheap} versions for IBM-compatible personal computers are available.

Additional

key

words: Network thermodynamics, compartmental systems, coupling

oj’,fluxes,

chemical reactions.

RÉSUMÉ SPICE, un

logiciel

de simulation pour

physiologistes,

Cet article présente un _{logiciel, appelé} SPICE, pour simuler un certain nombre de _problèmes de _physiologie

végétale. Bien _qu’ayant été _développé à _{l’origine pour simuler} et mettre au _point des circuits _électriques et

électroniques, il est _{depuis quelques} années utilisé en physiologie humaine et animale.

Après avoir présenté les principales caractéristiques et _{possibilités} de ce _logiciel, 3 exemples simples illustrant son

utilisation sont donnés : un modèle de transport d’eau à travers le _végétal, un _couplage eau-solutés à travers une

membrane et une _{simulation d’un processus suivant la loi Michaelis-Menten}

Des versions peu chères pour microordinateurs existent.

Mots clés additionnels : Thermodynamique en réseaux, modèles à compartiments, couplage

do flux,

réaction

enzymatique.

I. INTRODUCTION

To

_{varying degrees, experimental}

science

_requires

the construction of models and the use of a simulation

approach.

In fields such as _agronomy,

_{bioclimatology,}

or

_physiology,

_{many processes} can be viewed as an

exchange

of matter

_and/or

_energy between different

compartments

of a

_complex

_system.

_Among

several tools available to solve such

_problems,

a circuit simula-tion

_package

named SPICE has been

_increasingly

used. Its

_original

purpose was to

_help

electrical

_engineers.

Later it was

recognized

as a

practical

simulator for network

_{thermodynamic}

models and has been used

especially

in animal and human

_physiology.

SPICE has three main features :

1)

it is a _very suitable tool for

_dealing

with

_coupled

fluxes of different kinds as

_they

are formulated in the

language

of

_{thermodynamics}

of irreversible _{processes ;}

2)

it

_emphasizes

the connections between structures

and

_functions,

and allows one " to

put

together

into a

functioning

whole a lot of

_pieces

we _{may have} observed

as

_parts

of a

_complicated

_system

"

_(M

_IKULECKY

_,

_{1983) ;}

3)

it solves the

_system

of

_{equations corresponding}

to

the behavior of the

_system.

For these reasons we believe that SPICE can be of

great

interest to _{many researchers. The purpose of this}

and to

_provide

a few

_examples

illustrating

its use in the

field of

_physiology.

A. What is SPICE ?

SPICE is a _{program for the simulation of electric and} electronic circuits aimed at

_assisting

in the

_development

of such circuits. It was

developed by

the

Laboratory

of Electrical and Electronical

_Engineering

at the

Univer-sity

of California at

_{Berkeley (USA).}

New

_versions,

more flexible and more

_powerful,

have been

_developed

regularly

since 1972. Several similar programs are avai-lable

_today,

SPICE

_being

the most

_widely

distributed. It can be used on both mainframe and

_personal

computers

(

*

).

B. Use of SPICE for simulation of

_{physiological}

problems

At first

_sight,

it may seem

_strange

to

_apply

a _program

for electric circuits to simulations in

_physiology.

Howe-ver, a brief review of some historical and

methodologi-cal facts will demonstrate the

_validity

of this idea.

1. Use

_of

electrical

_analogies

Such

_analogies

have been used for a

_long

time in

animal and

_{plant physiology (DAINTY,}

_{1960 ;}

SANDERS

et

al.,

1971).

The same terms

_(e.g.

_resistance)

are even

applied

to different but

_{analogous phenomena}

expres-sed

_{by formally}

similar laws.

2.

_{Thermodynamics of}

irreversible _processes

_(TIP)

In the

_1950’s,

KEDEM& KATCHAt,sKV

_{(1958, 1963a,}

b,

c)

were

_{undoubtedly pioneers}

in the use of TIP for the

phenomenological description

of flow-force

relations-hips

in

_biological

membranes. This theoretical

_approach

has

_spread

widely

and has been

applied

in _many fields of

physics, chemistry

and

_biology.

3. Network

_{thermodynamics}

_(NT)

In the same _way that TIP can be considered a

generalization

of classical

_{thermodynamics,}

NT can be

*

Practical information

2G6, the latest SPICE program written in Fortran 5 (or 77) was

developed at the University of California at Berkeley in 1983. Several versions are available _depending on the _type of _computer and

operating system (CDC, DEC, VAX (UNIX or _{VMS), IBM, etc...)}

A new _{program in C language,} named SPICE 3A. 7 was _put on the market in March 1986. In _spite of its _efhciency, this program is of limited interest to _{physiologists} for it does not allow the use of

dependent polynomial sources, thus reducing its usefulness for physio-logical simulations.

These versions are available on _{magnetic tape} for about $ 100. SPICE programs are also available for _{IBM-compatible personal}

computers. For _example INTUSOFT _(P.O. Box 6607, San Pedro CA

90734, USA) sells versions of SPICE derived from 2G6 which are more efficient in both ouput and simulation types than the most recent

versions for _{computers developed} at the _University of California at

Berkeley. These versions for PCs sell for about $ 200-250.

viewed as an extension of TIP. This method of

analysis

developed along

two

_lines,

one based on line

_graphs

(P

EUSNER

,

1970)

and the other on &dquo; bond

_graphs

&dquo;

(O

STER

et

al.,

_1973).

NT has a double basis :

1)

A

_{generalization}

of the notions of fluxes J and

forces X used in

_electricity,

fluid

mechanics,

chemical

thermodynamics

and TIP

_(T

_HIERY

_,

_{1983 ;}

_C

_HAUVET

_,

1987).

&dquo;

Each _process

_involving

_energy can be

decom-posed

into a flow of material and the force

_responsible

for this flow. The

_product

JX takes the value of a _power, an instantaneous _energy, which _may be

stored,

consu-med or

_transported

_&dquo;(A

_TLAN

_,

_1976).

The

generalized

displacement

and the

_generalized

momentum

_(by

analogy

with the terms used in

_mechanics)

are also used.

2)

The use of network simulation programs.

NT

_is,

so to

_speak,

a

_powerful

formal

_language

that can be used for

_solving

many

problems

of mass and energy transfer,

particularly

in

_biological

and chemical

systems.

One

_proceeds

as follows :

- Choose

a

topological representation

of the

biolo-gical

system

in the form of a network

containing

the

various

_appropriate

elements.

- For each

element,

the

flows,

the constitutive

driving

forces and the

_relationship

between these two

quantities

are defined. An element can be a

_one-port

if it is crossed

_by

a

_single

_type

of mass or _energy, or an

n-port

in the case where there is more than one flow. For instance a resistor and a

_{semi-permeable}

membrane

allowing

no active

_transport

are

_one-ports.

On the other

hand,

when

_representing

_{the energy}

_budget

of a leaf

exchanging

energy

by

radiation,

convection or

conduc-tion and water _{vapor with the environment,} one will

regard

this leaf as an

_n-port.

The same will

_apply

to a

study

of

_coupled

flows of water and ions across a membrane or to a set of associated chemical reactions.

Translation into the

_system’s

NT

_language

leads to a

description

of the

_system

_by

means of the constitutive relations of the elements and those of electrical

networks in

_{particular (Kirchhoff’s}

_law,

_...).

It is thus

possible

to make an electrical

_description

of how the

system operates

by using

elements

_{R, C,}

L and others. Two alternatives are then available to understand how the _system works :

-

using

direct numerical

_analysis

to solve the set of

equations describing

the system ;

-

using

a circuit

analysis

_program.

A circuit

_{description generally}

involves

_{characterizing}

the elements in the circuit and

_indicating

their

_position

in relation to the network nodes as well as the

_boundary

conditions. Then the

_system

of

_{equations corresponding}

to the

_system’s

_operation

with initial conditions is

automatically

solved. This

_step

therefore does not need be handled

_directly.

_{Since many similar programs}

(SPICE

is one of the most

_extensively

_used),

both

powerful

and _easy to use, are

_currently

_available,

one can understand the

_{major practical advantage}

of

translating

a

_problem

in terms of NT. We

_purposely

will

not consider here the theoretical

_advantages

of this formalization

_(M

_IKULECKY

_,

_1983),

since

_they

are

inde-pendent

of the program used.

Comment

The electrical

_analogue

is not the

_only

_possibility,

nor

always

the best. For various reasons, it is the one most

commonly

used

_today.

_{However programs for}

_hydraulic

engineering analysis

could be as

_appropriate

in certain cases.

_Obviously,

it would be better to use a

_fully

network

_{thermodynamics-oriented}

_program rather than

translating

into the

_language

of

_specialized

simulation

programs. It is

hoped

that such a _program will be

developed

in the near future.

Now that we have

_presented

the

_{principles underlying}

the use of SPICE in

_physiology,

we will

_give

a brief

description

of this simulation program.

II. BRIEF DESCRIPTION OF SPICE

Using

SPICE

_{(i.e. writing}

a _{program in SPICE}

language)

involves

_describing

each element of the circuit in order to

_represent

the

_system

to be

_investigated

and the conditions it will be submitted to.

A. Elements

_(table

₁₎

The

_type

of each element is defined

_by

the first letter

(e.g.: (R)esistor,

(C)apacitor, ...).

The name

_(e.g.:

CAP

₂₎

can be _{any combination} of 7

_alphanumeric

characters. It is followed

_by

two node numbers

identi-fying

its branch and then

_by

the

_value(s)

_{of the}

para-meter(s) defining

the electrical characteristics of this element.

ex : VCAP2 8 0 12 2

This line sets a constant

_voltage

source of 12 volts between the connection element nodes 8 and node 0. Node 0 is

_ground.

The circuits studied later will show other

_examples.

The main elements of SPICE can be classified somewhat

_arbitrarily

into 3

_{categories :}

a)

Linear elements include

_resistors,

_capacitors,

inductors,

independent

voltage

and current sources. The latter can be _constant, or

_{time-dependent.}

There are five

independent

source functions :

_exponential,

_sinusoidal,

piece-wise

linear,

pulse

and

_{single frequency.}

It is also

possible

to _compose variable resistors and

_capacitors.

*

Voltage

sources, in addition to

_being

used for circuit

excitation,

are the &dquo; ammeters &dquo; for

_SPICE,

that is

_voltage

sources of value zero _may be inserted into the circuit for the purpose of

reporting

current in a

given

branch.

b)

Non linear elements

_essentially

include semicon-ductor devices. As the latter have not been used for

physiological

simulation

_yet,

we will not consider them here.

c) Dependent

or controlled sources are

_extremely

useful elements

_conferring

considerable simulation

power to SPICE.

_{They produce}

a

_voltage

or a current

that may

depend

on one or several

_voltages

of currents

existing

elsewhere in the circuit. Four

_types

of

control-led sources are available :

- current-controlled

voltage

sources

_(H),

-

voltage-controlled voltage

sources

_(E),

-

voltage-controlled

current sources

_(G),

- current-controlled current

sources

_(F).

There are 2

major

restrictions

_affecting

these func-tions :

- the functions

must be

_expressed

as

polynomials

in which the

_exponents

of variables must be

_positive

integers ;

- _a

dependent

source can

_depend

on one or several

voltages (up

to

_6),

or one or several currents. It cannot

depend simultaneously

on a

voltage

and a source. This

condition, however,

can be

easily

circumvented.

Moreover,

the

_availability

of the source code for academic research allows

_{incorporation}

of

_special

func-tions

_(e.g.

_hyperbola

for Michaelis-Menten

_equation,

Goldman’s

_equation

for membrane

_transport).

Usually,

the definition of a non-linear

_dependent

source takes the form :

F

G

! ,

G

NameN’ N

_POLY

_{(n) NC1’}

NCI

NC2’

t

H

NC2

NC2 _{all a,} a. _a,

J

/

NC? _{a<j a j a!} a ;

The current

_(F

or

_G)

or the

_{voltage (H}

or

_E)

between nodes

N+

4

and N is a

_polynomial

of n variables

(currents

or

_voltages)

situated between nodes

NC1

+ and

NC

1

the

polynomial

coefficients

_{being successively}

a o

, dl> az, a3, ...

B. Control statements

The control statements serve to

_specify

the desired

analysis

and

_{output :}

New versions of SPICE do not need

_explicit

control lines for the

_{output :}

_they

store all the

_output

and

_permit

one to trace _any combination of these. In the same way,

some CAE _programs allow one to

_generate

the state-ments

_directly

from the

_design

of the circuit on a

graphic

screen

using

a mouse.

C. Simulation

SPICE allows 3

_majors

_types

of simulation to be

performed :

- A direct current

analysis (DC Analysis)

which

corresponds

to a

_steady

state

_analysis.

- A transient

analysis giving

the

_{time-dependent}

response of the

system

over an interval to be

_specified.

- An

alternating

current

_{analysis (AC analysis).}

Several alternatives are available _among these 3

cate-gories (table

2).

For

_example

for

_large-signal

sinusoidal

simulations,

a Fourier

_analysis

of the

_output

waveform

can be

_performed.

The DC

_component

and the first 9 _components are determined.

Likewise,

different

signal

analyses

can be obtained. To

_{date, however,}

the

_sample

DC and transient

_analyses

have been most

_commonly

used for

_{physiological}

simulations.

III. EXAMPLES OF SIMULATION OF PHYSIOLOGICAL PROBLEMS

Three

_{simple examples}

will be

_presented

to

demons-trate how the most common elements of SPICE can be

applied

in 3 fields of

_physiology.

Detailed

_explanations

are

_provided

to initiate

_{physiologists}

into the use of this program.

A. Whole

_plant

flow model

_(fig.

_{I )}

Water enters vascular

_plants

in

_liquid

form

_through

the root

_system.

It moves in this form

_along

_very fine

capillaries (xylem vessels) through

the whole

_plant.

Upon

reaching

the

leaves,

it passes from the

liquid

to

the vapor

phase

and is released

_through

the stomata

into the

_atmosphere.

The main

_driving

force for this

water flow is

_{transpiration,}

which maintains a

_gradient

of water

_potential

from the

_ground

to the leaves

analo-gous to

voltage.

If the amount of water absorbed

_by

the

roots is

_equal

to that

_evaporated

from the

leaves,

the flux is conservative and can be

_{represented by}

Ohm’s law

_type

relations. If these amounts are

_unequal

(wil-ting, rehydration),

the flux is non-conservative. In

addi-tion to

_resistors,

the evolution of the water balance and its

_dynamics

can then be described in terms of water

present

example,

which

_only

considers the flow in

_liquid

form,

the

_plant

is

_{schematically represented by}

the flow

model in

_figure

1

_including

a soil

_(constant

level

reser-voir) supplying

water to the

_plant.

The latter is made up

of 3 resistors and one constant

_capacitance

reservoir

(simplified view).

Simulating

involves

_following

fluxes and

_voltages

as a

function of time in response to

_{imposed transpiration.}

This

_{transpiration}

increases

_{instantaneously,}

levels off for a

_while,

and decreases

_linearly

and rather

slowly.

Therefore,

the

_input

consists of the

_parameter

values and the initial conditions

_(voltage

at

boundaries,

trans-piration).

The

_output

consists of the instantaneous or cumulative currents

_along

with the

_voltage

at the

diffe-rent nodes.

The

_{corresponding listing}

is as follows :

**

MODEL: PIA DATE

** TIME CONSTANT ** PARAMETERS: 3 resistors RI = 90

bar . mg ’ .

CM

2

.

s ** R2 = 60 ** R3 = ₁₅₀ **

PARAMETERS: 1 constant

_{capacitance =}

2.5 mg -

CM

2

-

bar-’

* _{DESCRIPTION OF THE CIRCUIT}

Rl 1 1 2 90.0 R2 2 3 60.0 R3 5 6 150.0

C1 0 6 2.5 IC = 0

* _{MEASUREMENT OF THE FLUXES}

VABS 0 1 0.0 VDEP 5 2 0.0 VTRP 4 0 0.0 * _{IMPOSED TRANSPIRATION} GTR 3 4 7 1.0 VCLOCK 7 0 PWL

₍₀

0.0 300 0.0 + 301 0.006 3300 0.006 5400

_0.0)

RCLOCK 7 0 1.0

* _{MEASUREMENT OF THE CUMULATIVE}

FLUXES :

CABS, CDEP, CTRP

CCABS 100 0 1.0 FCABS 100 0 VABS 1.0 RCABS 100 0 1 E7 CCDEP 110 0 1.0 FCDEP 110 0 VDEP - 1.0 RCDEP 110 0 1 E7 CCTRP 120 0 1.0 FCTRP 120 0 VTRP - 1.0 RCTRP 120 0 1E7 * _SIMULATION .TRAN 30 6000 UIC

.PLOT TRAN

_{I(VABS) I(VDEP) I(VTRP)}

.PLOT TRAN

_V(

₁₀₀

₎

_{v(llo) V}

₍

₁

₂₀₎

.END

Comments

- The

capacitance

of the

_capacitor

is 2.5 farads for

SPICE,

but 2.5 mg .

bar

1

CM

-2

for the

physiologist.

IC = _{0 means} that its initial

voltage

is 0 volt. Therefore

the water

_{potentials (expressed}

in

_bars)

of both the soil

and the

_plant

are

equal

at the start.

- Three ammeters

(sources

of zero

_voltage)

are used

to measure the currents in the branches :

_VABS,

_VDEP,

VTRP.

- To

impose

the desired

_{transpiration,}

i.e. the

current

_flowing

from node 3 to node

4,

2 elements are

used :

a)

a source G

_{(voltage-dependent}

current

_{source) ;}

b)

a

_{special voltage}

source

(VCLOCK)

between nodes 0 and 7 of an

independent auxiliary

circuit whose value varies

_linearly

with time

(PWL

=

piece-wise

linear).

Here,

the

_{transpiration}

rises from 0 to

0.006 volts

_(0.006

mg .

CM

2

-

s ’ for

the

physiologist)

within 1 second

_(between

the

300

th

and the

301

S

‘

se-cond),

remains at this level until time t = _{3 300}

s, and returns

_slowly

to 0 within 2 100 s. As indicated

_by

the source

_G,

the current

_flowing

between nodes 3 and 4 will be

_equal,

at _any

_time,

to one times the

_voltage

between nodes 7 and 0.

GTR 3 4 7 0 1.0 controlled

_controlling

coefficient

nodes nodes

The cumulated fluxes are read

_by

means of 3 small

independent auxiliary

circuits in which a

capacitor

is

charged

with current

_equal

to one of the fluxes of the main circuit.

In each

_auxiliary

circuit,

one

_requires,

_through

a source

F,

that the current be

_equal

to a current measu-red in the main circuit

_(e.g.

between nodes 100 and

0,

it is the current measured

_by

the ammeter VABS that will

flow).

The

_integral

of the current over the time interval is

_equal

to the

_charge

accumulated within the

_{capacitor :}

at _any

_given

time,

the

_{potential drop}

across the

capa-citor

_{(ex: V(100))}

is

_equal

to V =

Q/C

=

Q

since C = _{l. From}

a node’s

voltage,

one can thus obtain at

each time the cumulated amount of water

_having

left a

reservoir

_{(or having}

entered this

_reservoir)

over a

given

time interval.

In the

_output,

the

_following

is

_{required :}

-

a transient

analysis

(.TRAN)

of the

system’s

evolution from time t = _{0 to} time _t = 6 000

s, with one

point printed

every 30 s from time t

_{= 0 ;}

-

a

graph

showing

the evolution of currents and

voltages

at different

_points

of the main circuit and of the

auxiliary

circuits. M OLTZ

et al.

₍₁₉₇₉₎

have

_given

another

_example

of the

use of SPICE &mdash; without the

_listing

of the program &mdash; for

_studying

_quantitative

water relations in

_plant

tissues.

B.

_Coupled

flow of water and solute across a membrane

(fig. 2)

This

_example

will show how

_equations

can be

different kinds of fluxes is

_expressed.

This will

_provide

better

_insight

into :

a)

the use of

_n-ports,

i.e. of elements

_having

several kinds of in- and

outflows

;

b)

the

_{&dquo; physical &dquo; separation}

of the various kinds of

flows :

_only

one kind of flux flows

_through

a

_single

branch. The theoretical

_coupling

_expressed

in the

equa-tions

_essentially

occurs

_through

the

dependent

sources

(F,

G, H,

E)

that allow the

_{voltage and/or}

the

_potential

at one

point

to be linked with the

_{potential and/or}

the

voltage

at another

_point.

Therefore,

instead of

_relying

on a schematic

_diagram

supposedly representing

the

_global

behavior of a _system

(previous

example),

we will start from two

_equations

describing

the

_coupled

flows of water and solutes across a membrane in a

simple

case

_(fig.

_3).

terms : solvent

_drag

diffusive active

_transport

branches = _d

c d

with :

Jy

= total volume flux.

4P =

P

I

- P

2

=

difference of

_hydrostatic

pressure

across the membrane.

AT[i - 711i

i IT2i, osmotic pressure difference due to

non-permeable

solutes

_(for

which 6 =

I).

Ajr! ! I

Tls IT2,’ osmotic pressure difference due to

permeable

solutes

_(for

which

_{a < 1 ). ).}

Lp

= coefficient of filtration or

_{hydraulic conductivity.}

C

o

-

(C

1 +

_C2)/2,

C 1 and C2

_being

the concentration of either side of the membrane

_{respectively.}

In

certain situations

_(THOMAS

&

MIKULECKY,

1978),

it can also

be,

as in the

_present

example,

the concentration of the outside

_compartment.

a, = reflection

coefficient

for

permeable

ions.

w = coefficient of solute

_mobility.

R = universal gas constant.

T = temperature

(Kelvin).

J

*

,

= active solute flux.

Thus we have 2 kinds of flux

(considering only

one

solute)

each of which can be considered as the

_algebraic

sum of 2 or 3

elementary

fluxes,

each

corresponding

to

a term of the

_right-hand

side of

_equations

1 and 2. Our

aim is to

_represent

these terms

_{by appropriate}

depen-dent sources or other elements.

l.

_{Representation}

_{of flux}

_J,,

This form of the flux

_equation

is

_commonly

used in TIP and often the most convenient for SPICE.

_Jy

will therefore be considered as the sum of 2 terms XiFi. Xi is a

coefficient,

Fi a

_force,

each of these 2

_components

_{of J!}

will flow

_through

a branch

(fig.

2a).

Branch a consists of:

- _a resistor RWP

taking

the value

_I/Lp.

The current

flowing

across this branch will have the value

_{Lp P}

*

,

with P* = _{4P -}

Avr,

if the

_voltage

across this branch

equals

AP*

(see

below) ;

- _an ammeter VW

measuring

the current in this branch.

AP *

is fixed

_by

a

_voltage

source VWP = AP* between nodes 0 and 1. This

_grounded

source

_puts

node I at a

potential

AP* = 1 volt

(I

bar for the

_{physiologist),}

resulting

in a

_potential

of 1 bar between nodes 1 and 2

since nodes 2 and 3 are

_grounded.

The element VWT

will be considered below.

Branch b consists of a

_single

active element

_(in

addition to an ammeter VW2 used to measure the

voltage):

element GW2 which is a current source controlled

_by

a

_{voltage equal}

to ₆₅

_{4!! :}

controlled nodes

_controlling

nodes

_Lp

The current

_{flowing through}

branch b will be

_equal

to

-

Lp

AS

L1

T

Cs’

The VWT element

_(like

_VSA,

see

_below)

is

_composite,

because it acts both as an ammeter and a non-null

voltage

source. Flux

_Jy,

the current

_{passing through}

VWT,

is the sum of the 2 fluxes

_{passing through}

bran-ches a and b

_{respectively :}

This first sub-circuit

_corresponds

to

_equation

1.

2.

_{Representation}

_{of solute flux}

_Y,

We follow the same

_procedure

as above. Flux

_J

_s

is made _up of 3

_elementary

fluxes

_{(solvent drag}

or convec-tive term,

diffusive,

active

_transport)

which must be written in the form of a

product

XiFi. As shown

by

equation

2’

_only

the solvent

_drag

term does not meet

this

_requirement,

because it is the

_product

of 2 different variables : a

_{voltage (C}

_j

₎

and a flux

_(J!).

As SPICE does

not

_accept

this kind of source, one must first use a small

auxiliary

circuit to fix a node

_voltage

&dquo;

_V(J

_v

₎

&dquo;

equal

to

the current

_Jy.

Then a

voltage-dependent

current source

multiplies V(J!) by C,

to

_give

the solvent

_drag

contri-bution.

Then,

to avoid

_multiplying

the

branches,

the

solvent

_drag

and active

_transport

terms are

_grouped

on one side

_{(branch d)}

while the diffusive term is

_put

on the other side

_{(branche c).}

Conversion of the convective term

_C,

_{(I -}

_{as) Jy.}

To

convert

_C,

into a _current, we create a

_{voltage-dependent}

current source, a source G named

_GTRA,

such that:

This is a

_polynomial

function of 6s _reduced to a

_single

term, which will be

_{accomplished by}

use of an

_auxiliary

circuit

_(see

_auxiliary

circuit n°

_1,

_{fig. 2a).}

This circuit

includes 2

nodes,

0 and

_21,

between which a current

(GTRA)

flows. This current is

_equal

to 5.33 E - 0.6 _n,, where !, is a

_voltage

source

_(VCE)

located between nodes 1 and 10 in the main

circuit,

representing

the osmotic _pressure of the outside

_compartment.

The element line

_designating

this source will be written : GTRA 0 21

_{POLY( 1 )}

10 0 0.0 5.33 E - 0.6 .

The convective term of

_equation

3.2 thus becomes a

current that

_depends

on _{2 currents,}

_{Jy (measured by}

the

ammeter

_VTRA),

rather than on a

_voltage

and a

current.

The creation of branch

_d,

which includes both the convective and the active terms,

requires

the use of a

sub-circuit

_{corresponding}

to flux

_J*!.

This will include :

-

a

voltage

source VSA

taking

the value

J*! (in

this

case,

J*, !

7.5 E - 12 mole ’

cm-

2

.

s ’)

between

nodes 0 and 22

_(numbers

are

_{randomly assigned,}

_except

for the

_ground

which must be

_zero).

Then this source can also act as an _{ammeter ;}

-

a resistor RSA

_equal

to l. The current

_flowing

through

this circuit will be :

In branch

_d,

the

_flowing

current will be the

_algebraic

sum of the convective term

_(function

of

_J&dquo;

and of

GTRA)

and the active

_transport

term

_(equal

to

_VSA).

This current is therefore a

_polynomial

function

_(with

only 2 terms)

of 3 variables

_being

currents. It is a source

_F,

named

FS3,

such that :

The

_expression

of such a 3-dimensional

_polynomial

(

Xi

,

X2, X3

being

the

variables)

as understood

_by

SPICE is written :

with GTRA =

x, :

J! =

x, and

J*! =

x!, FS3 becomes:

FS3 = _{a3 X3} + _{a5 x, X2 with as = a3 =} 1 all other

coefficients

_being

zero.

The element line

_{corresponding}

to FS3 is written :

FS3 10 13 _{POLY (3)} VRTA VWT VSA 0.0 0.1 0.1 I controlled nodes _controlling nodes coefficients

An ammeter

_(VS3)

is also used to measure current FS3. Branch c, which

corresponds

to the diffusive term, is

simpler.

The 2

_voltage

sources

_{corresponding}

to the

outside

_(VCE)

and inside

_(VST)

concentrations are located at nodes 10 and 12

_{respectively,}

in

_parallel

with

branch d. In the

_present

_example,

VCE = 2 bars and VST = 0.2. The other element between nodes 10 and 11 I

is a resistor

_(RS2)

with a value

_{1/w =}

I.OE12 ohm

_(or

mol !

C

M-

2

_

S-

1

-

bar ’).

There also is an ammeter VS2. Note that ammeters are not

_always

_{necessary. Their}

presence, however, allows the measurement of

elemen-tary

fluxes that may be desired in the program

output.

They

are also of

_great

assistance in

_verifying

calcula-tions.

The simulation

_requested

here is a

_steady

state

analy-sis

_(control

card

_starting

with DC with different values of the inside concentration

_(up

to 2 bars

_by

increments of 0.2

_bar).

In

_addition,

the use of another control card

(.ALTER)

allows variations in one or several other

parameters

to be

_produced

in the same simulation.

_E.g.

in the

_present

case, the outside

hydrostatic

pressure

respectively

takes the value 3 bars and then the value 10 bars. For each of these values the source VST varies from 0 to 2 bars

_by

increments of 0.2 bar.

The

_program’s

_listing

_{corresponding}

to this membrane is in

_figure

2b.

THOMAS & MIKULECKY

_(1978),

MINTZ et al.

₍₁₉₈₆₎

and THOMAS

₍₁₉₈₈₎

_give

other

_examples

of different kinds of

_transport

mechanisms

_{involving coupled}

fluxes

using

SPICE.

C. Process

_obeying

the Michaelis-Menten law

_(after

WHITE &

_{MIKULECKY, 1982)}

Enzymatic

processes

(conversion

of substrate to

product catalyzed by

an

_enzyme)

or

_{physiological}

processes

(uptake

of a substance

_by

excised

_organ),

frequently obey

the Michaelis-Menten law :

v =

velocity

of the reaction or rate of

_uptake,

S = _{concentration of substrate}

or substance in the outside

_solution,

Vmax = maximum

velocity

of

_transport

or rate of

uptake

km = Michaelis _constant.

The

_preceding

relation indicates that the

_velocity

of formation of the

_product

or substance

_(e.g.

an

ion)

is a

hyperbolic

function of the substrate or ion

concentra-tion in the medium. Translated into SPICE

_language,

the

_preceding

relation is the

_expression

of a

voltage-dependent

current.

_{Unfortunately,}

the current versions

of SPICE do not deal

_directly

with

_quotients.

Thus for any

expression including

a

_ratio,

one must write

_using

an

auxiliary

circuit that is not connected to the main circuit

_representing

the

_biological

_system

to be

investi-gated.

Let us assume here that a substrate S is converted

to a

_product

P. Two

_capacitors

are

_placed,

one at each end of the circuit. In the

_present

_case, their

_capacitance

represents

a volume

_following

the

_{equation :}

C

-

amout

of substance

(moles) =volume

concentration of substance

_{(moles/liter)}

(liter)

Initially,

compartment

S has a certain volume C and

compartment

P is empty. The rate of

_change

of concen-tration of material

_produced,

as measured

by

the

amme-ter

_VSP,

will

_correspond

to a

_{concentration,}

since it is

given by :

Rate of

_change

of concen- amount of substance added to or substracted from the

_compartment

tration in _compartment = .

volume

Capacitors

are used when the volume of

compar-tments is limited or when this restriction is involved in the

_{system’s regulation.}

To

_represent

a

_nearly

infinite

reservoir,

the easiest would be to

_replace

the

_capacitor

by

an

_{independent voltage}

source.

The flow of material

_(or

the reaction

_velocity)

is

_given

by equation

3. It is a function of the substrate

concen-tration,

i.e. of the

_voltage

at note

_1,

and must therefore be

_{represented by}

a

voltage-dependent

current source,

hence

_by

a G element. This

_{controlling voltage}

will be

that at node 10 of the

_auxiliary

circuit

_{(fig. 4b),}

thus

giving :

GMM 2 3 10 11 1 1

controlled nodes

_controlling

nodes

_{proportionality}

coefficient

The current

_{produced by}

GMM is

_{linearly dependent}

on the

voltage

between nodes 10 and 11

(or

10 and

0)

Auxiliary

circuit

_(fig.

4b).

The _purpose of this circuit is to calculate the

_quotient

v of

_equation

3. To this

effect,

the

_equation

must take the form :

Each term of this

_equation

will be a source G. GN will

represent

the left term

_(numerator

of

₃₎

and

_GQD

the

right

term

_(product

of

_quotient

and denominator of

_3).

If both sources are

_placed

in

series,

the current

genera-ted

_by

GN from

_ground

to node 10 must

_equal

the

current across

_GQD

since there is no accumulation element. The

_voltage

at node 10 will _vary as a function

of the current

_{flowing through. According}

to

equa-tion

_4,

the source GN is a linear function of the concentration

S,

and will be written :

GN 0 10 0 1 0 Vmax

controlled nodes

_controlling

nodes coefficient The current

_flowing

from node 0 to node 10 is Vmax-fold the

_voltage

between nodes I and 10

_(e.g.

substrate

concentration).

The

_GQD

source is a function of

2 variables : v =

velocity

of formation of the

_product

(voltage

between nodes 10 and

_{I I )}

and S = _substrate concentration

_(voltage

at node

₁

The

GQD

source will therefore be a

polynomial

function written as follows :

current

_{GQD -}

0 + Km v + OS +

_O(

_V

₎

₂

+ 1 vS . Hence the coefficients are

_{0, Km, 0, 0,}

1. As a

_result,

the SPICE

_description

of this source is written :

GQD

10 11

_{Poly (2)}

10 0 1 0 0 Km 0 0 I

controlled

_controlling

_polynomial

nodes nodes coefficients

The order in which the

_controlling

nodes are chosen

determines the order of the coefficients. Another form

equivalent

to

_GQD

_may be :

GQD

10 11 1

_POLY(2)

1 0 10 0 0 0 km 0 1

Hence the simulation program for this chemical

reac-tion,

in which the

_dynamics

of substrate formation

_(or

substance

_uptake

_by

the excised

_organ)

are

_investigated

as a function of time and value

_S,

Km and

_Vmax,

will be written :

** _{PROGRAM : simulation of}

transport

according

to

the Michaelis-Menten

_equation

* _{MAIN CIRCUIT}

CS 0 1 value of

_capacitance

IC : initial

_voltage

VSP 1 2 0.0

GMM 2 3 10 11 1 1.0

CP 2 0 value of

_capacitance

IC: initial

_voltage

* _{AUXILIARY CIRCUIT} GN 0 10 0 1 0 Vmax

GQD

10 11

_POLY(2)

1 0 10 0 0.0 Km 0.0 0.0 1.0

VQD

11 1 0 0.0 * _SIMULATION

.TRAN TSTEP TSTOP UIC

.PLOT TRAN

V(I) V(2) V(3) V(10)

.PLOT TRAN

I(VSP) I(VQD)

.END

It is also

_possible

to obtain the same results

_{by directly}

incorporating

the source

_coding

for a Michaelis-Menten

type

relationship.

Beside,

the evolution with time of the

different substances from the

_{following enzymatic}

reac-tion can also be simulated :

E + S <--> ES <--> E + P .

with E =

_{Enzyme ;}

S

_{= Substrate ;}

P =

Product ;

ES

_{= Enzyme}

substrate

_complex.

Other more

_{sophisticated examples}

for

_simulating

cellular

_{pharmacokinetics}

_(WHITE

&

Mocut,ECKY,

1982 ;

THAKKER et

al.,

1982)

or reaction diffusion

systems

(W

YATT

et

_{al., 1980;}

MAY &

MiKUt.ectcY,

1982)

can be

found,

illustrating

the

_capabilities

of

SPICE in this domain.

IV. STRENGTHS AND WEAKNESSES OF SPICE

A.

_Strengths

1) Equations, particularly

sets of differential

equa-tions,

are easier to formulate than to solve.

_Many

physiologists

can write and understand certain kinds of

equations,

but often lack the

_{specific knowledge}

of mathematics

_especially

in numerical

_{analysis required}

to

solve them. As mentioned

_previously,

SPICE is _very

convenient to use in this

_respect,

because it

_usually

relieves

_{physiologists}

_{from such tedious work. It may be}

assumed that a

_physiologist

would rather

_spend

his time

trying

to better understand his

_system

than

_learning

techniques

of

_solving

differential

_equations.

Remark : The

_integration

method used

_by

SPICE is the

_trapezoidal

method. As an

alternative,

gear’s

method is available.

_Concerning

this

_point

the reader may refer to

_Nagel

_{( 1975).}

2)

On the other

hand,

the process of

_expressing

a

physiological

or biochemical

_problem

in the

_language

of SPICE

_compels

one to

_explore

the various aspects of this

_problem,

and

_{place emphasis}

_upon the structure

representing

this

_system.

This is an essential

_point,

for

elementary

equations

are

_frequently

mistaken for struc-ture. For

_instance,

most

_global

models of water and mineral

_{uptake by}

the roots are based on

_equations

I and 2. Some

authors, however,

believe that &dquo; one should use a

_{3-compartment}

model rather than these

equa-tions &dquo;. This is not the relevant

_point.

The

_quantitative

approach

to the

_functioning

of a

_system

is obtained

_by

applying

basic

_equations

to a

particular

structure. So the purpose is to define the structure and basic

equa-tions rather than to

_replace

a structure

_by

a law. Hence the above

_equations

which are valid at the membrane level

_produce

different effects

_depending

on whether the

structure consists of one or several

_{compartments.}

B. Weaknesses

SPICE has a number of limitations which can be

disappointing

or even deterrent. Yet these weaknesses tend to be remedied in the new versions and should be considered as

_temporary.

1.

_Input

It was shown that the

_input

cannot deal

_directly

with

quotients

or

polynomials

with

negative

_exponents.

Like-wise,

exponential, logarithmic

and

_hyperbolic

functions

cannot be written

_{directly. They require}

the

construc-tion of an

_auxiliary

_circuit,

for the standard versions of SPICE. Other circuit simulation programs

(ASTEC

for

instance)

permit

a more direct

_description

of non-linear

relations. That

_is,

one enters the

_equations

for the constitutive relations

_directly

in a FORTRAN-like

expression.

2. Simulations

Many

additional

_types

of

_analysis

can be made on

electrical networks and would also prove useful in

physiological

simulation

_{(e.g. :}

_parameter

_{optimization,}

statistical

_analysis

for

_evaluating

circuit

_properties

for a certain range of

parameter

values,

etc...).

It is

_only

because these

_analyses

were

_already

available in the program

library

at the

_University

of

_Berkeley

that the authors of SPICE did not

_try

to insert them into their

program !

So if &dquo;

network

_{thermodynamics}

is

comple-tely independent

of electronics and electrical network

theory

&dquo; (M

IKUL

ecKY,

1983),

SPICE is still very

depen-dent on electrical

circuits,

due to its

_origin.

V. CONCLUSION

We have

_presented

some very

simple examples

of the use of SPICE for

_{simulating problems}

of transfer of

matter _{and energy in}

_biological

_systems.

The

_simplicity

of these

_examples

does not

_{fully exploit}

the real advan-tages of

SPICE,

which are

_only

realised in much more

complex problems.

Unlike

_{techniques using higher}

mathematics

_requiring

an

_explicit

formulation in terms

of

_equations,

SPICE rests on schematic

_depictions

of the _{system. The}

_present

use of SPICE in such different fields as classical

_physiology,

membrane

_biophysics,

pharmacokinetics, electrochemistry

of vision

_provide

a

good

illustration of its usefulness for

_biologists.

The

advantage

of this

_approach

to simulation is that the modeler need know

_nothing

of numerical

_analysis

and

can nonetheless treat

_complex,

non-linear

_systems.

In this

_regard

SPICE is not

_necessarily

the easiest simula-tion program to use

_(ASTEC

is more

user-friendly)

but it is _very

_powerful,

is

_universally

_available,

has been

adapted

for

_nearly

all kinds of

_{machines, and is very}

cheap

or free.

Reçu le 18 décembre 1987.

Acceplé le 8 mar.s 1988.

ACKNOWLEDGMENTS

We thank _Evelync Ron for the translation and M. Gingold for

rechecking the programs.

REFERENCES

Atlan H., 1976. Les modcles _dynamiques en reseaux et les sources

d’information en _biologic. In : « Struclures et _{Iues qi(I/l’//ld}des

systè-Ill

es ». _Maloine, Paris, 95 131. 1.

Chauvet G., 1987. Ti-ail! f/c _{phrsiologie théorique.} Vol. 1, Masson. 76-83.

Dainty J., 1960. Electrical analogues in Biology. In: &dquo;Model.s and

analogues in _BioloRr &dquo;. _Symposia of the Society for Experimental

biology, nx XIV, Cambridge Univ. press., 140-151. l .

Fiscus E. L., 1977. Determination of _hydraulic and osmotic _properties

of soybean root _systems. Plant Physinl., 59, 1013-1020.

Kedem 0., Katchalsky A., 1958. _{Thermodynamic analysis or}

permea-bility of _biological membranes to non _{electrolytes.} enr. hBioc _BioI/ill’.

Acta, vol. 27, 229-246.

Kedem _{0., Katchalsky} A., 1963a. _Permeability of _composite

membra-nes. Part 1. Electric current, volume flow and flow solute through

membranes. Trans. Faraday Soc., 59. 1918-1930.

Kedem 0., Katchalsky A., 19631z Permeability of _composite

membra-nes. Part 2. Parallel elements. Trans. _Faraday Soc., 59, 1931-1940. Kedem 0., Katchalsky A., 1963c. Permeability of composite

membra-nes. Part 3. _{Series array of elements. Trans. Faraday} Soc., 59, 1941-1953.

May J. M., Mikulecky D. C., 1982. The _sample model _{of adipocyte} hexose _transport kinetic features, effect of insulin and network

thermodynamic computer simulations. J. Biol. Chem., 257, 11601-I 1608.

Mikulecky D. C., 1983. Network thermodynamics : a candidat for a common _language for theoretical and _{experimental biology.} Am. J.

Phrsiol., 245 _{(Regulatory Integrative Comp. Physiol. 14) :} RI-R9. Mintz E., Thomas S. R., Mikulecky D. C., 1986. _Exploration of _apical sodium transport mechanisms in a _epithelial model by network

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