Bond graph representation of a photoreception model

Bond Graph Representationof a PhotoreceptionModel* by J.

SCHNAKENBERG

and J.

Institut fiir Theoretische Sefent-Melaten,

D 5100

TIEDGE

Physik,

Rheinisch-

Aachen,

Westfiilische

Technische

Hochschule,

West Germany

A bond graph network representation of a photoreception model of Limulus is derived. The admitted network elements are exclusively restricted to the microscopically realistic processes of storage, diffusion and chemical reactions of molecules. All kinds of coupling in the network are explicitly expressed in terms of the above processes. The model is numerically evaluated and the results are compared with experimental data.

ABSTRACT:

I.

Introduction

The current status of modeling in biophysics may be characterized by an increasing amount of precise experimental data obtained with the aid of sophisticated physical and chemical instrumentation. Often, however, these data are hard to use effectively, since in many cases we do not know how to associate an experimentally well-known biological phenomenon with a particular molecular structure. There is not only a problem of analyzing the structure of the great variety of biomolecules and their interactions but even more of selecting from the possibilities those molecules and interactions which principally are responsible for the phenomenon under consideration. This situation has led to the use of black-box models rather than molecular and microscopic models, at least in the starting phase of the model analysis. At this point, network thermodynamics enters the scene since it represents a very straightforward method of designing phenomenological or black-box models. From the viewpoint of a purely formal mathematical analysis, a network model is an elegant and intuitive graphical representation of a system of non-linear differential equations. Consequently, it seems that finding a black-box model for an experimentally observed phenomenon is eventually equivalent to constructing a non-linear system of differential equations with the aid of network thermodynamics in such a way that the analytically or numerically obtained solutions of the system of differential equations optimally approximate the experimental curves. A main point of this paper is to argue that such a program is inadequate for constructing physically meaningful models as required in biophysics. Even if we do not yet know how to relate an experimentally observed phenomenon to a particular molecular structure, a model for this phenomenon should be expressed in a language which at least allows for a physically meaningful molecular * This work was supported

by the Deutsche Forschungsgemeinschaft,

@TheFranldinInstituteOOE-0032/79/0901-0327$02,00/O

SFB 160.

327

J. Schnakenberg

and J. Tiedge

interpretation from the very beginning. The reason for this condition is the experience that the molecular analysis of a biological phenomenon is a continuous process involving a great number of subtle experimental steps. A successful single step during this process should not lead to a complete revision or even to a cancellation of an existing model but rather to an extension or an improvement. Clearly, this requires the processes involved in the model (i.e. the elements of its network representation) to be restricted to such mechanisms which are known to exist on the molecular level. Subject to such restrictions, a bond graph model for photoreception will be proposed in this paper. As regarding the comparison with experimental data we shall refer to the photoreceptor system of the Limulus (horseshoe-crab) which has been investigated extensively. The molecular mechanisms which will be included in this model are as follows: (1) Storage of molecules in the vohrme of the system or in subdivisions (2) Transport of molecules by diffusion including transport of electric in the case of ions. (3) Chemical reactions among molecules.

of it. charge

As regarding the network representation of the above listed mechanisms, storage of molecules of type X is a capacitance element to be denoted as

with the constitutive

X-f,

(1.1)

J=$,

U-2)

relation

x being the concentration of X. Note that we choose the concentration x as intensive variable instead of the corresponding chemical potential p. For dilute solutions, x and p are related by P-PLO

x=exp

(1.3)

RT

where p. is the reference potential and R and T are the gas constant and the absolute temperature, respectively. Eq. 1.3 turns Kirchhoff’s voltage law (KVL) for l-junctions into a product law for concentrations, i.e. 5s

x3

x2

2 ;r &=I.

52 4

JI

(1.4)

i=l

X?t

Xl

J,

’

For the case of ions, the concentration x+xexpE,

variable

x has to be replaced

.zFV

by

(1.5) Journal of The Frankiin Institute

328

Pergamon

Press Ltd.

Bond Graph Representation of a Photoreception Model

where z, F, V, are the valency of the ion, Faraday’s constant and the local electrical potential, respectively. For diffusion, we shall assume the validity of Fick’s law, i.e. J = D . (x -x’),

(1.6)

for the diffusion flux between two compartments with concentrations x and x’, D being a renormalized diffusion constant. Although (1.6) allows for a l-port representation of diffusion, we prefer the 2-port version given by x AD--’

J

J’

(1.7)

with (1.6) as its constitutive relation. The reason for this choice is the fact that our third molecular mechanism, namely a chemical reaction, e.g. X+ Y+Z, necessarily is represented by a 2-port element:

where x, y, z are the concentrations of X, Y, Z and use has been made of the concentration version of KVL as given by (1.4). As constitutive relation of (1.8) we assume that the well-known product law for the reaction flux J is valid, i.e. J = kxy - k’z,

(1.9)

where k, k’ are the forward and reverse rate constants, respectively. Comparing (1.6) and (1.9) we see that from a formal viewpoint Fick’s law is a special case of the constitutive relation (1.9). Hence, choosing the 2-port version (1.7) of diffusion we are capable of restricting the type of network elements for our biophysical models to not more than two, namely a l-port material capacitance and a 2-port reaction including transport by diffusion. To conclude our brief introduction we should mention that we obviously neglect transport of energy and volume; i.e. we assume our systems to be isothermic and isobaric which will be the case at least for photoreception. Also, we exclude any kind of signal flow or parametric coupling between the network elements of our models. Actually, biological systems do exhibit signal flows which, however, are always carried by flow of matter and thus to be represented explicitly by the corresponding elements of storage, diffusion and chemical reactions of the signal molecules. With the above restriction to the network elements capacitance (1.1) and reactance (1.7) and (1.8) for all kinds of the model processes we follow the line of the work of Oster, Perelson and Katchalsky (1). Actually this paper and likewise a general introduction into the network thermodynamics of biological systems by one of us [Schnakenberg (2)] have been stimulated to a great extent by the ideas and concepts of Katchalsky. Vol. 308, No. 3, September 1979 Printed in Northern Ireland

329

J. Schnakenberg

and J. Tiedge

Ll. The System of Invertebrate Photoreceptor Cells

A photoreceptor cell transforms light signals into electric signals, the socalled receptor potential, which depends on the light intensity. The information included in the receptor potential is passed on to other cells in the form of “action potentials” which are the information units of the nervous system. Considering photoreception as black box we would thus ascribe to it two entries or ports, namely one for reception of light and another for the receptor potential which is the voltage difference across the cell membrane. A typical answer of the membrane voltage to a bright light pulse is shown in Fig. 1. The light-voltage response scheme is not passive but includes amplification and thus requires energy supply which would require a further port of the system. Furthermore, in laboratory experiments one varies a lot of further parameters of the system besides intensity and duration of illumination, in particular the outside concentrations of Na+, K+ and Ca*+ and one observes the changes induced thereby in the light-voltage response scheme of Fig. 1. Also, from voltage clamp experiments one can obtain current-voltage characteristics of the receptor potential port of the system under various conditions for the other ports, of particular interest being the dark and light characteristics. To sum up, the photoreception system under laboratory conditions should be described by a multi-port network. Its complete analysis on the basis of the available experimental data would go far beyond the framework of a single paper like this. We therefore restrict ourselves to a network analysis of mainly the light-voltage response scheme of Fig. 1 whereas all other relations and parameters of the system will only be mentioned incidentally. For further physiological information on photoreception of invertebrates the reader is referred to the physiological literature on the subject e.g. Milecchia and Mauro (3), Langer (4), Stieve (5-7). Concerning model descriptions of photoreception other than network models we would like to mention Borsellino and Fuortes (8), Kramer (9) and Eckert and Kramer (10). At least to our knowledge, a network description of the full photoreception system has not yet been worked out. mV 0

-40

1

‘iii.

0

FIG. 1. Typical receptor

330

potential

after a 1.5 s light stimulus Stieve el al. (19)].

4

TIME/s

of high intensity.

[After

H.

Bond Graph Representation of a Photoreception Model

IZL The Rhodopsin Cycle It is well known from the investigations of Hubbard and Sant George (ll), Suzuki et al. (12), Hamdorf and Schwemmer (13), Wyman et al. (14) and others that the molecular mechanism by which light couples into the photoreception system is a number of light induced chemical reactions between different conformations of the rhodopsin molecule. Together with another number of dark reactions between the rhodopsin conformations, the complete rhodopsin reaction system forms a network consisting of one or more cyclic reaction paths such that the sum of the concentrations of all different conformations remains constant. The simplest version of the cyclic rhodopsin reaction system which will satisfy the requirements of our model is a single cycle of four different rhodopsin conformations to be denoted as X1, X,, X,, X, as shown in Fig. 2. In addition to the capacitances for Xi, X,, X,, X, an external I-port has been introduced where I= I(t) describes the light intensity as a function of time. Among the four reactions RI, R,, R,, R4, reactions R,, R, are lightinduced whereas R,, R, are dark reactions. Since the system is cyclic we can suppress the reverse rates such that according to (1.9) R,, R,, R,, R, are given by

J1 = k,x,l(t)

R,:I+X,+X,, R,:

X*-,

J2 = k,x,

X3

R3:I+X3--,X4,

J3 = k,x,l(

R,:

J4 = b‘s,

X4+X,,

(3.1)

t)

with xi, x2, x3, x4 denoting the concentrations of X,, X,, X,, X,. By making use of the constitutive relation (1.2) for the capacitances and of Kirchhoff’s current law (KCL) for the O-junctions of the network in Fig. 2, we obtain the differential equations

dx,

:=

dx,

dt=

-J1+J4,

-J2+J1, (3.2)

dx, ==

-J,+J,,

O-l-R,-0

1

% 1

I-O

0-R3-l-

2=

-J4+J3,

1

1

I

1

R2

I

4

XL FIG. 2. Bond graph network Vol.

308,

Printed

No. 3, September

in Northern

Ireland

x3 of the rhodopsin

cycle.

1979

331

J. Schnakenberg

and J. Tiedge

from which we immediately

derive

&+x,+x,+x3=0, i.e. the conservation relation of the sum of rhodopsin molecules. A number of only two rhodopsin conformations coupled by one dark and one light-induced reaction would not be adequate for our model since the experimental investigations have shown that there exist at least two rhodopsin conformations corresponding to X, and X3 which are thermally stable in the dark.

IV.

Coupling

Between the Rhodopsin Cycle and the Membrane Voltage

The receptor potential, i.e. the response of the system to a light pulse, is observed as a deviation of the membrane voltage V= Vi,,- V,,, from its resting value of V,- - 50 mV. This resting voltage is a consequence of a non-equilibrium distribution of mainly Na’ and K+ inside and outside of the cell membrane. Since the membrane in the dark state is mainly K’-conductive but Na+-insulating, the above mentioned value of V, is roughly the Nernst potential of Kf. The ionic non-equilibrium distribution is maintained by pumps acting across the membrane and driven by energy-carrying enzymes which thus represent one of various energy ports of the system. As a consequence, we could expect to obtain a reasonable photoreceptor model if we introduce some appropriate coupling between the light-induced rhodopsin conformations and the ionic membrane conductivity in order that reception of a light pulse causes a conductivity change in such a way that a receptor potential as shown in Fig. 1 is produced. A direct coupling between the membrane conductivity and perhaps the concentration x2 of the lightinduced conformation X2, however, would not reproduce the type of the observed receptor potential but just yield a steep increase of the voltage followed by an exponential relaxation. Particularly, the latency time between the beginning of the light pulse and that of the responding potential change would not be described by such a direct coupling. After quite a number of model attempts we found that we need two further intermediate steps between the rhodopsin cycle and the membrane conductivity in order to obtain realistic receptor potentials. The first step is a reaction cycle between three conformations Z1, Z,, Z3 of an activator molecule Z which in turn activates an enzyme E from its ground state El to an activated state E2 in the second intermediate step. Eventually, E2 is assumed to control the dynamics of a selective ionic conduction channel across the membrane as to be discussed in the following section. The two intermediate steps of Z and E are shown in the network in Fig. 3. The conformations Z1, Z,, Z3 of the activator are connected by a cycle of reactions R,, R,, R7. The rhodopsin conformation X2 is assumed to act as 332

Journal of The

Franklin Institute

I’ergamon

Press Ltd.

Bond Graph Representation of a Photoreception Model

x2

Zl

El

1

1

-O-Rii:/--i-;-~\

IA /o\ 1-o-

i" -0-1

TD-1

%

(l/rd

1

1 E2

22

FIG. 3.

catalyst

~~Od[C02+li

{IJ

Bond graph network for the coupling between rhodopsin conformation the enzyme E controlling the membrane conductivity. in the reaction

R, such that its flux is given

X2 and

as

Js = kw1,

(4.1)

where again the reverse rate has been suppressed. This type of catalytic coupling does not interfere with the conservation relation (3.3) of rhodopsin since a catalyst enters into and leaves a reaction at the same rate. Reactions R6 and R, are non-catalytic, its fluxes being given as J6 = kbz2, Similarly as for rhodopsin, mations read

2~ From (4.3) molecules:

we

-J,+

obtain

J, = k7z3.

the differential

J,,

equations

%zz -J6+J5,

a conservation

(4.2)

relation

$ (z1+ z* + z,) = 0.

for the activator

2=

-J7+J6. for

the

sum

confor-

(4.3) of

activator

(4.4)

Between the enzyme conformations El, E, in its ground and activated state, respectively, we have assumed three reactions R8, R,, R,, which may be classified as activation or decay, but this notation should not be taken literally since all of them may be driven by special enzymes. The main point is that the activator Z in its conformation Zx acts as a catalyst in the activation reaction R,, however, not with first order kinetics like X2 in R, but with a stoichiometric coefficient n > 1 such that the activation flux J, reads J8 = k,z;el.

(4.5)

This coefficient n is generated in the network by two transducers TD with moduli n and l/n, respectively. In the potential flux language a transducer is defined by transforming the pair (CL,J) of conjugated potential p and flux J into Vol. 308, No. 3, September 1979 Printed in Northern Ireland

333

J. Schnakenberg

and J. Tiedge

(np, J/n) with respect to its reference orientation. Due to (1.3) in dilute solutions, this transformation is translated into the concentration-flux language as (x, J) + (x”, J/n). This transducer rule directly leads to (4.5) where again the reverse rate has been suppressed. To the decay reaction R, we ascribe the usual flux J9 = kge,.

(4.6)

Finally, R,, is a second decay reaction, but different from R, it is catalyzed by the Ca2+-ions inside the system. This coupling to the internal Ca2+ -concentration is a crucial point of the whole model since it establishes a negative feedback coupling which is necessary to account for the overshooting phase in the receptor potential curve in Fig. 1. This feedback coupling acts in the following way: as to be shown in Section V, light-induced conduction channels across the membrane will give rise to Ca2+-influx which in turn causes an E, -decay and hence a reduction of the number of open light channels. Consequently, the dark conduction channels will take over and rapidly tend to restore the membrane potential from its actual value of + 10 - - * + 20 mV to its resting value of -50 mV. It is known from physiological experiments of Lisman and Brown (15,16) and Lisman (17) that Ca2’ inside the cell indeed suppresses its response activity. The resulting differential equations for the concentrations ei, e, of the enzyme conformations El, E, now read de1 dt=

-J8+J,+Jlo,

2=

-2,

(4.7)

such that the sum e, + e2 is conserved. Note also that again the catalytic type of coupling between 5 and J8 does not interfere with the conservation relation (4.4) for the activator molecule.

V. Membrane Conduction Channels The final step of the construction of our network model is concerned with the ionic conduction channels across the membrane. Interpreting the ionic transport through the membrane conduction channels as a diffusion process and assuming Fick’s law (1.6) to be satisfied for it, the ionic flux across the membrane is given as J=D[q

exp (z)-c=

exp (z)},

where ci, c, and Vi, V, are the internal and external ionic concentrations electric voltages, respectively and use has been made of the substitution (1.5). Sometimes it is convenient to transform (5.1) into

zF( v-V,) II’ 1 [

zF( v - V,) -exp 2RT 334

-

2RT

(5.1) and rule

(5.2)

Bond Graph Representation of a Photoreception Model where

Z is a mean

V, is the Nernst

concentration,

F = (tic,)+, and the voltage scale has been chosen argument in the exponential expressions mate (5.2) by

potential,

RT c, V, = 2 In c_, I

(5.3)

such that Vi = V/2, V, = - V/2. If the in (5.2) were small, we could approxi-

J= a(V-

V,),

(5.4)

for V- V,=60 mV, T = 300 K, z = 1, the argument in with (Y- E. Although the exponential expressions in (5.2) is of the order of unity, version (5.4) turns out to be a useful description for quite a lot of ionic transport processes across membranes including the photoreceptor membrane. If one hesitates to adopt this kind of matching theory and experiment one should refer to (5.2) in its original form. For a number of ionic transport processes across membranes there is experimental evidence that D in (5.1) or CYin (5.4) is voltage dependent. We could account for this dependence in our network representation by introducing signal flows into the ionic version of the bond graph (1.7), i.e. C,

ci

\

l-

cieqt

c eve Ll

/

JRJ I/ \\ /’ ‘\ \ / ‘. .-__.’ eve.______.-”

2

(5.5)

eqc

where qi,e = FV,,,/RT. As stressed in the Introduction, this kind of just simulating the experiments does not satisfy our requirements on a model. Instead, let us try to interpret any dependence of D on other network parameters by channel dynamics as shown in the bond graph network in Fig. 4. In the network of Fig. 4, we have introduced an explicit capacitance A for free transport channels acting as catalysts for the diffusion process of the ion. As a consequence, the diffusion flux becomes proportional to the concentration a of free channels A. On the other hand, the free channels A are generated by an activation reaction R* from a non-activated conformation A* and vice uersa decay from A into A” by R-. If further molecules take part in R*, the effective diffusion constant D is no longer constant but a dynamical variable depending on the state of the network. If the activation reaction R* involves a spatial change of some charged molecule or of a charged group of a molecule, the effective diffusion constant D may become a function of the membrane voltage. This latter type of voltage-dependent conductivity has been proposed by Hodgkin and Huxley (18)in their model for nervous excitation. Vol. 308. NO. 3, September 1979 printed in Northern Ireland

335

J. Schnakenberg

and J. Tiedge Ci

e9i

‘1

9e ocie9i \~_Qk_t_~,

ce .9e

FIG. 4. Bond graph network scheme for membrane

channel dynamics.

This type of activation and decay of transport channels is precisely what we need to couple the active enzyme conformation E, to the ionic conduction channels. This coupling is shown in the network in Fig. 5. This network contains K+-conducting dark channels with a constant number and independent of any other network variable. The light channels A are catalytically activated by the active enzyme conformation E,. Note that again this catalytic coupling leaves the conservation relation for E unchanged, cf. (4.7). As regards the conduction properties of the light channels, we have assumed that the free A are permeable to both Na’ and Ca2’. Since internal Ca2+ also acts as a catalyst in the decay of active enzyme E, into its non-active conformation El, we now have established the negative feedback loop, as mentioned at the end of Section IV. This feedback loop is mathematically formulated by setting up the differential equations which follow from the network in Fig. 5 $=

k*E2a*-

k-a,

da* -_= dt

-- “,y, i

[Ca2+]i = -J&-

J& = D& - a - ([Ca2+]ie2qc - [Ca2+]ee2qe).

n[Ca”],,

(5.6) (5.7)

In (5.6), we have added a term -r)[Ca2+li not explicitly shown in the network of Fig. 5. It describes a pumping mechanism which eliminates Ca” from the interior and which must be present in the system since the actual internal Ca2+-concentration is kept at very low values. The Ca2+-flux has been formulated in the original version corresponding to (5.1) and may easily be transformed into the versions of (5.2) or (5.4). We could have added the equations for the time changes of [Na+li and [K’], but since voltage-clamp experiments 336

Journal of The Franklin Institute Pergamon Press Ltd.

Bond Graph Representation of a Photoreception Model [K’J

2% La 2+ lie-l-

FIG. 5. Bond

graph

network

D K -

di A!J

J

D&-

LK+&&

\

1 -[cc12+!&!

29,

of the dark channels for K’ and for the dynamics light channels of Na’ and Ca’+.

of the

have shown that the Nernst potentials of Na’ and K’ do not vary considerably during illumination and since NaC and K’ have no feedback effect we have omitted the eqjuations. As far as the description of the receptor potential is concerned, our model is complete. In the next section we shall describe the numerical results and compare them with the experimental data.

VL Numerical Calculations For an evaluation of the receptor potential V of our model we start by setting up the bond graph network of the electric circuit of the membrane as shown in Fig. 6. In Fig. 6, I denotes the total electric current passing the system and 1, is its capacitive part defined by

(6.1) since due to KVL at the l-junction

we have (6.2)

C is the electric capacitance per membrane area. The symbol {D} comprises the total electric flux as carried by diffusion of Kf, Na+ and Ca*+ across the Vol. 308, No. 3, September 1979 Printed in Northern Ireland 12

337

J. Schnakenberg

and J. Tiedge

FIG. 6. Complete

membrane,

electric

circuit of the membrane.

i.e.

43 = w,, where corresponding

+2&a + JK),

(6.3)

to Fig. 5 and Eqs. (5.6) and (5.7), we write FJ,,~FJ~,=gf;;a.(V-V,,)

2FJ,, = 2FJ& = g$, * a * (V - Vc,)

(6.4)

In these equations, we have chosen the version (5.4) of the ionic diffusion fluxes with V,,, V,-,, V, being the Nernst potentials of Na+, Ca*+ K+, respectively, since then the variable V occurs only linearly. Note also that we have introduced an extra light flux J& of K+ which is not included in the network of Fig. 5 and accounts for experimental evidence that the light channels are not only permeable to Nat and Ca*+ but to some extent also to KC. Finally, KCL at the O-junctions in Fig. 6, I=&+& completes

our equations. mV

With I = 0 for the receptor

(6.5) potential

experiments,

I

TIME/s

FIG. 7. Receptor potential after a 0.5 s light pulse for different intensities - 1.0, - 2.0, - 2.5, - 3.0, - 3.5, - 4.0 (arbitrary units) from the superior curve.

I; log Z = 0.0, to the inferior

Bond Graph Representation of a Photoreception Model mV

FIG. 8. Receptor

potential after a 0.01 s light flash; log I= 0.0, - 1.0, -2.0, - 3.0, - 3.5 from the superior to the inferior curve.

-2.5,

Eqs. (6.1), (6.3), (6.4) together with the differential equations of Sections III-V now represent a closed set of equations which may be solved numerically. In particular, we are interested in the time behaviour of V for different illuminations. There are two typical experimental situations, firstly a short flash lasting a few ms and secondly a light pulse of a few s duration. Figures 7 and 8 show the computer solutions of our model for both of these cases. In Fig. 7 the model receptor is illuminated with a 0.5 s light pulse. Parameter of the curves is the light intensity with values of 1 (arbitrary units) for the superior curve down to lop4 for the inferior curve. The curves show the well hm

FIG. 9. Values of the membrane potential at the receptor potential peak, h,, and at the end of illumination, h,, relative to the resting potential versus log I. Vol. 308, No. 3, September 1979 Printed in Northern Ireland

339

J. Schnakenberg

and J. Tiedge

known characteristics of the receptor potential, namely a peak following a latency time and declining to a plateau [Milecchia and Mauro (3), Stieve (7)].‘ Curves at lower light intensities are enclosed by those at higher intensities, and the latency time increases inversely proportional to the light intensity. Figure 8 shows the voltage responses to light flashes of 0.01 s duration at intensities of 1 for the superior curve down to 0.3 x 10e3 for the inferior curve. For such short light flashes the plateau phase of the receptor potential is absent which is in agreement with the experiments. Note also that due to the feedback character of our model network the duration of the voltage response exceeds that of the light flash by an order of magnitude. Figure 9 shows the peak and the plateau values h, and h, respectively, of the receptor potential curves as functions of the logarithm of the light intensity for pulse illumination. While the h, curve resembles the experimental results quite well, the experimental h, curve is less sigmoidal but more linear than that of our model. In any case, the h,, h, characteristics demonstrate how the receptor manages to transduce light signals within a range of five orders of magnitude into physiologically acceptable voltage responses within a range of 50 mV with respect to the resting value.

VIZ Concluding Remarks In the preceding sections, we have evaluated our model only with regard to its receptor potential behaviour. Beyond this phenomenon, our model also displays a further well-known property of the photoreceptor: adaptation. By this expression one denotes the capability of the photoreceptor to shift its h, -log I characteristic (cf. Fig. 9) towards higher intensities following illumination at high intensities. The adapted state may last over a period up to a few minutes and is due to an accumulation of internal Ca’+. Within our network model we easily can account for this accumulation with the material capacity Ca’+, cf. Figs. 3 and 5. Further significant experimental information is the electric current-voltage characteristic of the membrane under voltage clamp conditions in the dark or under illumination. As far as we have presented our model in this paper it would produce linear current-voltage curves in contrast to the experiments which show superlinear curves of a more or less rectifying type. It causes little difficulty to include this behaviour in our model by making the membrane conductivity voltage dependent in just that way as described in Section V and Fig. 4, i.e. by assuming charged or polar gating molecules controlling the channels. An experimental phenomenon which would require a major extension of our model is the so-called “bumps” i.e. potential fluctuations at very low light intensities. It is suspected that bumps are locally restricted responses which are too weak to spread out over the whole membrane. Following this explanation, the usual receptor potentials at higher intensities could then be interpreted as non-linear, i.e. mutually supporting superpositions of a great number of local bumps. It is clear that our model does not include such a mechanism since it is

340

Journal of ‘I’he Franklin Institute Pergamon Pmss Llci.

Bond

Graph

Representation

of a Photoreception

Model

homogeneous with respect to the spatial state of the membrane. In order to include bumps we would have to introduce cell or membrane compartments which may exist in different states and which are coupled to each other by lateral flows of molecules and electric charge due to lateral concentration and potential gradients. For all such extensions of the model as noted above, we again can design simple partial networks as described in this paper or in more detail by Schnakenberg (2) elsewhere. It turns out that the complete network of the model becomes increasingly complex. This means that the very advantage of the bond graph language mainly consists in the possibility to construct realistic and physically meaningful models for partial mechanisms. References (1) G. F. Oster,

A. S. Perelson

and A. Katchalsky,

Quart. Reu. Biophys., Vol. 6, p. 1,

1973. (2) J. Schnakenberg,

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