Rhodopsin cooperativity in visual response

CiGort Rc~t. Vol. I?. pp 35- 48. Pergsmon Press 1975. Printed in Great Britain

RHODOPSIN

COOPERATIVITY

IN VISUAL

RESPONSE

G. WILSE ROBINSON Arthur

Amos Noyes Laboratory of Chemical Physics, ’ California Pasadena, California 91109, U.S.A.

Institute

of Technology,

(Received 31 October 1973; in revisedform 28 February 1974) Abstract-A theory of vertebrate scotopic vision is proposed. The essence of the theory is that absorption of one photon in the disc membrane gives rise to transmitter release through cooperativity among a number (10-50, depending on vertebrate species) of unbleached rhodopsin molecules. Conclusions based upon the theory are in good agreement with both psychophysical and electrophysiological threshold experiments carried out in the absence of background illumination. Empirical incorporation of a generalized Weber-Fechner law allows the theory to be applied in the case of certain backgrounds.

NOMENCLATURE (Equation

numbers

give location

K where first used)

A

ai

distribution

coefficient

for i;

B

: G

I

Ii 1, AI

AIll 61; Y”I0 J J rndl

AJ

d’Jo ii 6ji

distribution coefficient for b‘ji total ion flow from rectangular light pulse I n (log, lo)- 1 incident light intensity incident intensity at ith rod intensity of rectangular light pulse threshold increment of intensity; i.e. the least increment of intensity necessary to stimulate a perceivable visual difference in the presence of background I the absolute threshold of increment intensity; i.e. the required AI when I = 0 z t,AI, increment of intensity at ith rod -= AI0 the rod sensory photocurrent over an area of the retina the maximum rod sensory photocurrent obtainable for a given retinal area; achieved for uniform intensity I = k/(n - 1)K threshold increment of sensory photocurrent; i.e. the least increment of sensory photocurrent necessary to stimulate a perceivable visual difference in the presence of background J = q”BI,,, the absolute threshold of increment sensory current; i.e. the required AJ when J = 0 the sensory photocurrent produced at the ith rod increment photocurrent at ith rod

’ Contribution

k (15)

;

(7a) (9)

n

(7b) (20) (17)

;I

I:; (19)

p0 Pmi.

(2)

Q’ q,

(104

,,

(8) (2)

ii *

(3)

;’

(18)

‘li 5i a

(3)

rate of rhodopsin bleaching per incident intensity rate of rhodopsin regeneration size of “rod pool” average number of coincident absorptions in the rod pool to excite the associated nerve fiber number of rhodopsin molecules in rhodopsin pool of critical size fraction of unbleached rhodopsin fraction of unbleached rhodopsin in ith rod fraction of unbleached rhodopsin when1 = Oandt - co ; p0 is essentially unity minimum fraction of unbleached rhodopsin molecules necessary for finite threshold proportionality constant in WeberFechner law for sensory photocurrent proportionality constant in ordinary Weber-Fechner law see q”I, number of rhodopsin molecules per rod time variable duration of rectangular light pulse proportionality constant between photocurrent and photons absorbed per second coefficients for geometrical pattern of incident light intensity on the retina coefficients for geometrical pattern of increment light intensity on the retina saturation constant in Weber-Fechner law.

I:; (13)

(13) (6) (1) (6)

(19)

(13) (3) (2) (2) (13) (5) (20)

(6) (15) (8. 9) (17)

1. INTRODUCTION

(3)

(6) (7b)

no. 4489.

Nearly 40 yr ago Hecht (1937) proposed a photochemical theory of vision based on simple bleaching and regeneration of photopigment molecules. However, measurement of rhodopsin bleaching (Dartnall, Goodeve and

35

Lythgo,

1938) and

its concentration

in the liv-

36

G. W~LSERORINSON

human eye (Campbell and Rushton, 1955; Rushton. 1956) showed the theory to be in error by many orders of magnitude. To understand the fact that “considerable changes in threshold or in electrophysiological signals can be achieved with little bleaching of the visual pigment”. Wald (1954) introduced his compartment theory of vision. Unfortunately Wald equated the compartments to the entire disc membrane, containing around IO4rhodopsin molecules. In addition Wald did not clearly delineate the consequences of his ideas when applied to dark adaptation on the one hand and background adaptation on the other (Rushton, 1969).It was easy therefore to extinguish completely this theory of visual responsei(Rushton, 1961). The purpose of this paper is to show how the photochemical bleaching and regeneration of rhodopsin can be synthesized into the psychophysical and electrophysioiogical characteristics of vertebrate scotopic vision. The ideas do not depend on the assignment of specialized roles to the various intermediates in rhodopsi!~ bleaching (Sillman, Owen and Fernandez, 1972; Abrahamson and Ostroy, 1967; Donner and Rueter, 1967; Ernst and Kemp, 1972), but only upon the cooperative release of transmitters from and a concomitant irractivation of a region of the disc membrane containing a number of rhodopsin molecules. The intermediates in rhodopsin bleaching may very well mirror important changes taking place in the rhodopsin molecules during the various stages of visual response, but for the purposes of this paper it is not necessary to spell these out.

The existence of cooperativity among the chromophorie macromolecules may stem partly from the same evolutionary source as it does in photosynthesis (Clayton. 1971). In photosynthesis, photon energy is absorbed within a pkotosyutketic urtit containing a few hundred chIorophyl1 molecules. This energy is then transferred by the physical process of resonance transfer (Robinson, 1966) from one molecule to another in the photosynthetic unit to a specialized location called the reactiorr center. Excitation of the reaction center then triggers the chain of redox reactions that guide the chemistry of photosynthesis. Thus. the ~oop~rat~vity among the chloropllyI1 molecules in photosynthesis provides a means of combining the advantages of a space consuming “chemical apparatus” with the increased light absorbing power of more than a single pigment molecule. It will be seen that this same general theme is repeated in the proposed model for visual sensitivity, namely the utilization of photon energy absorbed within a pool of rhodopsij~ molecules. Unlike photosynthesis the exact mechanism for cooperativity among the visual pigment molecules is unknown. Increased light absorbing power would also bc afforded by a pool of rhodopsin molecules. However. -’ In an attempt to simplify the discussion and retain clarity. unbl~ched rhodopsin will be referred to in this paper as rkobpsin (italicized) while any more general use of the word will omit the italicization.

and probably more important, the cooperative effect will be seen to make possible a broad range of visual sensitivity from only modest photo~hemi~al changes of the active pigment molecule.

2. THE VISUAL MECHANISM

The following scheme for vertebrate scotopic vision is examined. (a) Ahsorptiun qf Ii& by rkodopsin’ irl tke disc metnhrarw giz:rs rise to hlrackiruj of pigment (Wald, 1968) according to a purely photochemical equation which in its simplest form is (Ripps and Weale, 1969; Alpern, 1971) -dpjdt

= h;&) - k(l - p)

(1)

where p is the fraction of rkodopsirz, KI and k are the rates (set- ‘1 of bleaching and of regeneration, respectively, and I is the j~~ide~~tlight intensity. [In the human eye K 2 I.2 x lo-’ td-’ set- ’ and k z 24 x 1O-‘3set- ‘. The intensity unit troland (td) is roughly eqtrivalent to about 3 photons/rod per sec.] (b) Th
Rhodopsin

cooperativity

Fuortes and O’Bryan, 1971) and between neighboring horizontal cells (Westheimer and Wiley, 1970) are indicated. (e) Purullel with the ordinary Weher-Fechner luw (Barlow, 1973) Al = q’l + 4”1,, a law for total sensory photocurrent

(2) is postulated,

AJ = Q’J + q”J,.

(3)

Here AJ is the least increment of sensory photocurrent necessary to stimulate a perceivable visual difference in the presence of background J; q”J, is an “absolute threshold’ of sensory current. In general, Q’ need not equal q” because, under many experimental conditions, the currents AJ, J, and JO may arise from different areas of the retina. In fitting some experiments this may be thought of as a mere redefinition of the absolute threshold. [However, complications can arise (see Section 3) when very small areas of excitation are used to produce J and AJ.] A generalization of the ordinary Weber-Fechner law is necessary in order to take into account the fact that the magnitude of the physiological quantity J is limited by saturation while the physical quantity I can be increased essentially without limit by the experimenter. (F) Evidence points against the ordinary and thertlfore the generalized Weher-Fechner law having a photochernical origin (Stevens. 1972; Enroth-Cugell and Shapley’ 1973). Hence in this paper these laws will be empirically added onto the photochemical part of vision in order to obtain a complete theory of the role played by rhodopsin bleaching in the early visual response mechanism. (g) Many sources of saturation exist. One source is in the neurons (Werblin, 1971). In addition, sources of saturation exist (I) in the disc membranes corresponding to depletion of the proposed functional rhodopsin pools, and (2) in the plasma membrane corresponding to depletion of open Na+ channels (Penn and Hagins, 1972). For sufficiently weak light intensities, however, J has just the spatial and amplitude properties of the absorbed light intensity from which it derives. In fact, for sufficiently low light levels, J and 1 are related through a proportionality factor, J proportional

to I

(low light levels).

(4)

Only under the conditions of low light levels then is (i = Q’ in equations (2) and (3). Since rhodopsin bleaching and regeneration, transmitter production, transmitter diffusion, Na+ channel closing and regeneration, and interneural communication may all be occurring simultaneously in a given situation, the overall time course of the visual mechanism is expected to be quite complicated. Experiments should, therefore, be designed to reduce these complications by isolating if possible the changes caused by a single event when all others are held constant. Dark adaptation threshold measurements are a

in visual response

37

type of experiment which probes the visual machinery with a test light while it is in a passive state, i.e. when it is not simultaneously being driven by background illumination. Such measurements might therefore be expected to show how transmitter release varies over a range of rhodopsin concentration when Na+ dark current and neural activity are kept constant. Transmitter release is assumed here to be an early physiological event caused by light absorption (Hagins, 1972; Cone, 1973a).

3. THE THEORY

It can be shown by integrating equation (1) that starting at zero time (t = 0) after a full rhodopsin bleach @ = 0) and under a constant background illumination I, the fractional rhodopsin concentration regenerates according to the equation p = 1 _ ,-(k+Kr)t 1 + U/k

-’

(5)

Consider now the situation where, in the disc membrane, absorption of one photon within a pool having critical size n releases a fixed number of transmitters (possibly equal to n, the number of rhodopsin molecules in such a pool). Then, assuming a random distribution of bleached rhodopsin molecules over the area of relevance, p”’ is the probability that a photon absorbed by a rhodopsin molecule leads to transmitter release. (This is simply the probability that all the other rhodopsin molecules in the pool are in the unbleached state.) Providing there is a linear relationship between transmitters released and Na+ channels closed, the sensory photocurrent produced by a randomly chosen rod, say the ith rod, is, ji = yKI&

(6)

since KI,p, is proportional to the number of photons absorbed by that rod per second. The proportionality constant y in equation (6) contains the product of two “amplification factors”: one having to do with the number of transmitters released per photon absorbed and the other having to do with the decrease in Na+ dark current flow per blocked channel. A relationship such as equation (6) might be expected to hold at sufficiently low levels of transmitter release, i.e. either low background light intensities as in certain experiments of Fuortes, Gunkel and Rushton (1961), and Alpern, Rushton and Torii (1970a) or in threshold experiments in dark adaptation where only the minimum number of transmitters required for threshold is produced by a test light (Rushton, 1961, 1964). Psychophysical and electrophysiological dark adaptation experiments are usually concerned with threshold response to test flashes during regeneration of rhodopsirz after a total bleach and in the absence of background illumination. The data are expressed as relative thresholds, which are the thresholds relative to

G. Wr~sa ROBINSON

38

the absolute threshold (measured at “infinite” time in the dark when all the rhodopsin has regenerated). Going back to equation (3) the implication there is that the test flash has to supply a least increment AJ of sensory photocurrent to meet the threshold requirements of the system; and in the absence of background illumination (i.e. in the absence of any background current J) this increment is equal to a constant q”5, irrespective of the fraction of rhodopsin present. However, as the fraction of r~oaopsi~~ increases during regeneration, test flashes of reduced intensity are able to produce this same increment threshold of sensory photocurrent. In fact, for different rhodopsin concentrations l’i, production of a constant increment photocurrent ?iji at the ith rod illuminated with test flash intensity 61, requires that the product p:6fi must be a constant equal to 8jij,i’^iK. To proceed further one must assume linear relationships between J, the rod sensory photocurrent over an area of the retina; and the j;s; and between AJ, over possibly a different area of the retina, and the 6j;s J = c Uiji

(7d)

i

AJ = Ch&. (7b) i where the coefficients ai, bj = + 1 for all i in the simple case of no inter-rod interactions; but these coefficients may take on different positive, and even negative, values in more general cases. We are now prepared to calculate dark adaptation thresholds using the Weber-Fechner law for the sensory photocurrent [equation (3)]. In the absence of background illumination I = 0 and therefore J = 0 SO that one has only to worry about AJ and J,) in equation (3). A complication immediately arises however since the experimentalist is able to control the area over which a AJ is produced but has no control over f,,, the “dark current”. This “dark current” must arise from all the rods in the retina. However, what may be important in equation (3) is just the rods in a “rod pool” (not to be confused with the “rhodopsin pool”). There is ample experimental evidence that rods work together in pools. [See Rushton, (1964); and Barlow ( 1964) for a good discussion of the significance of the “rod pool” or “summation pool” as it is called by Rushton.] If a given “rod pooi” drives a single optic nerve fiber then it could happen that the “rod pools” behave independently even though the rods themselves do not. If this is the case, equation (3) must then be thought of as a threshold criterion for the “excitation” by a rod pool of its optic nerve fiber. In this case Jo would be the “dark current” associated with the nerve fiber and all its associated neurons and rod cells. In any case equation (3) can only be true in a simple sense for dark adaptation when the increment photocurrent AJ is produced by the experimentalist over the same area that the “dark current” is being intrinsically produced i.e. if the A1 illumination just covers a “rod pool” or is large enough so that many rod pools are

covered and “edge effects” may be ignored. If the patch ofillumination is smaller than a “rod pool” then we are asking only part of the rods in the pool to make up for the “dark current” of the whole pool! Ignoring these complications for the moment, we can obtain the simplest dark adaptation threshotd relationship, valid for large test lights, by combining equations (3) and (7b), with J = 0 and Sii = yKplGli.

Equation (8) is just a restatement of equation (3~namely, that in order to be detected the sensory photocurrent summed over all its rod contributions must exceed a certain fraction q” of the dark current. The point at which AJ is just equal to q”Jo is the threshold for dark adaptation. If in a threshold experiment the rhodops~n in all the rods is equivalently bleached and if the regeneration rate is the same for all the rods then p, at any time t is the same for all rods; i.e. pi = p for all i. In addition, if the threshold experiment is carried out so that only the intensity of the test flash is changed, but not the distribution of intensity over the retina, then &Ii = &Al for all i, where the ti coefficients define the geometrical pattern of the incident light intensity on the retina. These required experimental conditions are met in the usual dark adaptation experiment. Factoring out AI and p” from the summation in equation (8), and noting that the quantity

must be equal to the proportionality factor of equation (4), one sees finally that for ordinary dark adaptation experiments the relationship Al = q”JoB ‘p-”

c?

must hold between the threshold intensity of the test light and the fraction of rhodopsitz. If Al is given the symbol AI, when p = I, relative threshold in dark adaptation will be given by, AI/Ai<, = pm” = (1 _ e-i;r)--ri.

(lOa) (lob)

Equation (lob) follows from equation (5). since I = 0 in dark adaptation experiments. Note that 4

hiZi

is a constant for identical experimental illumination conditions and therefore does not affect relative threshold results even in more complicated cases, providing of course, that the linear relationships hold and the illuminated area is not changed during the entire course of the experiment. By fitting equations (lOa) or (lob) to experimental dark adaptation thersholds (for sufficiently large test lights), and knowing the value of k from pigment regeneration measurements, the critical size n may be

Rhodopsin cooperativity in visual response

39

-2

-6

TIME (min)

Fig. I. Dark adaptation after bleaching in the skate retina. The solid lines are the rhodopsin regeneration (bottom) and relative threshold (top), using the values k = n = 9. The open symbols are experimentally measured rhodopsin regeneration while spond to experimental thresholds for ERG (b-wave) or ganglion cell discharge, all periments given in Fig. 12 of Dowling and Ripps (I 970).

For instance, when t is long compared with km ‘, the value of eeLr is small compared with unity, and the approximation

found.

log,,AI/Al,

= n(log, IO)- ’ (1 - p)

(11)

is valid. A theoretical basis for the well-known empirical relationship between log relative threshold and rhodopsin concentration in dark adaptation experiments (Dowling, 1960) is thus achieved. The slope of the straight line plot of log relative threshold vs (1 - p) should then yield the critical size directly. PI = (log, 10) x slope. (12) Experiments by Rushton (1961, 1964) on human rod monochromats do give the expected straight-line relationship for rhodopsin fractions greater than about 0.5. The slope of these plots is about 20 and thus the value for 17for rod monochromats is around 46. (The value of n appears to be lower for persons with normal vision.) Using equation (12), electroretinographic hwave thresholds on rat retina yield an n of about 10 (Dowling, 1960, 1963). In frog the value of n (log, lo)- 1 is reported to be 8.56 from these kinds of measurements (Baumann, 1967), and thus n seems to be around 20 in frog. Apparently the critical size varies rather considerably with species. Over lOyragoRushton(l963)severelycriticizedequation (l&12) as they applied to Wald’s compartment theory of vision: As an rmpirica/,fact equation (1 l),fits rscellerltly the relatior2s studied irl rats and in man. But in these relations hleuching is not small but very large, and it is upon the jit with equation (lOa), not equation (1l), that the compartment hypothesis must stand or fall.

theoretical curves for 3.3 x 10e4 set- ’ and the solid circles correfrom skate eyecup ex-

In.fact, relation (10d) diverges greatly from observution, and when applied to the e.ycellent linear results ~$Dowling and Wuld (1960) reaches 10-mil~ion~f~fbld in the dispurity of’ threshold. Since that time, however, it has

been shown that albino rats have a small population of cones, which makes the Weber-Fechner law for rods appear to extend for many more log units than it actually does (Green, 1973). The presence of cones clearly casts doubt on the linearity of the relationship between log threshold and low percentage rhodopsin. Some recent results of Dowling and Ripps (1970, 1971) confirm the perfectly reasonable expectation contained in equation (1 Oa) that thresholds become infinite as rhodopsin concentration tends to zero. Their data have been plotted in Fig. 1 where they are compared with the theoretical curves (solid lines) for rhodopsin regeneration (bottom curve) and log relative threshold (top curve). Here k = 3.3 x 1O- 4 set- ‘, and n was found to be around 10. The expected strong deviation from the linear relationship between log threshold and (1 - p) is seen to occur in this case when p < 0.50. Dowling and Ripps (1970) tentatively ascribed this deviation to a mechanism of “neural origin”, since a similar effect (Dowling, 1963; Frank and Dowling, 1968) was found after intense bleaches when there is no pigment regeneration. See also Frank (1961). However, when accompanied by pigment regeneration, the behavior is totally consistent with the results of the theory presented here. Deviations from the straight line relationship between log AI/AI0 and (1 - p) in equation (I 1) also occur for humans. Experiments (Rushton. 1964; Alpern, Rushton and Torii, 1970b), for example, show

40

G. WILSE ROBINSON

a subject’s rod vision to be absent when half or less of the rhodopsin is bleached. This interesting aspect is clearly a nonlinearity inconsistent with the form of equation (I l), and will be discussed more quantitatively in the next section. On concluding this section, a few observations should be made about experiments with small test flashes. For this purpose, compare dark adaptation thresholds for 17” test flashes and 10’test flashes (Craik and Vernon, 1941). The lo’ test flash covers only about 10mh part of the retina, and is thus illuminating at most only 50 rods, a small fraction of a “rod pool”. Because of what was said earlier about test flashes of a size smaller than the size of a rod pool, the threshold for the 17” flash is always smaller than that for the IO flash. The presence of cones greatly complicates the rod thresholds at low p in the work of Craik and Vcrnon. However it is still rather obvious that the rod branches are not parallel; i.e. they are not related by merely a shift of scale (see Rushton. 1969). One reason for this can be found by referring ahead to Section IV. For small test flashes the rods are forced to work below their full “rod pool” complement. When the rhodopsirz concentration is low the probability that the understaffed pool can muster the required number of coincident absorptions to “fire” the sensory system is different than when the full pool is being utilized. In other words, the pmin [in equation (13)] below which infinite threshold occurs is different for the fully illuminated and the partially illuminated “rod pools”. With the two threshold curves bending upward to infinity at different points along the abscissa, there is no way that they can be related by a simple translation of scale. This seems to provide a tentative explanation of the curves depicted by Craik and Vernon (1941) (their Fig. IO). but experimental data in the absence of cone interference is needed before a complete analysis of this effect can be made. 4. PERSISTENCE

OF BLINDNESS AFTER BLEACHING

As mentioned in the last section, blindness persists following a total bleach long after much of the rhodopsin has regenerated. But how high does the rhodopsin concentration have to become before the threshold becomes finite? Taking Rushton’s (1961) rod monochromat as an example, where ft 1 46, 50’!<,rhodopsirl regeneration corresponds to a concentration of less than one part in lOI of the critical sized pools. (p” = 1.4 x lo-“, when fr = 46 and p = 0.50). But this is essentially zero. since an entire rod ‘can never contain more than around 10’ such pools! It is obvious therefore that the subject remains blind (infinite threshold) even after more than .500(,of his rhotlopsin has regenerated. This result is in agreement with observation, which indicates at a minimum 65::) of the rhodopsirl must regenerate in the rod monochromdt before the threshold becomes finite (Rushton, 1961, 1964). These thoughts about blindness following a rhodopsin bleach can be carried farther by assuming that a

rod is active if it contains at least one ~,/rodop.\ir~pool having the critical size. A simple theoretical analysis (Appendix A) may then be performed by which the critical size II and the minimum fraction )I,,,,,,of rlzotlopsir! necessary for finite threshold may be r&ted. log p,,,,,,=

I I1

‘,;;;’

log i

)

(13)

Here M. the size of the “rod pool”. is the number 01 rods on the average connected to the same optic nerve fiber, fn < M is the average number of coincident absorptions in the rod pool necessary to excite the nerve fiber, and R is the number of rhodopsin molccules per rod. Taking ffr 2 7 (Hecht. Schlaer and Pirenne, 1942; Barlow, 1964) M - 1000 (Rushton, 1964). andR = 2 x IO’ (Wald. 1954; Hagins. Penn and Yeshikami, 1970) and /I”,~,,= hS”,, gives II 2 42. A factor ol 10 uncertainty in the rn/MR ratio changes lr by only about 13 per cent. This value of fr for the rod monochromat is very close to the one we just obtained from dark adaptation threshold experiments, a fact which would seem very suggestive considering the two analy ses are totally independent. As mentioned previously. for a person with normal vision, the value of fr appears to he lower. According to the work of Alpern. Rushton and Torri (197Ob) the slope of the log,,, AI/Al,, vs (I - ~1)curve is 12. yielding 11= 28. Substituting 17= 28 into equation (I 3) with all other factors the same as for the rod monochromat shows that I),,,,,, should be around 0.52 for the normal human eye. This is in approximate agreement with Fig. 4 of Alpern. Rushton and Torii (1970b). which shows the “last” point of their log threshold curve to occur at p = 0.47. It is interesting that when 11= 9. as it appears to be for the skate (see Section 3), one can ostimate that ths minimum percentage of unbleached rhodopsin necessary for finite threshold in skate is IO IS. This estimate is consistent with the existence of threshold data for very low values of /I( 20.18) in the skate experiments. and indicates that the long rhodopsin regeneration time [k(man)/k(skate) 2 7.31 is no hindrance for this species in regaining rod sensitivity after a total bleach. That the calculated log,,, Al!AI, curve from the “exact equation” (1Oa) is not completely in agreement wrth experimental data is evident from Fig. I (top curve). where it is seen that the experimental points (solid circles) lie fairly far below the theoretical curve in the region where the curve begins to bend upward. For instance, the error in Fig. I at Xmin regeneration time is nearly an entire log unit. According to the data of Alpern, Rushton and Torii (1970b). the error is more than two log Lmits for the normal human eye. [When p = 0.50, equation (IOa) giv,es log,,, Al/Al,, = 84. while the observed value is about 60. The deviation is in the same direction as for the skate data.] Providing this difference is not just experimental error at these high relative thresholds, and there is no reason to believe that it is, the conclusion must be that equation (10a) is itself an approximation. But we already know

Rhodopsin cooperativity in visual response

that equation (lOa) does not give the full story. It predicts infinite threshold only when p is identically zero, while for finite sized rods our analysis at the beginning of this section shows that infinite threshold occurs for values of p much greater than zero. The answer therefore may lie in the breakdown of the simple statistical argument used to derive equation (lOa) when the distribution of unbleached rhodopsin becomes sparse, which in turn may depend on the intricate details of the structure and mechanism of r~~odopsjncooperativity in the rod cell 5. THE PENN-HAGINS

EXPERIMENTS

The time course for changes in Na’ current in isolated pieces of rat retinas following stimulus by a light pulse of short duration has been measured by Penn and Hagins (PH) (1972). These workers find that the decrease in Naf dark current saturates when about 100 photons per rod are absorbed from a 2 psec pulse. As pointed out by PH, this small a number of absorbed photons bleaches only a negligible fraction of the _ IO7 rhodopsin molecules per rod present. Neither can it excite a significant fraction of functional rhodopsin pools in the rod. Thus this type of ~turation can have nothing to do with rhodopsin bleaching. That the experimental results of PH are perfectly consistent with the scheme of visual sensitivity proposed here is easy to see. First of all it must be remembered that unlike dark adaptation experiments, the PH work is not carried out at threshold but rather is a series of experiments under various flash intensities. This is a fundamental difference as stressed earlier in this paper since there is no guarantee that a sensory photocurrent or a neural response is linearly related to transmitter release over a wide range of illumination conditions. Depending on flash intensity, some or all

41

of the Na+ channels become closed during the course of the experiment. The experiments of PH therefore probe not simply the release of transmitters, but this in combination with transmitter diffusion to and utilization at the plasma membrane, as well as the regeneration of Nat channels following closure by the transmitters. It therefore may not be possible to draw conclusions about the transmitter release mechanism from this type of experiment. Psychophysical increment thresholds under background illumination, such as those giving rise to the Weber-Fechner relationship, or the more complicated “windmill surround” experiments of Alpern, Rushton and Torii (1970a) are also of this latter type, except there is an added complication Neural activity also varies for different levels or geometrical patterns of background illumination. Looking now in more detail at the results of Penn and Hagins (1972) it is possible to understand the general forms of their experimental curves within the framework of the mechanism outlined at the beginning of the paper. The photovoltage response curves in Fig. 2 of PH contain most of the pertinent data. For nonsaturating light pulses the response curves show a steep rise followed by a slower decay, For saturating light pulses (2 100 photons absorbed/rod per pulse), the photovoltage response flattens out following the initial rise and remains constant for a time that lengthens with increasing flash intensity. The photovoltage then decays back to its dark current value. In the present model for transmitter release, groups of transmitters originate from functional r~odopsin pools which randomly happen to absorb a photon. The rise time for the photovoltage response curves therefore must have to do with the time required for creation of transmitters following the absorption act, with the diffusion time of transmitters from their origin in a disc membrane to the plasma membrane, and with

MINUTES

Fig. 2. Theoretical curves from equation (16) (solid lines) compared with experimental data (symbols) for relative thresholds measured during regeneration of rhodopsin in the presence of various background intensities (Rushton, 1961).

G.

42

WILSE

the time required to find the Na+ channels and to close them. Even though the Na+ channels are in low concentration (perhaps one per disc), all the relevant diffusion times are well within the observed rise time for dark current decrease (see Appendix B) indicating that the transmitter model as described here is tenable. Cone (1973a) has also discussed the plausibility of this model. One kind of saturation occurs when there are SO many transmitters in the vicinity of the plasma membrane that all the Na” channels become closed. The greater the intensity of the light, the greater is the number of transmitters in the vicinity of the plasma membrane. Above saturation, surplus transmitters must in some way “stand by” waiting for a Na+ channel to reopen. The Na ‘- channels therefore continue to remain effectively closed until all of the available transmitter is depleted. This apparently is what gives rise to the fiat-topped response curves of PH whose duration depends on flash intensity. As the transmitters are used up, the photovoltage decays back to its dark value. That there is an auxiliary mechanism for transmitter decay is evidenced by the time-integrated response curves of PH at higher Aash intensities, which show that the total charge flow in the photocurrent response to a bright Rash is less per absorbed photon than ii is for dimmer fishes. In uiuo an auxiiiary mechanism for transmitter decay might have to be highly effective in order to avoid prolonged bright afterimages resulting from intense stimuli and excess transmitter production. 6. BACKGROUND

tion (16) is the ordinary Weber-Fechner law except for the interesting factor p -” in the last term; p” is very near unity except in bleaching experiments or under high intensity backgrounds (I > 100 td), in which cases equation (16) deviates from the ordinary law of equation (2). The y”l,,p-” term in equation (16) displays a possible theoretical origin of the statement (Barlow and Sparrock, 1964): ‘The intrinsic noise of the retina, which is thought to limit the absolute threshold, can be expressed in terms of-dark light’, and it is natural to think of the equivalent background as an elevation of this dark light”. The effective dark light intensity I,, is indeed elevated to I,,/) IIwhen rhodnpsir~ is bleached. It is not possible at this time to offer an explanation of the origin of the positive after-image accompanying bleaching. Suffice to say that the noise in the receptors is raised from I,, to lt,p I’ as rhodopsin is bleached. What causes the noise and why this is interpreted as an after-image are not known. Various saturations. including excess transmitter production (Section 5). must play a partial role. The fact that “this positive after-image from bleaching (I,@“) is as bright as a real luminous background I that raises the threshold as much as this bleaching does“ follows because I and I,@” both enter equation (16) on the same footing. The AI required for threshold in this equation does not distinguish between “noise” coming from the background light and that coming from the bleaching. Equation (16) should be compared with an empirical one.

ADAPTATION

Even though the emphasis of this paper so far has been concerned with experiments where the background intensity is zero, use of the Weber-Fechner law for sensory photocurrent [equation (3)J allows experiments with certain backgrounds also to be described. One will he interested in the visual excitation under background illumination and the increment of total sensory photocurrent caused by a flash of light. From equations (6) and (7) and the same conditions that led to equation (9).

where analogous

ROBINSON

A.J = BpnAl

(14)

f = .4p”f

(15)

to B, A = ?.‘K~U,~,. I

Equations (14) and (15) are based, of course, on the assumption that the sensory photocurrent is proportional to the number of transmitters released (i.e. when there is no Na’ channel saturation). Combining equations (14) and (15) in equation (3) one can write Al = q’l + y”l,p-”

(16)

where y’ = Q’AIB and q”I, is the absolute threshold. It has been assumed here that pfl + Al) = p(I). Equa-

suggested by Alpern, Rushton and Torii (197Ob) to explain “all” bleaching-background threshold experiments including “windmill surround” experiments (Alpern. Rushton and Torii. 197Oa) as well as the equivalent background experiments (Barlow and Sparrock, 1964). For ease of comparison, equation (17) has been rewritten from equation (2) of Alpern, Rushton and Torii ( 1970b) by changing H--t 1. 0,) -+ y”l,,/y’. d + AI. log, () h = G( 1 - p). R:+ q”I,/cr for thresholds, and by noting that alwaysq” I,,/n $ 1. Equations(l6)and (17) are identical under experimental conditions where R : 10”“~~p’and c[l;n : 0. The applicability of the P first approximation has already been discussed in Section 3. The second approximation breaks down under a sufficiently high background intensity where the ordinary Weber-Fechner law “saturates”. Figure 2 shows how well theoretical curves from equation (16) fit some data obtained by Rushton (1961) where relative thresholds are measured during rhodopsin regeneration in the presence of background light. Another point should be raised here. Professor Rushton (private communication) has reminded the writer of an important observation in the experiments on the rod monochromat (Fuortes, Gunkel and Rushton, 1961). If 3.5 log td of background “saturates” and 2.5 log td does not. and there is a very strong test flash

Rhodopsin cooperativity in visual response every 2 set, then interposing a 1.0 density filter successively in and out of the 35 log td beam causes the monochromat to say “yes, no, yes.. .” with respect to seeing the test flash. The responses are immediate, so there is no possibility of this effect being associated in any way with the regeneration of rhodopsin molecules, which is much too slow. Even though the rho&sin concentration may not have time to reach any sort of photostationary state, it is nonetheless being bleached at a rate consistent with equation (1). The fraction of rhodopsin consumed may be quite small depending upon the light intensity and length of the experiment. However, what is of importance here is that the sensory photocurrent must change essentially instantaneously as the light intensity varies. One can see that this is the case theoretically because of equations (14) and (15).(J is dependent not only on p, whose changes may not be significant, but on I itself, which can be rapidly changed by the experimentalist.) The result in the above experiment with the rod monochromat is thus merely a consequence of the generalized WeberFechner law [equation (3)], which says that for an increment AJ to be detected it must be at least equal to Q’ x J (ignoring the absolute threshold term). When the background intensity is “instantaneously” raised from 2.5 log td to 3.5 log td, J increases “instantaneously” [limited only by the temporal response of the eye], and AJ is simply not sufficiently large to be detected anymore. When the background light is switched back to 2.5 log td, J drops “immediately”’ and AJ is again detectable, etc., etc. If the experiment continues for a very long time (of the order of minutes) it could happen that p would finally reach a photostationary state corresponding to a background intensity between 2.5 and 3.5 log td. Under such a condition, it may then not be possible for AJ to be detected above the 2.51og td background. Even at background intensities below Weber-Fechner saturation, experimental results of the type just discussed would be found. Switching background intensity back and forth between 0 and + 1 log td when the test light is just above threshold for 0 log td would be an example. 7. SOME SATURATION

43

to the relationship equation (I 5). The intensity may be administered in two ways. It may come from a constant background which implies that the visual pigment system has reached the photostationary state, dp/ dt = 0; or it could come from a brief flash, short compared with the time resolution of visual response and therefore also short compared with rhodopsin regeneration times. The constant background case yields data that pertain to the Weber-Fechner law. Using the photostationary state value of p one can easily see that, AI J = (1 + KI/k) where I is a time-independent background intensity. The equation shows that at low I, J is linear with I having proportionality constant A. This occurs in the linear range of the Weber-Fechner law. As I increases further, the linear dependence bends over, J reaching a maximum when I = k/(n - l)K. Then J decreases with further increase in background intensity. This is illustrated in Fig. 3. Equation (16) which relates AI and I, does not hold at background intensities so high that the maximum in J is approached, since the approximation p(I + AI) _ p(I) used to derive this equation breaks down. In order to find the relationship between AI and I under such conditions one must use equation (18) or better a plot of equation (18) as shown in Fig. (3). He then should choose the appropriate background intensity, and see which value of J this corresponds to. He can then determine, say from the plot, how much more intensity is required to give a AJ which meets the threshold condition on J[equation (311. A saturation in the

,

I

I

I

400

600

800

EFFECTS

The q’l/u term in the denominator of equation (17) empirically takes care of the upturn caused by saturation in the log AI vs log I curve (Aguilar and Stiles, 1954). This occurs in the rod monochromat near log I(td) = $2 (Fuortes, Gunkel and Rushton, 1961). Penn and Hagins (1972) have suggested that this saturation effect on the Weber-Fechner law might be saturation of the rod signal due to closure of all the Na+ channels. It is conceivable though that the effect is caused by a depletion of critical-sized rhodopsin pools when the background light intensity becomes sufficiently high that a moderate amount of rhodopsin is bleached. That equation (3) leads to saturation in the WeberFechner law is best seen by plotting J vs I according

0

PO0

I(td) Fig. 3. The theoretical sensory photocurrent J(I) plotted against I in trolands in the photostationary state with n = 45. The value of AI necessary to obtain the required fractional increase in sensory photocurrent for threshold may be obtained graphically from this type of curve. See text.

G. WILSE ROBINSON

44

ration of rhodopsin, the experimental tl should be much less than a millisecond. PH used 2psec for all but their most intense flashes. During the time when the flash is on, rhodopsirt bleaches according to, I’ = p<,e- hr”t

(19)

and the total ion flow from equation J dt = (,+$/nK)(l

I

-m

I

-4

I

-2

I

I

0

+2

logI (td) Fig. 4. Theoretical log AI vs log I curve with II = 45. The solid circles are data for a rod monochromat (Fuortes, Gunkel and Rushton, 1961). Note that the log-log plot allows the origin of the ordinate to be arbitrarily adjusted. The part of the curve at low intensities [log I(td) < 1.51 is given by the ordinary Weber-Fechner law, equation (2). It is therefore the agreement between theory and experiment in the high intensity region that is most relevant. At higher background intensity than that given by the dashed vertical line. the increment threshold is predicted theoretically to become

infinite. law is thus seen to arise, since because of the maximum in J there eventually comes a point where the AJ for threshold can no longer be met no matter what increase in I is chosen. The rods are then blind to increments and lose contrast, giving rise to a “dazzling” sensation. The rods are not blind, however, to the background itself until much higher intensities arc reached, since J is non-zero. Figure 4 illustrates the theoretical results (solid line) of these concepts for a rod monochromat (Fuortes, Gunkcl and Rushton, 1961) where k 2 2.4 x lO-” see 1 and K : 1.2 x lO_’ td-’ set-‘. J,,, occurs saturation hem when I 2 450 td, thus Weber-Fechner must occur at a lower intensity. Comparison of this theoretical curve with the experimental data (Fig. 4) should be noted, particularly with respect to the onset of saturation, which is dependent upon the value of n chosen. While the above analysis provides evidence that the saturation effects in the Weber-Fechner law are caused by depletion of rhodopsin pools having critical size, the saturation observed by Penn and Hagins (1972) on sensory photocurrent produced by flashes of light is certainly not caused by the latter. Consider, for mathematical simplicity, the flash to have rectangular shape, turning on abruptly at time zero, remaining on for a short duration with constant intensity I,, then shutting off abruptly at a time t,. To avoid complications because of regeneration of Na+ channels or regeneWeber-Fechner

(15) is given by

- e ~“K’,~~r~). (20)

For a fixed pulse width t,, a plot of c vs I,tf, the photon energy administered to the system per unit area, would yield an initial linear dependence; then (j?bends over and saturates with half maximum occurring when nKl,t,. = 0.69. Since K, which depends on the absorption probability of the ROS, is of the order of IO-’ rods per quantum for all species of vertebrates, then I,tf would have to correspond to hundreds of thousands of quanta per rod. The PH experiments show that saturation is reached at light levels around three orders of magnitude lower than this. Thus saturation in the flash experiments cannot be caused by rhodopsin bleaching, but must arise from something else. presumably Na’ channel depletion. A similar conclusion was stated by PH. Why isn’t Weber-Fechner law saturation also caused by Na+ channel depletion’! One must remember that the steady background experiments are quite different from the flash experiments in that many more quanta are being delivered to the system in the steady background experiments, and a photostdtionary state with respect to both rhodopsin concentration and Naf channels is set up. One should not simply compare the peak intensities at which saturation occurs in the two cases. Regeneration of Na’ channels is apparently fairly fast; the experiments (Penn and Hagins, 1972) imply k 2 1 set-‘, while the transmitter production rate (in the photostationary state when I -. 200 td) is comparatively slow (< 0.1 set- ‘). Thus the repair of Na+ channels is fast enough to keep up with the flow of transmitters from the discs at these intensities, and the Weber-Fechner saturation is very likely caused by the depletion of critical-sized rhodopsip pools as implied in Fig. (4).

8. CONCLUSION3 The author realizes how difficult it would be to resurrect an old theory, such as Wald’s compartment theory of vision (Wald, 1954). which died so many years ago. Simply fitting the corpse with new clothes cannot instill it with life. The theory presented here, however, is quantitatively quite different from Wald’s theory, even though cooperativity among rhodopsin molecules is the essence of both theories. The size of rhodopsin pools is much smaller here than in Wald’s theory. Empirical incorporation of the Weber-Fechner law into the theory, rather than an attempt to explain it photochemically, is a different feature. Even though an explanation of the origin of

Rhodopsin cooperativity in visual response the Weber-Fechner law would be an important advance in itself, it is not the purpose of this paper to concern itself with this. It is felt, however, that a precise delineation of the photochemical effects in vision is a necessary step in the eventual understanding of this important law. The clear differentiation between dark adaptation and background adaptation is crucial to this paper, but was not made in Wald’s work. The smooth synthesis of the photochemistry into the psychophysics of certain complex backgrounds (Alpern, Rushton and Torii, 1970a, 1970b) on the one hand and the electrophysiological results of Penn and Hagins (1972), and Dowling and Ripps (1970, 1971) on the other, speaks well for the theory. A partial understanding of the “equivalent background” concept (Barlow and Sparrock, 1964) could be an important result, as is the theoretical origin of the relationship between relative threshold and unbleached pigment concentration in dark adaptation. The usual functional form of this latter relationship changes at low rhodopsin concentration, and the explanation of this deviation by the theory removes any necessity for a more complex picture (Dowling and Ripps, 1970). Saturation of the Weber-Fechner law at high background intensities may be explainable by the theory. The reason for infinite thresholds (blindness) even when rhodopsin concentrations are relatively high is also an interesting result, being sharply dependent on the critical size of the rhodopsin pools. Finally, the near exact agreement between critical size obtained from such considerations and that obtained totally independently from the threshold-pigment relationship in dark adaptation is certainly very suggestive. Thus the present theory does much more than just fit the critical size II of the rhodopsin pools to a few curves. Since the inception of this theory early in 1971, many relevant results have been reported in the literature. The recent experiments (Brown, 1972; Cone, 1972) indicating not only rotational but also translational diffusion (Cone, 1973b) of rhodopsin molecules in the disc membranes destroy any simply solid state picture of cooperativity as envisioned by Changeux and Thiery (1968).It is not felt, however, that the Cone results destroy the idea of cooperativity. The cooperative release of transmitters from a pool of rhodopsin molecules followed by a period of inactivity until bound transmitters are reestablished certainly seems a plausible chain of events whether associated with a fluid or a rigid membrane. In fact the diffusion of rhodopsin may be directly related to the cooperative process either (1) because diffusion provides the necessary many-body intermacromolecular interaction or (2) because disorder and melting result from the cooperative interaction following the absorption of a photon. Since 1971, the curve of log threshold vs rhodopsin concentration has taken the upward turn which had been predicted at low rhodopsin concentration. The conclusion of Dowling and Ripps (1971) from their Spotential studies that the seat of dark adaptation

45

resides in the rods is, of course, essential. See also the recent paper by Witkovsky, Nelson, and Ripps (1973). In addition, biochemical evidence for some type of cooperativity has appeared in the literature (Bowndq Dowes, Miller and Stahlman, 1972), where in the frog ROS, absorption of a photon leads to the utilization of l&50 ATP molecules. Clearly the slow time course for phosphorylation eliminates it as a step in the primary visual response mechanism, but the experiments certainly raise the possibility that one photon does something to a number of rhodopsin molecules (uptake of protons? release of bound transmitters?) necessitating later repair by ATP. See also .Emrich (1971). Hints are also being made (Yoshikami and Hagins, 1973) that perhaps l&100 Ca2+ ions are released per photon absorbed in the ROS. If these results can be traced to the action by a single photon over a region containing a number of rhodopsin molecules, this is of course just what the theory is talking about. Note added in proof: Recently Aaron Lewis and his group at Cornell University have obtained tunable laser resonance Raman spectra ofrhodopsin in the live eye of a New Zealand white rabbit while simultaneously monitoring the electrophysiological responses of the animal They found that there are discrete structural changes in unbleached (!) rhodopsins when an electroretinogram can be detected. These changes are directly correlated to the magnitude of the electroretinogram and appear to be reversible.

Acknowledgements-The author wants to thank the John Simon Guggenheim Memorial Foundation for support that allowed him to separate himself from the mundane duties of everyday academic life so that he could interact privately with his own thoughts. The 1971 Guggenheim Fellowship took him to near antipodes of the earth-Sheffield, England and Christchurch, New Zealand-where many of the ideas entered here were crystallized. The hospitality of the Chemistry Departments at both Sheffield and Canterbury Universities is sincerely acknowledged. REFERENCES

Abrahamson E. W. and Ostroy S. E. (1967) The photochemica1 and macromolecular aspects of vision. Prog. Biophys. molec. Biol. 17, 179-215. Aguilar M. and Stiles W. S. (1954) Saturation of the rod mechanism of the retina at high levels of stimulation, Optica Acta 1, 59-65.

Alpern M., Rushton W. A. H. and Torii S. (1970b) The Physiol., Lond. 217, 447-471.

Alpern M., Rushton W. A. H. and Torii S. (1970a) The size of rod signals. J. Physiol., Land. 206, 193-208. Alpern M., Rushton W. A. H. and Torii S. (1970b) The attenuation ofrod signals by bleachings. J. Physiol., Loud. 207,449-461.

Barlow H. B. (1964) The physical limits of visual discrimination. In Photophysiology (edited by Giese A. C.), Vol. II, Chap. 16, pp. 163-202. Academic Press. New York. Barlow H. B. (1973) Dark and light adaptation: psychophysics. In Handbook of Sensory Physiology, Vol. VII/4 Chap. 1, pp. l-28, Springer, Heidelberg.

46

G. WILSE ROBINSON

Barlow H. B. and Sparrock J. M. B. (1964) The role of afterimages in dark adaptation. Science, Wash. 144, 1309-l 3 14. Baumann C. (1967) Bleaching visual purple and rod function in the isolated frog retina-III: Dark adaptation of the scotopic system after partial bleaching of visual purple. F’jliiycrs Archs yes. Physiol. 298, 70-81. Baylor D. A., Fuortes M. G. F. and O’Bryan P. M. (1971) Receptive fields of cones in the retina of the turtle. J. Physiol., Loud. 214, 265-294. Bownds D., Dawes J., Miller J. and Stahlman M. (1972) Phosphorylation of frog photoreceptor membranes induced by light. Nature, New Biol. 237, 125-127. Brown P. K. (1972) Rhodopsin rotates in the visual receptor membrane. Nature, New Biol. 236, 3538. Campbell F. W. and Rushton W. A. H. (1955) Measurement of the scotopic pigment in the living human eye. J. Physiol., Lond. 130, I3 l- 147. Changeux J.-P. and Thiery .I. (1968) On the excitability and cooperativity of biological membranes. Regularity Functions ofBiological Membranes, Sigrid Juselius Foundation Symposium on Regulatory Functions of Biological Membranes (edited by Jarnefelt J.), pp. 116138. Elsevier Publishing Company. Amsterdam. Clavton R. K. (1971) Light and Living Matter, Vol. 2: The biological part. Chap: 1. MacGrawIHill, New York. Cone R. A. C1972) Rotational diffusion of rhodoosin in the visual receptor ‘membrane. Nature, New Biol. 236, 39-43. Cone R. A. (1973a) The internal transmitter model for visual excitation: Some quantitative implications. Biochemistry urzd Physiology of Visual Pigments, pp. 275-282. Springer, New York. Cone R. A. (1973b) In preparation. Craik K. J. W. and Vernon M. D. (1964) The nature of dak adaptation. Br. J. Psychol. 32, 62-S I. Crank J. (1956) The Mathematics qf’D@usion. Chaps. 5 and 8 Clarendon Press, Oxford. Danielson J. T. (1973) In search of the neural channel which codes real light and equivalent backgrounds. Vision Res. 13, 383.-391. Dartnall H. J. A., Goodeve C. F. and Lythgoe R. J. (1936) The quantitative analysis of the photochemical bleaching of visual purple solutions in monochromatic light. Proc. R. Sot. Land. A, 156, 158-170. Donner K. 0. and Rueter T. (1967) Increment threshold and dark adaptation. Vision Res. 7, 17-41. Dowling J. E. (1960) Chemistry of visual adaptation in the rat. Nuturr. Lornl. 188, 114 118. Dowling J. E. (1963) Neural and photochemical mechanisms of visual adaptation in the rat. J. gen Physiol. 46, 1287-1301. Dowling J. E. and Ripps H. (1970) Visual adaptation in the retina of the skate.~j. gen. Physiol. 56, 491-520. Dowling J. E. and Rioos H. (1971) S-notentials in the skate retina Intracellular’ recordings during light and dark adaptation. J. gen. Physiol. 58, 1633 I X9. Dowling J. E. and Wald G. (1960) The biological function of vitamin A acid. Proc. n&n. Acud. Sci. U.S.A. 46, 587 608. Emrich H. M. (1971) Primarv processes in vertebrate photoreception-cooperativity -in the disc-membrane. First European Biophysics. Congress, Sept 1417, Baden, Austria. Abstract YIIA/3, pp. 6571. Enroth-Cueell C. and Shaolev R. M. (1973) Adaotation and dynamics of cat retina< ganglion cells. J. physiol.. Lorld. 233, 271-309. Ernst W. and Kemp C. M. (1972) The generation of the late

receptor potential: an excitation-inhibition phenomenon. Vision Res., 12, 1937-1946. Frank R. N. (1971) Properties of “neural” adaptation in components of the frog electroretinogram. Vision Res. 11, 1113~1123. Frank R. N. and Dowling J. E. (1968) Rhodopsin photoproducts: Effects on electroretinogram sensitivity in isolated perfused rat retina. Science. Wash. 161, 487-489. Fuortes M. G. F., Gunkel R. D. and Rushton W. A. H., (1961). Increment thresholds in a subject deficient in cone vision. J. Physiol., Lond. 156, 179-192. Green D. G. (1973) Scotopic and photopic components of the rat electroretinogram. J. Physiol., Lond. 228, 781-797. Hagins W. A. (1972) The visual process: excitatory mechanisms in the primary receptor cells. Ann. Rev. Biophys. Bioengng 1, 131. 158. Hagins W. A., Penn R. D. and Yoshikami S. (1970) Dark current and photocurrent in retinal rods. Biophys. J. 10, 3X0-4 12. Hecht. S. (1937). Rods, Cones, and the chemical basis of vision. Physiol. Ret:. 17, 239-290. Hecht S., Shlaer S. and Pirenne M. H. (1942). Energy, quanta and vision. J. yen. Plzysiof. 25, X19-840. Knox R. S. (1968) On the theory of trapping of excitation in the photosynthetic unit, J. theor. Biol. 21, 244-259. Marmarelis P. Z. and Naka K.-I. (1972) White noise analysis of a neuron chain: an application of the Wiener theory. Science, Wash. 175, 127&-1278. Ostroy S. E. (1969) Photolysis and regeneration of visual pigments. E.~pl Eye Res. 8, 243. Penn R. D. and Hagins W. A. (1972) Kinetics of the photocurrent of retinal rods. Biophys. J. 12, 107331094. Ripps H. and Weale R. A. (1969) Flash bleaching of rhodopsin in the human retina. J. Physiol.. Lond. 200, 151~ 159. Robinson G. W. (1966) Excitation transfer and trapping in photosynthesis. Brookhave>c Symp. Biol. 19, 16-48. Rushton W. A. H. (1956) The difference spectrum and the photosensitivity of rhodopsin in the living human eye. J. PliysioL, Lorid. 134, 1 I -29. Ruston W. A. H. (1961) Rhodopsin measurement and darkadaptation in a subject deficient in cone vision. _r. Physiol., Loud. 156, 1933205. Rushton W. A. H. (1963) The effects of rhodopsin decomposition on PI11 responses of isolated rat retinae. J. opt. Sot. .4m. 53, 104-109. Rushton W. A. H. (1964) Vision as a photic process. Photophysiology (edited by Giese A. C.) Vol II, Chap. 15 Academic Press, New York. Rushton W. A. H. (1969) Light and dark adaptation of the retina. In The Retina (edited by Straatsma B. R.. Hall M.O., Allen R. A. and Crcscitelli F.). University of California. Berkeley and Los Angeles (1969). See also the exchange between G. Wald and W. A. Rushton, pp. 253. 255. Sillman A. J., Owen W. G. and Fernandez H. R. (1972) On the mechanism of the visual threshold and visual adaptation. Vision Rrs. 12, 1519.m1531. Stevens S. S. (1972) A neural quantum in sensory discrimination. Science, Wash. 177, 749-762. Wald G. (1954) The photochemical and macromolecular aspects of vision. Science, Wash. 119, 887-892. Wald G. (1968) Molecular basis of visual excitation. Science. Wash. i62, i3Cb-239. Werblin F. S. (1971) Adaptation in a vertebrate retina: intracellular recording in Necturus. J. Neurophys. 34, 228-241.

Rhodopsin

cooperativity

Westheimer G. and Wiley R. W. (1970) Distance effects in human scotopic retinal interaction. J. Physiol., Lond. 206, 129-143. Witkovsky P., Nelson J. and Ripps H. (1973) Action spectra and adaptation properties of carp photoreceptors. J. hen. Physiol. 61, 401-423. Yoshikami S. and Hagins W. A. (1973) Control of dark current in vertebrate rods and cones. J. gen. Physiol. 61, 245% 255. Zuckermann R. (1972) Photoreceptor dark current: role for an electrogenic sodium pump. Biophys. Sot. Abstr., 16th Annual Meeting, Toronto, Canada, February 24-27, 1972. Abstract Sa Am-B8. p. 101a.

APPENDIX A

Equation (13) is derived by determining the probabilities that a barely sufficient number of functional rhodopsin pools exists in the entire rod pool. When the fraction of unbleached rhodopsin molecules is P,,,~., then p,$, is the fraction of functional rhodopsin pools, and I - p& is the fraction of nonfunctional pools. Thus the fraction of nonfunctional rods is (1 - p&.)R’“, and the fraction of functional rods is 1 - (1 - p&,)R/“. In order to excite m out of M rods in the rod pool, pmin would have to be large enough such that m/M = 1 - (1 - pz,,)R’“. from which equation

(13) follows.

APPENDIX B

Hagins (1972) suggests that in the dark the extradisc cytoplasm in the rat ROS is c 10-s M in Ca’+ ion, which works out fo about 35 Ca” per rod assuming 10 per cent extra disc cytoplasm content by volume per rod. After a half-saturating light flash of 3&50 photons absorbed per rod, the Ca’+ concentration in the extradisc cytoplasm is supposed to jump IO-fold. Taking Ca2+ to be the transmitter, this implies that around IO transmitters are produced per absorbed photon. The Ca *+ ions then diffuse to the plasma membrane where they “hunt for” available Na+ channels. From the saturation data it can be presumed that there are only around .lOOO Na’ channels in the entire rod, roughly one per disc. The question then is: “If diffusion, i.e. ‘hunting’, were the slow step in the overall process, how fast could the transmitters combine with the Na+ channels and disappear?” If this time turns out to be substantially slower than the rise time in the photovoltage response measured by Penn and Hagins (1972), then rather serious doubts about the integrated picture presented in this paper would arise. Estimates will show that the overall “hunting” time is comparable or shorter than the photovoltage rise time indicating no lack of selfconsistency in the model. It is pretty well accepted that the stimulus for the late receptor potential is generated during the meta I to meta II transformation (Ostroy, 1969). This is essentially terminated in 100 psec (Abrahamson and Ostroy, 1967) and therefore does not affect these calculations. D$fiaiofl

to rquilihrium

The limit where radial diffusion is very rapid compared with longitudinal “hunting”, is easy to justify because the ROS is much longer than its diameter. Diffusion of transmitters to the plasma membrane from a localized point

in visual response

47

between two discs may then be approximated by radial Fick’s law diffusion in a cylinder of radius R. with impermeable surface. The solution of

ac _ = DV2C

at

is (Crank,

1956)

C(r, t) = 2R;’

I<,, r’F(r’) dr’ [i 0 r’F(r’)JO(r’r,) dr’

(B2)

where F(r) = C(r, 0); - DdC/drl,, = 0; and the Bessel function arguments u,, are found from the roots of J,(R,cc.) = 0. (R,,a, = 3.8317, R,a, = 7.0156, R,n, :, R,,r, for n > 2.) For a rat rod R. - 0% pm (Hagins, Penn and Yoshikami, 1970). Using the typical diffusion coefficient D = lo-’ cm’ see-’ for an ion, one can see that the slowest time dependent term [n = 1 in equation (B2)] approaches within 5 per cent ofequilibrium in 10m4 sec. This is much faster than the transients reported by Penn and Hagins (1972) and therefore cannot be rate determining in Na+ channel closing. Using similar methods it can be shown that it requires about 100 times as long for 95 per cent equilibrium to be reached by axial diffusion. The length of the ROS in rat was taken to be 25 pm (Hagins, Penn and Yoshikami, 1970). The time to reachequilibrium by axial diffusion is faster by about a factor of ten than the transient times reported by PH. Of course, one might want to reduce the axial diffusion coefficient by the ratio of the area of free axial space to total cross section of the ROS to take into account the impeding effects of the discs. It is unclear just what this ratio should be (10 per cent, perhaps less). One is, however, not so much interested in axial diffusion times as in the time required for transmitters to search out and react with Na+ channel sites. This time will now be estimated. For this estimate we make the assumption that transverse diffusion is infinitely fast. This assumption causes the problem to revert to a two-dimensional random walk trapping problem, such as the one arising in the trapping of an exciton in the photosynthetic unit (Robinson, 1966; Knox, 1968). Take the active site of the Na’ channel to be 5 A dia (probably too small a value, but chosen since this approximates the diameter of a hydrated Cal+ ion). Now visualize the plasma membrane surface as a checkerboard rolled into a cylinder. The squares of the checker board are chosen to be 5 A on a side to match the size of the active site. There are thus about 5 x 10’ squares in the checkerboard, of which 1000 are active sites. It is assumed that the active sites are evenly distributed over the checkerboard. The number of random jumps required for l/e of the transmitters to react with Na’ channels is then given by (Robinson, 1966). 0720 .,+’ log,,

+ + 0.259 f ‘.

(B3)

Here &” is the ratio of total sites to the active sites: in this case _+^ = 5 x 105. The time r per jump can be obtained from the diffusion constant, T = (A.u)‘/2D = 1.25 x lo-”

set,

(B4)

and thus the time to react with l/e of the transmitters is 2.7 x 10e4 set, providing the Na+ channels do not become substantially depleted. Near saturation, the channels do get depleted, of course, and reaction times slow down, but even when half the channels are closed, reaction times are still less than a millisecond.

4x

G. WILSE ROBINSON

These estimates then indeed provide evidence that the transmitter idea as used in this paper is consistent with dif-

fusion times. Similar Cone (1973a).

arguments

have been presented

propose une theorie de la vision scotopiques des vertebres. L’essence de cette theorie cst que l’absorption dun photon dans la membrgne dun disque produit l’emission dune substance transmettrite par cooperation dun certain nombre de molecules de rhodopsine non dtcoloree. de 10 a 50 selon l’espece de vertebre. Les conclusions bakes sur cette theorie sont en bon accord avec les experiences tant psychophysiques qu’electrophysiologiques de seuil en l’absence d’eclairage du fond. En incorporant RCumbOn

empiriquement fonds.

une loi de Weber-Fechner

generalisee,

on peut appliquer

la thtorie

dans le cas de certains

Zusammenfassnng-Es wird eine Theorie des skotopischen Sehens der Wirbeltiere aufgestellt. Der Kern dieser Theorie ist, dass die Absorption eines Photons in der Rezeptormembran durch das Zusammenwirken einer Anzahl (l&50, je nach Vertebratenart) ungebleichter Rhodopsinmolekiile eine Au&sung des Ubertrags bewirkt. Schlussfolgerungen aus dieser Theorie sind in guter Ubereinstimmung sowohl mit psychophysiscehen als such mit elektrophysiologischen Schwellenexperimenten. die ohne Hintergrundbeleuchtung durchgeftihrt wurden. Die empirische Hinzunahme eines verallgemeinerten Weber-Fechner-Gesetzes gestattes es, die Theorie im Fall bestimmter Hintergrunde anzuwenden.

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