MEASURING AND INTERPRETING CALCIUM CONCENTRATION CHANGES IN VERTEBRATE PHOTORECEPTOR OUTER SEGMENTS
ALAMANACH
INTRODUCTION: The dynamics of calcium concentrations in the vertebrate photoreceptor are thought to be well understood. The channels that allow for the influx of calcium ions have been identified, as have the chemical processes that govern these channels (for example, see Hsu and Molday, 1993). The sodium-powered pumps that eject these same calcium ions are very familiar to us (Yau and Nakatani, 1985), and calcium’s various interactions inside the photoreceptor are also clearly identified (Kock and Stryer, 1988, Kawamura and Murakami 1991). Experiments by Sampath et al. have measured the changes of calcium concentration in photoreceptors over time as a dark adapted retina is suddenly exposed to bright light (Sampath, A.P. et al. 1998, 1999). As we would expect from our knowledge about photoreceptors, the calcium concentration drops dramatically as the photoreceptor adapts to its new condition.
Mathematically, the precise manner in which calcium concentration drops can yield further insights into photoreceptor physiology. Because the mathematical descriptions that have come out of experiment are of one general form and not another, models of the photoreceptor also have to take a particular form. This paper will show, through mathematics alone, that photoreceptors are compartmentalized systems, capable of possessing dramatically non-uniform concentrations of Ca++ ions.
MECHANISMS OF DARK ADAPTATION IN PHOTORECEPTORS: The sensitivity of photoreceptors is not static; biochemical changes inside the photoreceptor can render it more or less sensitive to light stimuli. There are four mechanisms of dark adaptation operating in the vertebrate photoreceptor:
- Activity of guanylate cyclase
- Hydrolyzation of cGMP by phosphodiesterase
- Direct modulation of cation channels
- Modulation of rhodopsin
Ca++ ions play a major role in each of these mechanisms, as will be shown. In all cases, high Ca++ concentrations are present in a dark-adapted photoreceptor, and Ca++ concentrations fall away as the photoreceptor responds to light.
Activity of guanylate cyclase: Cyclic guanosine 3’-5’-monophosphate, also known as cyclic GMP or simply cGMP, acts as a ligand to photoreceptor cation channels, and they open in its presence. In the absence of a light stimulus, photoreceptors are continuously depolarized, and continuously releasing neurotransmitter. This dark depolarization requires an ample supply of cGMP, and the photoreceptor hyperpolarizes without it.
Synthesization of cGMP requires the presence of the enzyme guanylate cyclase. Experiments in toads have shown that the activity of guanylate cyclase is in turn an inverse function of intracellular Ca++ concentrations; high Ca++ concentrations correlate with low guanylate cyclose activity. (Koch & Stryer, 1988) Because this will tend to reduce availability of cGMP and close cation channels, it impedes the ability of new Ca++ ions from entering the cell. This negative feedback prevents the Ca++ concentration from growing without bound.
Further experiments with bovine photoreceptors have revealed a ~20kDa membrane-bound protein that appears to be required for proper guanylate cyclase action. This guanylate cyclase activating protein (GCAP) is very sensitive to changes in calcium concentration, especially at low levels. In the absence of GCAP, guanylate cyclase shows very little activity, regardless of Ca++ concentrations. (Gorczyka, et al. 1994)
In short, guanylate cyclase could be thought of as moderating a photoreceptor’s baseline tendency to hyperpolarize. Reduced guanylate cyclase activity will push a photoreceptor closer to hyperpolarization, effectively making it more sensitive to incoming light stimuli.
Hydrolyzation of cGMP by phosphodiesterase: As noted, cGMP synthesis is moderated by guanylate cyclase activity. Breakdown of cGMP is acomplished through hydrolyzation by phosphodiesterase (PDE). Experiments with rod outer segments (ROS) taken from frogs has shown that there is a 26K regulatory protein called S-modulin that binds to ROS disk membranes at high Ca++ concentrations and becomes soluble at low concentrations. At high concentrations, this protein increases PDE efficiency by as much as 50%, accelerating breakdown of cGMP. (Kawamura & Murakami 1991) Again, this has the effect of making the photoreceptor more sensitive to light stimuli.
Direct modulation of cation channels: The speed with which cGMP acts on a cation channel can be described in terms of this reaction’s Michaelis constant. (A higher Michaelis constant indicates a slower reaction.) A pair of intermediate proteins, including calmodulin and an unnamed 240K protein, are involved in the reaction between the cation channels and their ligand, cGMP. Calmodulin, when in the presence of Ca++ ions, excites the 240K protein (as is typical in most reactions involving calmodulin), which in turn raises the Michaelis constant for cGMP and the cation channel. (Hsu & Molday, 1993) Consistent with the other reactions discussed, high Ca++ concentrations affect cation channels in a manner that increases the photoreceptor’s sensitivity.
Modulation of rhodopsin: Rhodopsin is the visual pigment that absorbs incoming light. This light creates a conformal change that puts rhodopsin in an active state, and it is while in this state that rhodopsin stimulates PDE to hydrolyze cGMP. A single active rhodopsin molecule can result in the hydrolysis of 105 cGMP molecules per second. Calcium modulates this process in two ways.
Experiments with mechanically truncated tiger salamander ROS have shown that low calcium levels substantially reduce the catalytic rhodopsin activity. There is some evidence that an intermediary protein, perhaps similar to S-modulin and calmodulin, may be involved. (Lagnado & Baylor, 1994)
Earlier experiments have shown that low Ca++ concentrations accelerate the deactivation of rhodopsin. This deactivation happens in two steps. First, opsin kinase phosphorolates rhodopsin. Once phosphorylated, the rhodopsin becomes the target of the regulatory protein arrestin, which deactivates it. Calcium impedes this process by inhibiting the opsin kinase-mediated phosphorylation. (Kamura, S. 1993)
MEASURING Ca++ CONCENTRATIONS: Having established the central role played by Ca++ in the dark adaptation of photoreceptors, we turn to the measurement of calcium kinetics. Ca++ concentrations inside a cell can be measured by means of the laser confocal technique. This was originally developed in conjunction with work on muscles cells; researchers sought the precise locations on the sarcoplasmic reticulum from which Ca++ ions are released.
This technique involves the use of a calcium indicator called fluo-3. This indicator fluoresces when exposed to Ca++ ion and light. By introducing the dye into the cell and aiming a laser and a photodiode at the point of interest, the calcium dispersal along different parts of the sarcoplasmic reticulum could be measured. (As it turns out, most calcium was released by the terminal cisternae.) (Escobar, et al., 1994)
This technique has been applied to the photoreceptors of the tiger salamander. Precise measurements have been made in which a dark-adapted photoreceptor (with a high Ca++ concentration) is suddenly exposed to a strong light stimulus, and the Ca++ concentrations are measured over time. A least-squares technique has fit the resulting data to curves that turn out to be the sums of two exponential time constants. Specifically, those results are as follows:
Rods: [Ca++](t) = 0.5e(-t/260ms) + 0.5e(-t/2200ms)
Red-sensitive cones: [Ca++](t) = 0.37037e(-t/43ms) + 0.62963e(-t/640ms)
Blue-sensitive cones: [Ca++](t) = 0.5e(-t/140ms) + 0.5e(-t/1400ms)
All of these equations describe a sharp drop in Ca++ concentration, as expected. (Sampath et al. 1998 and 1999)
INTERPRETING Ca++ CONCENTRATIONS: These equations for calcium kinetics are all of the general form
[Ca++](t) = A1e-t/τ1 + A2e-t/τ2
It is a sum of two exponentials.
This general form is important to a wide range of physical systems because it is a solution to many second-order partial differential equations. For example, we consider a spring-mass-damper system. A mass is simply a lump of matter; it resists acceleration. Springs are also commonly familiar; they resist whenever stretched or compressed. A damper is formally defined as a device that removes energy from a physical system. Dampers do not particularly resist acceleration or position, except insofar as they do resist movement. A piston moving through syrup experiences damping, as does a block of wood sliding across a sheet of rubber. (For a detailed treatment of spring-mass-damper systems, see Den Hartog 1934 or Shabana 1997.)
When a mass is attached to the ground by means of a spring and a damper, the motion of the mass relative to the ground will depend on the strengths of the spring and the damper, and on the quantity of mass. A moderate mass mounted on a strong spring and weak damper can be made to oscillate. But a strong damper and a weak spring can result in very little motion at all. The general equation for a mass-spring-damper system is
mx” + cx’ + kx = F
where F is an externally applied force. x represents position as a function of time. x’ and x” are the first and second derivatives of x with respect to time. This implies that x represents displacement, x’ represents velocity, and x” represents acceleration. m is the quantity of mass, c is the strength of the damper, and k is the strength of the spring.
Because of the presence of x”, this is a second order differential equation When presented with such equations, it is usually necessary to find an expression for x in order to have all information concerning the system’s motion.
Because in a moving system the displacement x depends on time, it is common to speak of “x at time t,” which can be written x(t). The expression for x that solves this second order differential equation is in the form
x(t) = Ae-t/τ
If the system has two degrees of freedom, this expression becomes
x(t) = A1e-t/τ1 + A2e-t/τ2
Note that this is identical to the calcium ion concentration expression given by Sampath, et al. The constants A1 and A2 depend on the initial position and velocity of the system. In any real world application, these are always specified and are not of great concern. The constants τ1 τ2 depend entirely on system parameters m, c, and k, and their values will ultimately determine the behavior of the system.
Because both systems possess the same mathematical form, one can model the other. Setting x = [Ca++], k would represent (in some way) the buffering capacity of a photoreceptor; buffers seek to maintain a fixed concentration, just as springs seek to maintain a fixed position. The mechanisms of dark adaptation already described act as a form of buffer, stabilizing Ca++ concentrations. m would represent the Na+/Ca++, K+ pumps that continuously remove Ca++ ions from the cell. There are no known mechanisms modulating the actions of these pumps, and so they tend to reduce [Ca++] at a constant rate, just as mass favors a constant velocity. The variable c is something inversely proportional to activity of the cation channels. These channels, when open, allow calcium to flow into the cell and for [Ca++] to increase. But these channels are not always open, and to the degree to which they are closed they are a resistance, c, to [Ca++] change.
It was noted that the system has two degrees of freedom. For a spring-mass-damper system, this implies two masses joined by springs and/or dampers, with one of the masses joined by springs and/or dampers to the ground. For the photoreceptor, it implies something that was confirmed by observation during the experiments of Sampath, et al.: calcium concentration (and by corollary, dark adaptation) is a very local phenomenon. Despite the small size and large surface area of a photoreceptor outer segment, it is possible for Ca++ to be virtually depleted in one part of the cell while concentrations remain high nearby. There was concern among the experimenters that their results reflected bleaching of the fluo-3 indicator, rather than Ca++ depletion. But repeating the experiment with lower laser intensity resulted in the same time constant. Furthermore, shifting the laser to a previously unilluminated portion of the outer segment would result in fresh fluorescence. (Sampath et al., 1998 and 1999)
This can be understood through a different physical model, the mixing tank. We imagine a tank containing some solution at concentration x, with a drain at the bottom and inlet at the top. Through the inlet comes solute-free fluid, and the drain allows for constant volume in the tank. For a single tank, any combination of concentration strengths, inlets, outlets, and fluid flow rates all lead to the same form of equation. The concentration of solute x in a tank at time t is given by
[x](t) = Ae-t/τ
There is a variation of this classic problem in which two tanks exchange fluid between each other at a constant rate, and in which one tank has an inlet and the other an outlet. For each of the tanks, equations describing the local concentration [x] at time t will be of the form
[x](t) = A1e-t/τ1 + A2e-t/τ2
and depending on the parameters, it is quite possible for the two tanks to possess markedly different concentrations.
CONCLUSION: Though many mechanisms act on Ca++ concentration inside the photoreceptor, the combined effect of these mechanisms can be easily understood if they are conceived of as a second-order system. Modeled in this way, it quickly becomes apparent that the photoreceptor outer segment is a complicated, heterogeneous environment, where conditions at one location may be very different from conditions at another location nearby. Experimental results hint at this, but mathematical analysis demands it. The potential for mathematics to reveal such hidden phenomena should not be underestimated.
REFERENCES
Den Hartog, J.P. Mechanical Vibrations. Dover Publications, New York, 1934.
Escobar, Ariel L., Jonathan R. Monck, Julio M. Fernandez, and Julio L. Vergara. 1994. Localization of the site of Ca2+ release at the level of a single sarcomere in skeletal muscle fibres. Nature 367:739-741.
Gorczyca, Wojciech A., Mark P. Gray-Keller, Peter B. Detwiler, and Krzysztof Palczewski. 1994. Purification and physiological evaluation of a guanylate cyclase activating protein from retinal rods. Neurobiology. 91:4014-4018.
Hsu, Y.T., and R.S. Molday. 1993. Modulation of the cGMP-gated channel of rod photoreceptor cells by calmodulin. Nature. 361:76-79.
Kawamura, S. 1993. Rhodopsin phosphorylation as a mechanism of cyclic GMP phosphodiesterase regulation by S-modulin. Nature. 362:855-857.
Kawamura, S., and M. Murakami. 1991. Calcium-dependent regulation of cyclic GMP phosphodiesterase by a protein from frog retinal rods. Nature. 349:420-423.
Koch, K.W., and L. Stryer. 1988. Highly cooperative feedback control of retinal rod guanylate cyclase by calcium ions. Nature. 334:64-66.
Lagnado, Leon, and Denis A. Baylor. 1994. Calcium controls light-triggered formation of catalytically active rhodopsin. Nature 367:273-277.
McCarthy, S.T., J.P. Younger, and W.G. Owen. 1996. Dynamic, spatially nonuniform calcium regulation in frog rods exposed to light. J. Neurophysiol. 76:1991-2004.
Sampath, A.P., H.R. Matthews, M.C. Cornwall, and G.L. Fain. 1998. Bleached pigment produces a maintained decrease in outer segment Ca2+ in salamander rods. J. Gen. Physiol. 111:53-64.
Sampath, A.P., H.R. Matthews, M.C. Cornwall, and G.L. Fain. 1999.Light-dependent changes in outer segment free-Ca2+ concentration in salamander cone photoreceptors. J. Gen. Physiol. 113:267-277.
Shabana, A.A. Theory of Vibration: An Introduction. Springer-Verlag, New York, 1997.
Yau, K.W. and K. Nakatani. 1985. Light-induced reduction of cytoplasmic free calcium in retinal rods outer segment. Nature. 313:579-582.
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