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Intelligent Design, the best explanation of Origins » Molecular biology of the cell » Principle of Design of a Cellular Eye

Principle of Design of a Cellular Eye

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1 Principle of Design of a Cellular Eye on Wed May 25, 2016 1:53 pm

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Principle of Design of a Cellular Eye 1

The arguments just given do not mean that cellular eyes are impossible, but they must use principles other than geometric optics or intensity comparisons. In order to identify the geometric characteristics of a cellular eye, we will begin with principles and postulate that cells use the most optimal design. As mentioned earlier, an eye may be defined as an instrument that maps, in a one-to-one fashion, different directions toward an object space onto different points of an image space (see Fig. 5).




Fig. 5. 
Generalized definition of an eye to achieve a one-to-one mapping of different directions in an object space onto different points in an image space.


If the different points of the object space (extra ocular space) are sources of a certain radiation, then the task of providing a one-to-one map of their direction can be formulated as the task of locating the direction of the sources. The general situation of an eye locating a source is illustrated in Fig. 6. 



Fig. 6. 
The most general design of a cellular eye to locate signal sources in its environment. The dotted lines indicate some of the directions of propagation of the signal toward signal receptors around the periphery of the eye. Even with an opaque interior (shaded area), the design does not yield a one-to-one mapping of source directions (object space) onto signal receptors (image space).


The object space is considered as a number of signal sources outside the cellular eye. Only one of them is depicted in Fig. 6. The ideal eye has to be rotationally symmetrical or it would be biased and prefer the detection of certain directions over others. It must contain receptors for the type of radiation that the sources emit (black dots in Fig. 6); these receptors must be spaced evenly to avoid directional bias. The receptors provide the points of the image space onto which the different directions of the signal sources are to be mapped. The design of Fig. 6. however, does not provide a one-to-one mapping because more than one receptor is exposed to the signals of the source. If the inside of the eye were transparent for the signal, the situation would be worse because every receptor would receive the signal. In order to minimize the ambiguity, we take the first step toward the final design of the cellular eye by filling the inside of the cellular eye with a material that is designed to absorb the signal (indicated by the gray area in Fig. 6).

Use of Blinds

Because optical methods fail to provide the one-to-one mapping between directions of signal sources and the set of receptors, the ambiguity of the signal reception must be reduced by other means. Perhaps the simplest and most effective block of the signals would be by blinds that are attached to one side of each receptor as indicated in Fig. 7.



Fig. 7.
Reduction of the ambiguity of the mapping function of the design shown in Fig. 6 by attaching blinds to the side of each receptor. The blinds are supposed to absorb the signal in their thickness. The improved design reduces the ambiguity, but does not eliminate it, because more than one receptor is exposed to the signal.


The blinds are supposed to absorb the signal by their thickness. Obviously, the use of the blinds reduces the ambiguity, although a one-to-one mapping is still not achieved. However, if we close the blinds in the manner shown in Fig. 8, only one receptor is exposed to the source, thus yielding the desired one-to-one mapping function. Optimally, the angle of the blinds is such that they extrapolate to the next receptor in the counterclockwise direction (see Fig. Cool.



Fig. 8. 
Final design of the geometry of a two-dimensional cellular eye. After closing the blinds in the depicted way, one and only one receptor is exposed to the signal source, insuring a one-to-one mapping.


If the angle is more radial, the ambiguity of Fig. 7 reappears. If it is more tangential, the design generates blind spots where signal sources cannot reach any of the receptors.

Three-Dimensional Shape of the Eye


The design as developed thus far works as long as the signal source remains within the plane of the ring of receptors. However, we must expect that the signal sources around the real cell are also located above and below this plane, as shown in Fig. 9.



Fig. 9.
Demonstration that the design of Fig. 8 is ineffective if the signal source moves out of the plane of the two-dimensional cellular eye. From the position of the source above the plane, all receptors are exposed to its signal, excluding a one-to-one mapping.

In this case, the full extent of the ambiguity of signal reception would reappear, since all receptors are exposed to the signal. Therefore, the next step in the construction of a cellular eye is the extension of the planar design of Fig. 8 into a three-dimensional one. Considering that the optimal design must remain rotationally symmetrical, there are only two possibilities for its spatial structure: a cylinder and a sphere. In the case of a sphere, it is apparent that it would not allow the attachment of blinds in a slanting fashion all over its surface. At least at the poles of the sphere the slanting directions would contradict each other. (This fact is known to mathematicians as the Theorem of the Well-Combed Hedgehog.) Therefore, the sphere has to be capped as shown in Fig. 10.




Fig. 10. 
The only two possibilities for extending the two-dimensional cellular eye into the thirddimension are a straight cylinder or a capped sphere (a bulging cylinder).

In other words, the three-dimensional structure of the cellular eye has to be a straight or bulging cylinder, but a cylinder.


Requirement of Two Orthogonal Cylinders

The design as developed so far permits a one-to-one mapping between the set of signal receptors and the angle of the signal source as measured around the axis of the cylinder. Calling this angle the longitude of the signal source, it is obvious that we need to map its latitude as well if we wish to identify the direction of the signal source. Since the longitude and latitude of a point are angles measured around axes orthogonal to each other, the design requires a second cylinder at right angles to the first, as shown in Fig. 11.



Fig. 11.
Necessary use of a pair of orthogonally oriented cylinders as a cellular eye to determine simultaneously the longtitude and latitude of the signal source (see text).


Therefore, we conclude that the cellular eye would be a pair of cylinders at right angles to each other, each having the cross section shown in Fig. 8. It hardly needs mentioning that this design bears striking resemblance to a pair of centrioles. If the arguments as presented are correct, then the structure of centrioles and basal bodies is the expression of a physical necessity and not the result of an accident of evolution, which may explain the high degree of conservation of centriole structure throughout nature. The right-angle orientation can be expected to be found whenever the signal sources are distributed in three-dimensional space. If under special circumstances the relevant signal sources are distributed in a plane, only one cylinder is required that is oriented perpendicular to this plane. We speculate, therefore, that the array of basal bodies that are oriented perpendicular to the surface of a ciliated cell not only form nucleating structures for the cilia but also act as monitoring devices for signal sources at the cell surface.

Refinement of Angular Resolution

The similarities between a hypothetical cellular eye and the structure of centrioles and basal bodies extend even further if we try to optimize the angular resolution of the device. The design developed so far limits the angular resolution by the number of receptors and blinds around the perimeter of the cylinder. If a signal source is located anywhere in the sector defined by the angle between adjacent blinds, it will be detected by one and the same receptor. In other words, if the cross section of the cylinder has N blinds, then the limit of angular resolution is Θ = 360°'IN. Therefore, it would be desirable to improve the design in such a way that it can distinguish various directions in each sector. The most direct method to improve the angular resolution would be to increase the number of receptors and blades (N). However, each blade must have a certain thickness in order to absorb the signals. Therefore, the number N cannot be increased at will. A much more elegant way to achieve better angular resolution is to attach the blinds in such a way that they lean across one sector as they run from the top of
the cylinder to the bottom (see Fig. 12).


Fig. 12. 
Pitch of the blades as a means to improve the angular resolution of the design. The percentage of receptors on blade n exposed to the signal source S is a measure of the location of the signal source in the sector between two adjacent blades (see text).


The blades of actual centrioles and basal bodies are, indeed, pitched in this way (cf. Fig. 1). Assume that the viewer is positioned at the location of the signal source S. As shown in the perspective drawing of Fig. 12, the viewer would see the outside of some blades (gray shading), the inside of other blades (white areas), the opaqueness of the inside of the cylinder (black areas), and the rows of receptors exposed to the direction of the source (dots lining the blades). The main difference from the earlier design is the exposure of some receptors along blade n and some receptors along blade w+1 to the source. Therefore, at first sight, the pitched blades may seem to defeat our initial intention of providing a one-to-one mapping between receptors and directions of signal sources. However, the exposed receptors on adjacent blades differ by their position along the blades. More importantly, the apparent ambiguity offers the possibility to determine the location of the direction of the signal source in the sector of blade n. By determining the ratio

                                                               number of receptors exposed on blade n
                                              r   =        -------------------------------------------------
                                                                 number of all receptors on blade n

(see Fig. 12), the angle φ of the signal source reaching receptors on blade «becomes φ = (n + r)6 This angle is no longer an integral multiple of the former limit of resolution Θ = 360°'/N, but contains fractions of Θ, thus improving the angular resolution. It is still limited, though, because of the finite number of receptors lining the inner edge of each blade. The explanation of the calculation just presented is obvious if one considers a
series of cross sections along the cylinder axis, as shown in Fig. 12. As we move up the axis, the cross sections rotate by the angle of one sector. Depending on the direction of the source within the sector, the blade will cut off the signal source at some distance from the bottom of the cylinder, giving rise to the ratio calculated.


Likely Wavelength of Signal Sources

The next important question about our speculation concerns the physical nature of the postulated signal sources. Considering that the function of the cellular eye is to locate objects (signal sources) in the environment of the cell, one may argue that the wavelength emitted by the sources must be linked to the size of these objects. If the objects were smaller than the wavelength, then the cellular eye would not be able to resolve two objects closer than the wavelength. It seems reasonable to assume that the interesting objects around probably include other cells; therefore, we can expect that the wavelength of the sources is smaller than a cell diameter or λ < 10 μπι. This wavelength excludes sound as a signal, because the frequency of a sound wave with 10-μιη wavelength in water is n = (1200 m/sec)/(10 μιη) = 1.2 x 108 Hz. At this frequency, water is a completely absorbing medium for ultrasound and, therefore, unsuitable as medium of transmission of signals. Consequently, we are left essentially with electromagnetic radiation as the suspected signal. The range of the wavelength places the radiation in the range of visible or infrared light. There are various reasons to exclude visible light as potential signal carrier: (1) Most animal cells are located deep inside organisms and are thus cut off from visible light. (2) Centrioles and basal bodies are surrounded by cytoplasmic organelles such as mitochondria and lysosomes that are up to 1 μιη in size. In order to reach the cellular eye without much scattering by such obstacles in the line of sight, the signal sources should emit wavelengths that are larger than these objects. In other words, this argument places the wavelength outside the visible range. Thus we are left with infrared light of wavelengths shorter than 10 μπι as the most likely signal radiation to be detected and located by centrioles. To be sure, it is quite possible that cells or cell parts emit such radiation, because at room temperature these wavelengths belong to the higher harmonics of molecular vibrations and rotations. Further, it is conceivable that cells can receive such signals because certain animals, including snakes, are able to see infrared sources, suggesting that infrared-sensitive receptors are part of the genome of some animals; the postulate of infrared-sensitive receptors in centrioles cannot be dismissed.

The following argument seems to narrow the range of wavelengths even more. As shown in Fig. 13, the main wavelengths of black body radiation at 27°C range from 3 μηι to > 10 μιη. A very similar curve applies to room temperature or 37°C. A cellular eye sensitive to these wavelengths would detect a constant bright glow around the cell, stemming from the ambient temperature, that might bury the signals from surrounding sources in high background noise. Therefore, the wavelength of the sources would be placed best above the visible range and below 3 μιτι (Fig. 13).




Fig. 13. 
Black body radiation at and around room temperature (continuous curve; right ordinate shows the logarithm of the spectral emission; 300 K = 27°C) and extinction coefficient of water (dotted curve; left ordinate) as functions of wavelength in the visible and infrared range (redrawn after Wolfe and Zissis, 1989).

Unfortunately, in this range of infrared radiation, water is already rather opaque and one would suspect that the relevant wavelengths are located in relative "water windows." As shown in the water absorption curve in Fig. 13, there are three areas of relative transparency: 0.8-1.3 μπι, 1.6-1.7 μιτι (marked A), and 2.2-2.4 μπι (marked B). Using a specially designed microscope, we tested the ability of 3T3 cells to locate distant infrared light sources. The experiments, indeed, identify the window between 0.8-0.9 μιη as the most active range (Albrecht-Buehler, 1991).



2) The Centrosome,  Vitauts I. Kalnins

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2 Are centrioles the 'eye' of the cell? on Wed May 25, 2016 2:58 pm

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this is probably not a true hypothesis. I have not read anywhere beside the authors claim below that centriole has photoreceptor cells that can catch light.

Cellular eyes cannot use lenses to locate light sources.1

Cellular eyes cannot resemble the eyes of familiar organisms in the macroscopic world. For example, cellular eyes cannot not use lenses, because the typical cell size of approximately 10 µm is too small. Obviously, such lenses would have to be much smaller than the cell itself. Therefore, let us assume that the lens diameter is 1 µm. Lenses can only focus light whose wavelength is smaller than about 1/1000th of their diameter. Otherwise, the light would simply diffract around the lens and ignore it. In other words, cellular lenses can only work with light whose wavelength is smaller than 1/1000 µm = 1 nm, i.e .with X-rays. This means that there are no materials from which to grind cellular lenses because no materials exist which are able to refract X-rays to any extent.


Cellular eyes cannot compare signal intensities to locate light sources.


Many bacteria are fast swimmers. They can afford to do many trial-and-error runs in order to find the source of a chemoattractant. As long as the signal strength increases they continue their tack; whenever it decreases they tumble and change direction. Animal cells cannot use the same trick. Compared to bacteria they are extremely slow. By the time a cell has found a target in this way, the embryo would long be finished everywhere else.


There is a further problem. Due to the ubiquitous and violent thermal fluctuations the world of cells is extremely noisy in every respect. Whatever concentrations or intensities a cell on a trial-and-error run may wish to compare from place to place, they are all very unreliable. To be sure, a mathematical average over the signal strength could eliminate the fluctuations, however, in practical terms it would not work.Imagine yourself in a howling hurricane trying to average the strength of coffee aroma in the air order to find your cup!

The ideal eye has no directional preference.


Consider a signal source (e.g.a source of light or anything else that propagates along straight lines, and let us design the ideal eye for it. (Of course, this is not how centrioles and their function evolved. In fact, we have no idea how they evolved. However, for didactic reasons I found it simpler to convince the reader that a structure may have a certain function, if the reader him-/herself, imagining to be an engineer with that particular function in mind, would have ended up constructing this very structure). Here is how we could proceed: Unlike our own eyes which can only see in the foreward direction, an ideal cellular eye would have no such directional bias. Consequently, it would be rotationally symmetrical like the circle on the figure below.




For the same reasons of symmetry the receptors for the particular kind of signal (depicted as blue spots on the figure) would have to be evenly spaced. As pointed out in the main text, the definition of an eye is a device that maps the directions of light sources in a one-to-one fashion. Therefore, the above structure would not work because it is not capable of mapping different source direction to different receptors.

'Blinds' can accomplish the one-to-one mapping of source directions


Considering that neither lenses nor intensity sensors can help us eliminate the ambiguity of the above source mapping device, we may try to attach blinds to each receptor that are able to block the signal. If we attach them radially as shown in the figure below we reduce much of the ambiguity, but not all.




Every source can still reach at least 2 receptors. (In anticipation of the result of all these considerations each blind is drawn similar to the blades of centrioles.) However, if we attach the blinds in a slanting fashion as shown in the figure below, each source direction can reach one and only one receptor.




In order to eliminate the ambiguity of the mapping process the blinds have to be attached at a special angle: The backward elongation of each blind intersects the foot of the previous one.

Each 'blind' must be curved


In the previous figures the blinds were drawn slightly curved. It was necessary, because straight blinds could not prevent the ambiguity of the mapping for certain directions: Signals from a source in the very direction of the slanting blinds could still reach 2 receptors as shown on the left hand side of the figure below.[/left]



In contrast, curved blinds eliminate this last possibility of ambiguous mapping as shown on the right hand side of the figure: The blind's curvature casts a shadow on its own receptor if illuminated from this special direction.


In order to work in 3 dimensions, the structure has to be a cylinder


The above structure can only work as long as the sources and the 'eye' are both located in the same plane. If the source rises above the plane of the 'eye' as shown below, the blinds can no longer protect the receptors, and the 'eye' fails to map the source direction.



Therefore, the structure of the 'eye' has to be extended into the third dimension while still complying with the condition of rotational symmetry. Only 2 structures can fulfill this condition: the circle with the slanting blinds has to be stretched into a cylinder or around a sphere. The extension into a cylinder poses no problem as shown in the figure below.


In contrast, stretching the blinds around the surface of a sphere is not possible: At the poles the directions of the slanting blinds would contradict each other unless we cap the sphere at its poles, as shown in the figure below.


In other words, the basic 3-dimensional structure of the 'eye' has to be either a cylinder with straight sides or with bulging sides, but a cylinder in any case.
In order to map longitude and latitude of a source we need 2 cylinders perpendicular to each other.


Even the cylindrical structure of the 'eye' can only map the angle of the source in a plane perpendicular to its axis (e.g. the longitude). Therefore, we need a second cylinder at right angles to the first in order to determine the latitude of the source, too. The figure below illustrates how a pair of centrioles measures the longitude (yellow lines) and the latitude (green lines) of a source.[/left]





Pitched blinds achieve continuous angular resolution.


So far the structure guarantees that each source direction can irradiate one and only one receptor. However, it does not guarantee that each receptor can detect only one source direction. In other words, the angular resolution of the structure is rather crude: If it has N blinds, then the angular resolution is 360°/N.

Unfortunately, one cannot increase the number N at will in order to achieve finer and finer resolution because each blind has to have a certain minimal thickness in order to absorb or reflect the signal. But there is a much more elegant way to refine the resolution to such a degree that it is practically continuous: One may pitch the blinds as shown in the figure below.


As the blinds run from the bottom of the cylinder to the top, they cross one sector of 360°/N. In order to assess the result, the figure below lets us take the position of the source and 'look' at the cylinder.



(The illustration is animated.Click here for a minimal strip of frames.)Whichever receptors we are able to see from our positions are the receptors that will be irradiated from our source direction. As shown by the animated figure, we can see receptors on the inside of 2 consecutive blinds.They are shown in red and yellow. And as we advance sector by sector around the cylinder, their relative lengths change in a well-defined way. In other words, the 'eye' can measure in a continuous fashion the location of a source within each sector by measuring how many receptors are irradiated on the base of consecutive blinds.

Back to the drawing board?





Pitched blinds may offer continuous angular resolution, but can the 'eye' still map source directions in a one-to-one fashion? After all, each source can now reach the receptors of 2 consecutive blinds. Conversely, 2 sources in consecutive sectors should be able to reach receptors attached to the same pitched blind.

It is true, after adding a pitch to the blinds 2 sources can reach receptors on the same blind, but at different positions on the cylinder axis. Pitch or no pitch, any receptors at a specific position on the axis of the cylinder can still be reached by only one of the sources.
A problem arises, though, if the 2 sources are located in the same sector, and some of the receptors of the same blind are irradiated by both at the same time. Obviously, in this case the cell cannot tell the sources apart unless they differ in at least one characteristic. It would be rather simple to distinguish between different sources in the same sector if their emission would pulsate at specific frequencies. In this case the cell could compute the location of either source by (cross-)correlating only receptors that received signal pulses at the same time.
Pulsating source intensities would also offer a solution to the problem of increasing the signal-to-noise ratio that is particularly important in the thermally very noisy world of cells. Indeed, the experiments show, that cells are able to detect pulsating near-infrared light sources but not sources with constant intensity.


Significance for cell intelligence:





All the geometric properties of the ideal cellular eye are found in actual centrioles.

[/center]

The ideal cellular 'eye' looks remarkably like actual centrioles. Like the hypothetical 'eye' whose properties we described above

Centrioles are built as a pair of cylinders perpendicular to each other.
In cross-section they have slanting blades.
The angle of the blades is such that the backward elongation of each blade intersects the foot of the previous one.
Each blade is slightly curved.
Each blade is pitched .

Furthermore, since the postulated properties of the 'eye' was based on necessary and optimized conditions of its intended function, there are not many other structures that would fulfil the same conditions. This feature seems to be matched by actual centrioles, as well. The geometric features of centrioles belong to the best conserved properties in nature. Independent of the place on the evolutionary tree, if a cell has centrioles, they have this particular structure. Their structure is, therefore, likely to point to an apparently universal function.

Function and Formation of Centrioles and Basal Bodies 2



THE FUNCTION OF CENTRIOLES AS CELLULAR EYES:

Assume that animal cells during their morphogenetic movements require an instrument to locate distant signal sources in order to orient the direction of their migration. What would be  the best possible design for such an instrument ?  What would it look like? Amazingly, it would look precisely like a pair of centrioles.

An instrument to locate the direction of distant signal sources may be called an eye. As illustrated in Fig. 3, an eye mediates the one-to-one mapping of different directions of an object space (objects outside the eye) to certain points of an image space, which in the case of the human eye is the retina. For example, the two different directions toward the object points O and O' are mapped onto the two different image points I and Γ. If different object points lie in the same direction, such as 0 ' and O", then they are mapped onto the same image point (Γ = I"), albeit with some modification. In the case of the human eye, the image points Γ and T'may be blurred to different degrees due to differences in focus.



Fig. 3. 
Schematic illustration of the function of a generalized eye to map different directions of the object space (extraocular space) onto different image points of the image space (retina) in a one-to-one fashion. The different directions are represented by dotted lines connecting object points O, O', O" with the center of the lens (shaded area). If object points lie in the same direction (e.g., O' and O"), they are mapped to the same image point, Γ = I".


Impossibility of Using Cellular Optical Lenses

Could cells have eyes with lenses like those of large organisms? As shown in Fig. 4, the use of lenses in cellular dimensions is entirely impossible. Given a normal cell size of 10 μπι, the cellular lens can be expected to be considerably smaller (e.g., D - 1 μπι). Likewise, its focal length must remain inside the cell body (f « 10 μπι). The wavelength of the signals focused by this lens is denoted as λ. Consequently, the diameter of the image point (Airy disk) behind the lens is approximately d = λ f/D. In order to provide a useful optical image, each image point must be much smaller than the lens diameter (e.g., d = D/1000). As a result, λ = D2/1000/« 10"10 m = 1A. In other words, cellular lenses would require X-rays as the form of radiation that they focus. The essential result remains the same if we change the numbers used in this estimate, as long as they remain compatible with cellular dimensions. Obviously, no materials for lens construction are available with a refractive index large enough for X-rays; therefore, cellular lenses are not possible.



Fig. 4. 
Diagram relating the diameter D and focal length / of a lens (shaded area) and the wavelength I of a light wave with the diameter d of the Airy disk of the image point. The profile on the right indicates the intensity distribution of the focused light.


Impossibility of Using Signal Intensities for Mapping


If radiation cannot be focused by a cellular eye, perhaps an instrument could be designed that compares the various intensities of the signals from different sources in space, similar to bacterial chemotaxis. Unfortunately, the large thermal noise in cellular dimensions prohibits this imaging method also. Comparing intensities at different points is a feasible method to navigate toward a signal source, but the large thermal noise in cellular dimensions would make it impossible to generate a one-to-one map of the locations of multiple sources in a reasonable amount of time required by the function of an eye.

Principle of Design of a Cellular Eye

The arguments just given do not mean that cellular eyes are impossible, but they must use principles other than geometric optics or intensity comparisons. In order to identify the geometric characteristics of a cellular eye, we will begin with principles and postulate that cells use the most optimal design. As mentioned earlier, an eye may be defined as an instrument that maps, in a one-to-one fashion, different directions toward an object space onto different points of an image space (see Fig. 5).




Fig. 5. 
Generalized definition of an eye to achieve a one-to-one mapping of different directions in an object space onto different points in an image space.


If the different points of the object space (extra ocular space) are sources of a certain radiation, then the task of providing a one-to-one map of their direction can be formulated as the task of locating the direction of the sources. The general situation of an eye locating a source is illustrated in Fig. 6. 



Fig. 6. 
The most general design of a cellular eye to locate signal sources in its environment. The dotted lines indicate some of the directions of propagation of the signal toward signal receptors around the periphery of the eye. Even with an opaque interior (shaded area), the design does not yield a one-to-one mapping of source directions (object space) onto signal receptors (image space).


The object space is considered as a number of signal sources outside the cellular eye. Only one of them is depicted in Fig. 6. The ideal eye has to be rotationally symmetrical or it would be biased and prefer the detection of certain directions over others. It must contain receptors for the type of radiation that the sources emit (black dots in Fig. 6); these receptors must be spaced evenly to avoid directional bias. The receptors provide the points of the image space onto which the different directions of the signal sources are to be mapped. The design of Fig. 6. however, does not provide a one-to-one mapping because more than one receptor is exposed to the signals of the source. If the inside of the eye were transparent for the signal, the situation would be worse because every receptor would receive the signal. In order to minimize the ambiguity, we take the first step toward the final design of the cellular eye by filling the inside of the cellular eye with a material that is designed to absorb the signal (indicated by the gray area in Fig. 6).

Use of Blinds

Because optical methods fail to provide the one-to-one mapping between directions of signal sources and the set of receptors, the ambiguity of the signal reception must be reduced by other means. Perhaps the simplest and most effective block of the signals would be by blinds that are attached to one side of each receptor as indicated in Fig. 7.



Fig. 7.
Reduction of the ambiguity of the mapping function of the design shown in Fig. 6 by attaching blinds to the side of each receptor. The blinds are supposed to absorb the signal in their thickness. The improved design reduces the ambiguity, but does not eliminate it, because more than one receptor is exposed to the signal.


The blinds are supposed to absorb the signal by their thickness. Obviously, the use of the blinds reduces the ambiguity, although a one-to-one mapping is still not achieved. However, if we close the blinds in the manner shown in Fig. 8, only one receptor is exposed to the source, thus yielding the desired one-to-one mapping function. Optimally, the angle of the blinds is such that they extrapolate to the next receptor in the counterclockwise direction (see Fig. Cool.



Fig. 8. 
Final design of the geometry of a two-dimensional cellular eye. After closing the blinds in the depicted way, one and only one receptor is exposed to the signal source, insuring a one-to-one mapping.


If the angle is more radial, the ambiguity of Fig. 7 reappears. If it is more tangential, the design generates blind spots where signal sources cannot reach any of the receptors.

Three-Dimensional Shape of the Eye


The design as developed thus far works as long as the signal source remains within the plane of the ring of receptors. However, we must expect that the signal sources around the real cell are also located above and below this plane, as shown in Fig. 9.



Fig. 9.
Demonstration that the design of Fig. 8 is ineffective if the signal source moves out of the plane of the two-dimensional cellular eye. From the position of the source above the plane, all receptors are exposed to its signal, excluding a one-to-one mapping.

In this case, the full extent of the ambiguity of signal reception would reappear, since all receptors are exposed to the signal. Therefore, the next step in the construction of a cellular eye is the extension of the planar design of Fig. 8 into a three-dimensional one. Considering that the optimal design must remain rotationally symmetrical, there are only two possibilities for its spatial structure: a cylinder and a sphere. In the case of a sphere, it is apparent that it would not allow the attachment of blinds in a slanting fashion all over its surface. At least at the poles of the sphere the slanting directions would contradict each other. (This fact is known to mathematicians as the Theorem of the Well-Combed Hedgehog.) Therefore, the sphere has to be capped as shown in Fig. 10.




Fig. 10. 
The only two possibilities for extending the two-dimensional cellular eye into the thirddimension are a straight cylinder or a capped sphere (a bulging cylinder).

In other words, the three-dimensional structure of the cellular eye has to be a straight or bulging cylinder, but a cylinder.


Requirement of Two Orthogonal Cylinders

The design as developed so far permits a one-to-one mapping between the set of signal receptors and the angle of the signal source as measured around the axis of the cylinder. Calling this angle the longitude of the signal source, it is obvious that we need to map its latitude as well if we wish to identify the direction of the signal source. Since the longitude and latitude of a point are angles measured around axes orthogonal to each other, the design requires a second cylinder at right angles to the first, as shown in Fig. 11.



Fig. 11.
Necessary use of a pair of orthogonally oriented cylinders as a cellular eye to determine simultaneously the longtitude and latitude of the signal source (see text).


Therefore, we conclude that the cellular eye would be a pair of cylinders at right angles to each other, each having the cross section shown in Fig. 8. It hardly needs mentioning that this design bears striking resemblance to a pair of centrioles. If the arguments as presented are correct, then the structure of centrioles and basal bodies is the expression of a physical necessity and not the result of an accident of evolution, which may explain the high degree of conservation of centriole structure throughout nature. The right-angle orientation can be expected to be found whenever the signal sources are distributed in three-dimensional space. If under special circumstances the relevant signal sources are distributed in a plane, only one cylinder is required that is oriented perpendicular to this plane. We speculate, therefore, that the array of basal bodies that are oriented perpendicular to the surface of a ciliated cell not only form nucleating structures for the cilia but also act as monitoring devices for signal sources at the cell surface.

Refinement of Angular Resolution

The similarities between a hypothetical cellular eye and the structure of centrioles and basal bodies extend even further if we try to optimize the angular resolution of the device. The design developed so far limits the angular resolution by the number of receptors and blinds around the perimeter of the cylinder. If a signal source is located anywhere in the sector defined by the angle between adjacent blinds, it will be detected by one and the same receptor. In other words, if the cross section of the cylinder has N blinds, then the limit of angular resolution is Θ = 360°'IN. Therefore, it would be desirable to improve the design in such a way that it can distinguish various directions in each sector. The most direct method to improve the angular resolution would be to increase the number of receptors and blades (N). However, each blade must have a certain thickness in order to absorb the signals. Therefore, the number N cannot be increased at will. A much more elegant way to achieve better angular resolution is to attach the blinds in such a way that they lean across one sector as they run from the top of
the cylinder to the bottom (see Fig. 12).


Fig. 12. 
Pitch of the blades as a means to improve the angular resolution of the design. The percentage of receptors on blade n exposed to the signal source S is a measure of the location of the signal source in the sector between two adjacent blades (see text).


The blades of actual centrioles and basal bodies are, indeed, pitched in this way (cf. Fig. 1). Assume that the viewer is positioned at the location of the signal source S. As shown in the perspective drawing of Fig. 12, the viewer would see the outside of some blades (gray shading), the inside of other blades (white areas), the opaqueness of the inside of the cylinder (black areas), and the rows of receptors exposed to the direction of the source (dots lining the blades). The main difference from the earlier design is the exposure of some receptors along blade n and some receptors along blade w+1 to the source. Therefore, at first sight, the pitched blades may seem to defeat our initial intention of providing a one-to-one mapping between receptors and directions of signal sources. However, the exposed receptors on adjacent blades differ by their position along the blades. More importantly, the apparent ambiguity offers the possibility to determine the location of the direction of the signal source in the sector of blade n. By determining the ratio

                                                               number of receptors exposed on blade n
                                              r   =        -------------------------------------------------
                                                                 number of all receptors on blade n

(see Fig. 12), the angle φ of the signal source reaching receptors on blade «becomes φ = (n + r)6 This angle is no longer an integral multiple of the former limit of resolution Θ = 360°'/N, but contains fractions of Θ, thus improving the angular resolution. It is still limited, though, because of the finite number of receptors lining the inner edge of each blade. The explanation of the calculation just presented is obvious if one considers a
series of cross sections along the cylinder axis, as shown in Fig. 12. As we move up the axis, the cross sections rotate by the angle of one sector. Depending on the direction of the source within the sector, the blade will cut off the signal source at some distance from the bottom of the cylinder, giving rise to the ratio calculated.


Likely Wavelength of Signal Sources

The next important question about our speculation concerns the physical nature of the postulated signal sources. Considering that the function of the cellular eye is to locate objects (signal sources) in the environment of the cell, one may argue that the wavelength emitted by the sources must be linked to the size of these objects. If the objects were smaller than the wavelength, then the cellular eye would not be able to resolve two objects closer than the wavelength. It seems reasonable to assume that the interesting objects around probably include other cells; therefore, we can expect that the wavelength of the sources is smaller than a cell diameter or λ < 10 μπι. This wavelength excludes sound as a signal, because the frequency of a sound wave with 10-μιη wavelength in water is n = (1200 m/sec)/(10 μιη) = 1.2 x 108 Hz. At this frequency, water is a completely absorbing medium for ultrasound and, therefore, unsuitable as medium of transmission of signals. Consequently, we are left essentially with electromagnetic radiation as the suspected signal. The range of the wavelength places the radiation in the range of visible or infrared light. There are various reasons to exclude visible light as potential signal carrier: (1) Most animal cells are located deep inside organisms and are thus cut off from visible light. (2) Centrioles and basal bodies are surrounded by cytoplasmic organelles such as mitochondria and lysosomes that are up to 1 μιη in size. In order to reach the cellular eye without much scattering by such obstacles in the line of sight, the signal sources should emit wavelengths that are larger than these objects. In other words, this argument places the wavelength outside the visible range. Thus we are left with infrared light of wavelengths shorter than 10 μπι as the most likely signal radiation to be detected and located by centrioles. To be sure, it is quite possible that cells or cell parts emit such radiation, because at room temperature these wavelengths belong to the higher harmonics of molecular vibrations and rotations. Further, it is conceivable that cells can receive such signals because certain animals, including snakes, are able to see infrared sources, suggesting that infrared-sensitive receptors are part of the genome of some animals; the postulate of infrared-sensitive receptors in centrioles cannot be dismissed.

The following argument seems to narrow the range of wavelengths even more. As shown in Fig. 13, the main wavelengths of black body radiation at 27°C range from 3 μηι to > 10 μιη. A very similar curve applies to room temperature or 37°C. A cellular eye sensitive to these wavelengths would detect a constant bright glow around the cell, stemming from the ambient temperature, that might bury the signals from surrounding sources in high background noise. Therefore, the wavelength of the sources would be placed best above the visible range and below 3 μιτι (Fig. 13).




Fig. 13. 
Black body radiation at and around room temperature (continuous curve; right ordinate shows the logarithm of the spectral emission; 300 K = 27°C) and extinction coefficient of water (dotted curve; left ordinate) as functions of wavelength in the visible and infrared range (redrawn after Wolfe and Zissis, 1989).

Unfortunately, in this range of infrared radiation, water is already rather opaque and one would suspect that the relevant wavelengths are located in relative "water windows." As shown in the water absorption curve in Fig. 13, there are three areas of relative transparency: 0.8-1.3 μπι, 1.6-1.7 μιτι (marked A), and 2.2-2.4 μπι (marked B). Using a specially designed microscope, we tested the ability of 3T3 cells to locate distant infrared light sources. The experiments, indeed, identify the window between 0.8-0.9 μιη as the most active range (Albrecht-Buehler, 1991).


1) http://www.basic.northwestern.edu/g-buehler/append.htm
2) The Centrosome,  Vitauts I. Kalnins

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