Chemotaxis CheZ position experiments

From freeversity.org

Bacteria swim by their noses. That is, they smell food and swim towards it; they smell waste and swim away. The molecular basis of this amazing feat is the most well studied molecular signal transduction system and as such serves as a model for other lessor-studied bio-molecular signaling.

There's one little detail of the chemotaxis system that caught my attention a few years ago. The signal is transmitted from sensor to motors via a diffusing molecule called "CheY". When CheY has a phosphoryl group attached to it, it activates the motors in a certain way; when it loses that group it reverses the motors.

There's a very interesting subtlety to this system. The part of the system which "charges" the transmitter (to borrow electrical engineering terminology) is a kinase called CheA. The "discharging circuit" is the enzyme CheZ. It turns out that these two enzymes are, counter-intuitively, co-located. That is, it seems odd that the enzymes responsible for pulling-up a signal are co-located with the enzymes that pull it down. It would appear that the system is spinning it's wheels -- undoing what it just did. Why shouldn't it have the pull-down phosphatase evenly distributed or co-located with the motors?

When I first read this detail a few years ago in Eisenbach's book "Chemotaxis" it mentioned this counter-intuitive fact and I thought to myself, "I bet I know why -- it reduces saturation and evens out the signal." I wrote a little simulation years ago and convinced myself that this was indeed the case (at least for my toy simulation). Then I got distracted for years didn't get around to improving the simulations.

My hypothesis is that by co-locating CheZ and CheA the signal will saturate less near the transmitter and become more spatially uniform. My intuition is that when the external signal is rising and transmitter wishes to control the motors it needs a supply of free un-phosphorylated CheY in order to communicate this. Because CheY is produced at one end and diffuses to the other regions, there's a 1/r^2 distribution of it as it produces it. If it turns on the transmitter at some moment then a few moments later there will be an excess of CheY~P near the transmitter but a lot less further away. But, if CheZ is located nearby the transmitter then it is right where it is needed most -- where there's an excess of CheY~P.

Honestly, it's easier to see the effect than to describe it.

In the following figures, the top row has the CheZ co-located with CheA on the left side. The bottom row has the same amount of CheZ evenly distributed. These are space-time plots. Space is on the X axis with the transmitter on the left. Time progresses from the bottom of the graph towards the top. The right plot is the power spectrum of the right most spatial position which simulates the most distant motor's response. In the top row we see two things. First, the distribution is much smoother from left to right than it is in the bottom row. This is good for the bacteria as the motors are scattered throughout the cell and the controller depends on them to synchronously changing state as it switches from laminar movement to chaotic tumbling. Second, the non-linear clipping harmonic (the little spike on the right) is taller in the bottom row and the primary response (the big peak) is a little smaller. This indicates that there's a (mild) fidelity improvement in the co-location of CheA and CheZ. Given the simplicity of this argument I submit that such co-location is probably a common motif in other kinase/phosphatase (and similar pull-up/pull-down type) systems.



Caveats -- this is a scale-free simulation. I made no attempt to model actual parameters but rather went on the assumption that the bacteria probably operates near peak efficiency so I just twiddled the parameters until I saw what appeared to be peak efficiency. Of course, this is hardly rigorous so the next step will be to try to get accurate rate and diffusion constants. If anyone knows where they are conveniently located in one paper, that would be nice. :-)

Antenna analogy

An old fashioned radio without an amplifier.
Image from http://homepage.mac.com/msb/163x/faqs/radio-spark-crystal.html


How can a radio pull power out of the thin air sufficient to hear a radio broadcast many miles away? And what does this have to do with how bacteria swim? They both require an understanding of antennas.

I've been developing an analogy to with my friend John to help demystify antennas; the analogy is also intended to help my chemist friends see how the concepts of antenna design apply equally to all communication systems -- even biochemical ones.

Imagine a boat out at sea on a moonless night. Suppose there are waves rolling by on the ocean. How could we build a device on the boat to detect the invisible waves? A simple solution is to trap a marble into a slot and attach a switch a either end.



As the boat buoys up and down with the waves the ball will roll back and forth hitting the switches on either end. Consider why this works -- when the wave passes it lifts one side of the boat before the other. This lifting creates gravitational potential energy between one side of the boat and the other. Whenever there's a potential difference, anything free to change state under that potential will do so. In this case the marble will convert that potential energy into kinetic energy (and friction) which is detectable as the marble hits the switch.

There's a few non-obvious aspects of this that are easy to see when using this boat analogy and harder to see when talking about other kinds of antenna. Understanding these subtleties permits one to have much greater intuition for antenna design in other domains be that electrical or biochemical.

Aspect #1) It only works if the boat is rigid. If it was not rigid, say like a inflatable raft, the boat would simply deform to the shape of the wave and the marble could just sit in one place as the wave passes by.


Non-rigid ship.


#2) The length of the boat relative to the wavelength is important. Imagine how our marble-based water-wave receiver system would behave under two extremes: the boat being very short compared to the wave (top picture below) and the boat being very long compared to the wave (bottom).



When the boat is very short compared to the wave length, it hardly feels any potential energy difference between the bow and stern and thus the marble will not respond well to the wave. Similarly in the other extreme. If the boat is so long that many waves can pass under the it at the same time then again there will be little potential difference between bow and stern and the marble will not roll. We can therefore see that there is an optimal length range for our boat-antenna that is approximately equal to the receiving wave-length.

#3) The friction of the marble is important. A marble traveling through a denser medium will be slower to accelerate than will a marble going through a low viscosity medium. Imagine the marble moving through honey so that as the wave passes by it hardly has a chance to move at all before the wave has passed. Obviously this would be a bad detector because the ball wouldn't hit the switches.

Conversely if the ball were able to move too quickly, it would get all the way to the end of the rail very quickly and it would just sit on the switch. Although that might not matter for a simple-minded wave detector that only wished to detect the absence or presence of the wave (a binary sensor), it would matter if you were using the marble's velocity to drive some other system; for example, if we were making a recording of the marble's position to make a picture of the invisible waves. When the marble is able to travel so quickly to the end of the detector that it just sits uselessly at one end or the other it is called "saturation" or "clipping" and introduces a very particular kind of distortion called "clipping harmonics" and can be easily seen in the power spectrum with and without the clipping as seen below.


#4) You don't need a "ground" to detect a wave. The reason that a pilot can use a radio in the air is that that detecting a wave has only to do with detecting the difference in potential between the "top" of the wave and the "bottom". Indeed you need to be careful not to ground yourself in many cases because by "grounding" yourself you're creating an antenna that is the size of the earth!! For example, consider a boat mooring.



If you were to connect your boat to the ground then you'd be detecting the potential difference between ground and any wave even waves the size of the whole earth -- the tides. This can be very dangerous as the potential might be so great that it could destroy a ship. That's why in the picture below you see that boats that are moored to docs have to have rollers on them to isolate them from ground.



If there rollers weren't there the moored boats would be dangling from the ropes when the tide went out or deluged when the tide came in!


All of this stuff about length and friction boils down to a time constant. You need the marble "sensor" to have roughly the same time constant as the wave you're trying to detect. If the detector responds too slow (because it is too long or too burdened by friction for example) then you won't detect the waves very well. If your detector responds too fast (because it is too short or too frictionless for example) then the sensor will saturate.

This analogy demonstrates that an antenna is a "rigid" device that has a tuned response to a changing potential energy. This is true no matter the technology. And this lesson teaches us that antennas of any variety be they electrical, chemical, or anything else must be tuned to respond at a time scale in the ballpark of the speed that the signal of interest changes.

For example, an electrical antenna is a metal rod inside of which there are mobile electrons that are analogous to the marbles. As an electromagnetic field passes by the antenna the front and back of the antenna have different electrical potential so that electrons rush from end to end to try to cancel that potential just like the marbles did. And just as was the case with the boat-and-marble antenna, the length and tuning of detector circuit matters to optimize the response of the system to the wave.





All sensors of any technology are driven by changing potential energy. Consider a beautiful example of a biochemical "antenna" -- the bacterial chemotaxis sensor.



The receptor/sensor is a trans-membrane enzyme complex that undergoes a conformational change when it binds to a ligand of interest. The concentration of the ligand is variable in space and time and thus the bacteria needs to have a tuned antenna that responds at the same time scale. Imagine two extremes. Suppose the kinetics of the receptor enzyme were extremely slow to release the ligand. In that case, the bacteria would believe that the ligand was a high concentration even after it swam somewhere it wasn't. Conversely, imagine that the kinetics of the system were such that the motor was quickly saturated with signal. In that case, you'd get clipping distortion as described earlier. Both situations would reduce the performance of the chemotaxis system thus we'd expect that the bug would have evolved a circuit that is tuned to the time constants in the same rough proportion to the speed at which ligands change in the environment it is searching.

The above argument applies to any and all biochemical reactions. Ultimately every informatic aspect of a cell comes down to communicating information from place to place using diffusing metabolites. Therefore there's a lot to be said for thinking of the kinetics of these systems as "antenna" that are transmitting and receiving chemical messages at particular speeds with tuned circuits to optimize those communications.

[Revision 20 Jun -- thanks to my friend Sean Dunn for pointing out that I had incorrectly used a mass analogy where I should have used a viscosity analogy.]

ScienceNews

A few points from my talk yesterday made it into the web edition of ScienceNews today.

Talk at Council for the Advancement of Science Writing

Just gave a talk for a conference of science writers. They shot some (not so high quality) video of it viewable here.

The slide deck is here.

Molecular computers -- A historical perspective. Part 2

We left off last time discussing the precision of an analog signal.

Consider a rising analog signal that looks like the following ramp.


Notice that there's noise polluting this signal. Clearly, this analog signal is not as precise as it would be without noise. How do we quantify this precision? The answer was described in the early 20th century and is known as the Shannon-Hartly theorem. When the receiver decodes this analog variable what is heard is not just the intended signal but rather the intended signal plus the noise (S+N); this value can be compared to the level of pure noise (N). Therefore the ratio (S+N)/N describes how many discrete levels are available in the encoding.



The encoding on the left is very noisy and therefore only 4 discrete levels can be discerned without confusion; the one in the middle is less noisy and permits 8 levels; on the right, the low noise permits 16 levels. The number of discrete encodable levels is the precision of the signal and is conveniently measured in bits -- the number of binary digits it would take to encode this many discrete states. The number of binary digits need is given by the log base 2 of the number of states, so we have log2( (S+N)/N ) which is usually algebraically simplified to log2(1+S/N).

It is important to note that although Shannon and Hartley (working separately) developed this model in the context of electrical communication equipment, there is nothing in this formulation that speaks of electronics. The formula is a statement about information in the abstract -- independent of any particular implementation technology. The formula is just as useful for characterizing the information content represented by the concentration of a chemically-encoded biological signal as it is for the voltage driving an audio speaker or the precision of a gear-work device.

We're not quite done yet with this formulation. The log2(1+S/N) formula speaks of the maximum possible information content in a channel at any given moment. But signals in a channel change; channels with no variation are very dull!


(A signal with no variation is very dull. Adapted from Flickr user blinky5.)

To determine the capacity of a channel one must also consider the rate at which it can change state. If, for example, I used the 2 bit channel from above I could vary the signal at some speed as illustrated below.


(A 2-bit channel changing state 16 times in 1 second.)

This signal is thus sending 2 bits * 16 per second = 32 bits per second.

All channels -- be they transmembrane kinases, hydraulic actuators, or a telegraph wires -- have a limited ability to change state. This capacity is generically called its "bandwidth" but that term is a bit over simplified so let's look at it more carefully.

It is intuitive that real-world devices can not instantaneously change their state. Imagine, for example, inflating a balloon. Call the inflated balloon "state one". Deflate it and call this "state zero". Obviously there is a limited rate at which you can cycle the balloon from one state to the other. You can try to inflate the balloon extremely quickly by hitting it with a lot of air pressure but there's a limit -- at some point the pressure is so high that the balloon explodes during the inflation due to stress.


(A catastrophic failure of a pneumatic signalling device from over-powering it. From gdargaud.net)

Most systems are like the balloon example -- they respond well to slow changes and poorly to fast changes. Also like the balloon, most systems fail catastrophically when driven to the point where the energy flux is too high -- usually by melting.


(A device melted from overpowering it. Adapted from flickr user djuggler.)

Consider a simple experiment to measure the rate at which you can switch the state of a balloon. Connect the balloon to a bicycle pump and drive the pump with a spinning wheel. Turn the wheel slowly and write down the maximum volume the balloon obtains. Repeat this experiment for faster and faster rates of spinning the wheel. You'll get a graph as follows.


(Experimental apparatus to measure the cycling response of a pneumatic signal.)


(The results from the balloon experiment where we systematically increased the speed of cycling the inflation state.)

On the left side of the graph, the balloon responds fully to the cycling and thus has a a good signal (S). But, on the left side very few bits can be transmitted at these slow speeds so there's not a lot of information able to be sent despite the good response of the balloon. But, further to the right the balloon still has a good response and now we're sending bits much more rapidly so we're able to send a lot of infrmation at these speed. But, by the far right of the graph, when the cycling is extremely quick, the balloon response falls off and finally hits zero when it popped so that defines the frequency limit.

The total channel capacity of our balloon device is an integral along this experimentally sampled frequency axis where we multiply the number of cycles per second at that location by the log2( 1+S/N ) where S is now the measured response from our experiment which we'll call S(f) = "The signal at frequency f". We didn't bother to measure noise as a function of frequency in our thought experiment, but we'll imagine we can do that just as easily and we'll have a new graph N(f) = "The noise at frequency f". The total information capacity (C) of the channel is the integral of all these products across the frequency samples we took up to the bandwidth limit (B) where the balloon popped.



If you want to characterize the computational/communication aspects of any system you have to perform the equivilent of this balloon thought experiment. Electrical engineers all know this by heart as they've had it beaten into them since the beginning of their studies. But, unfortunately most biochemists, molecular biologists, and synthetic biologist have never even thought about it. Hopefully that will start to change. As we both learn more about biological pathways and we become more sophisticated engineers of those pathways we will have an unnecessarily shallow understanding until we come to universally appreciate the importance of these characteristics.

Next, amplifiers and digital devices. To be continued...

Molecular computers -- A historical perspective. Part 1

I've been having discussions lately with Andy regarding biological/molecular computers and these discussions have frequently turned to the history of analog and digital computers as a reference -- a history not well-known by biologists and chemists. I find writing blog entries to be a convenient way to develop bite-sized pieces of big ideas and therefore what follows is the first (of many?) entries on this topic.


In order to understand molecular computers -- be they biological or engineered -- it is valuable to understand the history of human-built computers. We begin with analog computers -- devices that are in many ways directly analogous to most biological processes.

Analog computers are ancient. The first surviving example is the astonishing Antikythera Mechanism (watch this excellent Nature video about it). Probably built by the descendants of Archimedes' school, this device is a marvel of engineering that computed astronomical values such as the phase of the moon. The device predated equivilent devices by at least a thousand years -- thus furthering Archimedies' already incredible reputation. Mechanical analog computers all work by the now familiar idea of inter-meshed gear-work -- input dials are turned and the whiring gears compute the output function by mechanical transformation.


(The Antikythera Mechanism via WikiCommons.)

Mechanical analog computers are particularly fiddly to "program", especially to "re-program". Each program -- as we would call it now -- is hard-coded into the mechanism, indeed it is the mechanism. Attempting to rearrange the gear-work to represent a new function requires retooling each gear not only to change their relative sizes but also because the wheels will tend to collide with one another if not arranged just so.

Despite these problems, mechanical analog computers advanced significantly over the centuries and by the 1930s sophisticated devices were in use. For example, shown below is the Cambridge Differential Analyzer that had eight integrators and appears to be easily programmable by nerds with appropriately bad hair and inappropriately clean desks. (See this page for more diff. analyzers including modern reconstructions).


(The Cambridge differential analyzer. Image from University of Cambridge via WikiCommons).

There's nothing special about using mechanical devices as a means of analog computation; other sorts of energy transfer are equally well suited to building such computers. For example, in 1949 MONIAC was a hydraulic analog computer that simulated an economy by moving water from container to container via carefully calibrated valves.


(MONIAC. Image by Paul Downey via WikiCommons)


By the 1930's electrical amplifiers were being used for such analog computations. An example is the 1933 Mallock machine that solved simultaneous linear equations.


(Image by University of Cambridge via WikiCommons)

Electronics have several advantages over mechanical implementation: speed, precision, and ease of arrangement. For example, unlike gear-work electrical computers can have easily re-configurable functional components. Because the interconnecting wires have small capacitance and resistance compared to the functional parts, the operational components can be conveniently rewired without having to redesign the physical aspects of mechanism, i.e. unlike gear-work wires can easily avoid collision.

Analog computers are defined by the fact that the variables are encoded by the position or energy level of something -- be it the rotation of a gear, the amount of water in a reservoir, or the charge across a capacitor. Such simple analog encoding is very intuitive: more of the "stuff" (rotation, water, charge, etc) encodes more of represented variable. For all its simplicity however, such analog encoding has serious limitations: range, precision, and serial amplification.

All real analog devices have limited range. For example, a water-encoded variable will overflow when the volume of its container is exceeded.



(An overflowing water-encoded analog variable. Image from Flickr user jordandouglas.)

In order to expand the range of variables encoded by such means all of the containers -- be they cups, gears, or electrical capacitors -- must be enlarged. Building every variable for the worst-case scenario has obvious cost and size implications. Furthermore, such simple-minded containers only encode positive numbers. To encode negative values requires a sign flag or a second complementary container; either way, encoding negative numbers significantly reduces the elegance of the such methods.

Analog variables also suffer from hard-to-control precision problems. It might seem that an analog encoding is nearly perfect -- for example, the water level in a container varies with exquisite precision, right? While it is true that the molecular resolution of the water in the cup is incredibly precise, an encoding is only as good as the decoding. For example, a water-encoded variable might use a small pipe to feed the next computational stage and as the last drop leaves the source resivoir, a meniscus will form due to water's surface tension and therefore the quantity of water passed to the next stage will differ from what was stored in the prior stage. This is but one example of many such real-world complications. For instance, electrical devices, suffer from thermal effects that limit precision due to added noise. Indeed, the faster one runs an electrical analog computer the more heat is generated and the more noise pollutes the variables.


(The meniscus of water in a container -- one example of the complications that limit the precision of real-world analog devices. Image via WikiCommons).

Owing to such effects, the precision of all analog devices is usually much less than one might intuit. The theoretical limit of the precision is given by Shannon's formula. Precision (the amount of information encoded by the variable, measured in bits) is log2( 1+S/N ). It is worth understanding this formula in detail as it applies to any sort of information storage and is therefore just as relevant to a molecular biologist studying a kinase as it is to an electrical engineering studying a telephone.

.... to be continued.