I created a number of cheat sheets for a class I did this quarter. The cheat sheets cover the topic of metabolic control. They include: 

1. General Metabolic Control 
2. Advanced Metabolic Control 
3. Linear Chains 
4. Branch Points 
5. Moiety Conserved Cycles

They are available a this GitHub repo:

https://github.com/hsauro/CheatSheets

Both the original LaTeX and pdfs files are provided.

MCA Rediscovered

It looks like someone has rediscovered metabolic control analysis (MCA).

A structural approach to understanding enzymatic regulation of chemical reaction networks Biochem J (2022) 479 (11): 1265–1283. Atsushi Mochizuki

The analysis is exactly the same as MCA but uses different symbols and they focus on the unscaled sensitivities instead. eg the r symbols are the unscaled elasticities. The core equation (4) can be found in equation (1) of the appendix of the following paper, and I am sure its been published elsewhere too:

Taking enzyme kinetics out of control; putting control into regulation. Jan-Hendrik S. HOFMEYR, Athel CORNISH-BOWDEN and Johann M. ROHWER, Eur. J. Biochem. 212, 833-837 (1993).

However, unlike the original MCA, the latest reincarnation doesn't include support for conserved moieties so as it stands it's somewhat limited. Note equation (1) in the above paper includes additional terms to take into account any conserved moieties. 

What is more concerning is that the reviewers of the paper never spotted this duplication of work. 

Some textbooks on MCA: 

Systems Biology: An Introduction to Metabolic Control Analysis (2018)

Understanding the Control of Metabolism (Frontiers in Metabolism Ser. ; No. 2)) 1st Edition (1996)

Fundamentals of Enzyme Kinetics 4th Edition (2012)

PID Control Demonstration

Those of you who are interested in feedback control may have come across the infamous PID controller. To many engineers, PID control is the answer to all their problems. What is it for? PID controllers are a sophisticated way to stabilize a system that is under continual threat of disturbance. The simplest example is the cruise control in a car. Most of us probably set the cruise control and forget about it. What’s happening behind the scenes is however much more interesting. As we go up a hill we notice the engine revving up to maintain the set cruise speed and when we go down a hill the engine slows down. The simplest feedback mechanism is where the controller compares the speed of the car with the set cruise speed and adjusts the engine accordingly. The secret to a good controller such as this is what it does with the error it finds between the actual speed of the car and the desired speed. The simplest solution is to adjust the speed of the engine directly in proportion to the difference, this is called proportional control and is most familiar to engineers. The problem with proportional control is twofold:

1. Too much proportional control and the response overshoots, perhaps so much so that it goes into uncontrolled oscillations, not a good thing in a car.

2. You usually have to set the cruise control slightly higher than the speed you actually want because a proportional controller finds it difficult to actually match the set speed unless the proportional control is strong enough (See problem 1!)

Both these problems can be solved by combining proportional control (The P in PID) with two others, integral (The I in PID) and derivative control (The D in PID). Derivative control is perhaps the easiest to understand. What derivative control does is measure how fast the error between desired and actual engine speed changes. If the difference is changing too fast then we want to slow down the response to avoid overshooting. Derivative control will therefore help get rid of the instability (Too much derivative control can however also result in instability).

The second problem is that the speed of the car doesn’t actually settle to the desired speed. This is fixed by using integral control. This is where the error is continually added up to form a net error and it is the net error than is then used to control the speed of the engine. So long as there is even a little bit of error (ie Desired speed 50 mph, actual speed 49 mph), the net error will increase and drive the car speed to the desired speed until the actual error is zero. Integral is particularly useful if there is a continual disturbance at work, for example, a headwind.

To demonstrate these capabilities, I have written a small PID control demonstration tool (for windows only I am afraid) that simulates each of the three elements in a PID controller. The screenshot below shows the application. A user can control the strength of the different controlling elements or remove them all together. To test the response a user can apply a step change to the desired car speed and see how the system responds. The simulation is continuous so it is easy to see the effects.

Download the PID Controller Demo here

Partly supported by the NSF

Front-Loading of Flux Control

Originally Posted on December 20, 2019 by hsauro

There is an interesting property that straight chain pathways have which I call front-loading. The phenomenon has been known for some time and I describe it in detail in my recent textbook on Metabolic Control Analysis. Let’s say we have a straight chain of reactions, something like:

  

There are no long-range feedback loops (other than short-range product inhibition) and it is assumed that for a given reaction, increases in substrate cause the reaction rate to increase and increases in the product causes the reaction to decrease. Each reaction is reversible using a simple reversible mass-action rate law: . I assume that  and  are fixed species. Given suitable rate constants and equilibrium connstants for each reaction, the system will admit a steady state. A question to ask is what is the distribution of flux control across the chain? There are two ways to answer this, we can use theory, this gives us what the distribution is and insight into why or we can do simulation but this will just tell us what the distribution is likely to be. For now let’s do a simulation.

The code is shown below. The code defines a four-step straight-chain, runs 10,000 simulations, randomizing the rate constants for each simulation. Each time it does a simulation we compute the flux control at each step, store this information and then at the end, plot histograms of the distribution of the flux control coefficients. Scroll down to see the plots. The average values for the flux control coefficients is shown in the histogram:

The distribution of flux control is shown below. The vertical axis is the frequency and the x-axis the value of the flux control coefficient from zero to 1.0. The red curve corresponds to the flux control for the first step, on average this has the highest control. The yellow distribution corresponds to the last step of the pathway, you can see it is least likely to have any significant flux control. The conclusion is that given a straight chain with a random set of parameters, on average the first step will have the highest control, progressively decreasing as we work our way down the pathway. Why is this? The explanation is a thermodynamic one which in turn boils down to have easy it is for perturbation to travel up and down the chain. So long as the thermodynamic gradient is from left to right, it is easier for perturbations to propagate downstream compared to upstream. Since flux control is really a measure of how much a perturbation has on the steady-state flux, the easier a perturbation can travel the higher the flux control.

This is not to say that it isn’t possible to get excellent flux control in downstream steps, it’s just that fewer parameter combinations will achieve that.

# Monte Carlo simulation of a straight chain pathway
# Samples parameter values while keeping Keq constant
# Plots the distribution of control coefficients
# Not the most elegant code but I wrote it quickly
 
import tellurium as te
import roadrunner
import random
import pylab as pl
 
r = te.loada("""
   J1: $Xo -> S1; k1*(Xo - S1/Keq1);
       S1 -> S2;  k2*(S1 - S2/Keq2);
       S2 -> S3;  k3*(S2 - S3/Keq3);
       S3 -> $X1; k4*(S3 - X1/Keq4);
     
     k1 = 0.1; k2 = 0.1;
     k3 = 0.1; k4 = 0.1;
     
     Keq1 = 4;
     Keq2 = 3;
     Keq3 = 2;
     Keq4 = 1;
     Xo = 5;
     X1 = 0.1;
""")
 
m = r.simulate (0, 10, 100);
 
aC1 = 0; aC2 = 0; aC3 = 0; aC4 = 0;
aC1a = []; aC2a = []; aC3a = []; aC4a = [];
n = 1000
a = 0
upperLimitK = 50
for i in range (0,n):
    r.setValue ('k1', random.uniform(0, upperLimitK))
    r.setValue ('k2', random.uniform(0, upperLimitK))
    r.setValue ('k3', random.uniform(0, upperLimitK))
    r.setValue ('k4', random.uniform(0, upperLimitK))
    
    try:
      r.simulate()    
      r.steadyState()
      C1 = r.getCC ('J1', 'k1')
      C2 = r.getCC ('J1', 'k2')
      C3 = r.getCC ('J1', 'k3')
      C4 = r.getCC ('J1', 'k4')
      aC1 = aC1 + C1
      aC2 = aC2 + C2
      aC3 = aC3 + C3
      aC4 = aC4 + C4
      aC1a.append (C1)
      aC2a.append (C2)
      aC3a.append (C3)
      aC4a.append (C4)
    except:
        a = a + 1
 
print (aC1/n, aC2/n, aC3/n, aC4/n) 
 
bins = 100
 
pl.hist(aC1a, bins=bins, histtype='stepfilled', density=True, color='r', alpha=0.5, label='C1')
pl.hist(aC2a, bins=bins, histtype='stepfilled', density=True, color='b', alpha=0.5, label='C2')
pl.hist(aC3a, bins=bins, histtype='stepfilled', density=True, color='g', alpha=0.5, label='C3')
pl.hist(aC4a, bins=bins, histtype='stepfilled', density=True, color='y', alpha=0.5, label='C4')
print ("fails = ", a)

Inverse Laplace Transform of e^(2 s)/((s^2 + 1)

I recently needed to compute the inverse Laplace transform of: $$\frac{e^{-2s}}{s^2 + 1}$$ For something like this we'd use the second time shifting theorem: $$ \mathcal[L] f(t-a) u(t-a) = F(s) e^{-as} $$ where in this case $ F(s) = 1/(s^2 + 1) $ We just need to take the inverse transform of $ 1/(s^2+1) $ which is $ \sin (t) $ and then substitute in the time delay so that we arrive at: $ \mathcal[L] f(t-a) u(t-a) = F(s) e^{-as} = u(t-2) \sin (t-2) $ Not too difficult. Out of curiously, I tried to use Mathematica to compute the inverse transform and most surprisingly it failed.

Bode Plots using Python

 I needed a quick way to plot some Bode plots for a second-order system. I didn't have access to Matlab, instead, I searched for a solution using Python, and I found one. Documentation is a bit sparse so this example might be helpful.

The signals package supports the signal.bode method which turned out to be quite easy to use. Signal is part of the scipy package and is something we bundle with our Tellurium platform.

There is a more comprehensive discussion of Python and Control Theory here.

[code lang="python" fontsize="10"]
#
import numpy as np
from scipy import signal
from matplotlib import pyplot as plt

# Coefficients in numerator of transfer function
num = [1]
# Coefficients in denominator of transfer function
# High order to low order, eg 1*s^2 + 0.1*s + 1
den = [1, 0.1, 1]

# Scan over zeta, a parameter for a second-order system
zetaRange = np.arange(0.1,1.1,0.1)

f1 = plt.figure()
for i in range(0,9):
    den = [1, 2*zetaRange[i], 1]
    print den
    s1 = signal.lti(num, den)
    # Specify our own frequency range: np.arange(0.1, 5, 0.01)
    w, mag, phase = signal.bode(s1, np.arange(0.1, 5, 0.01).tolist())
    plt.semilogx (w, mag, color="blue", linewidth="1")
plt.xlabel ("Frequency")
plt.ylabel ("Magnitude")
plt.savefig ("c:\\mag.png", dpi=300, format="png")

plt.figure()

for i in range(0,9):
    den = [1, zetaRange[i], 1]
    s1 = signal.lti(num, den)
    w, mag, phase = signal.bode(s1, np.arange(0.1, 10, 0.02).tolist())
    plt.semilogx (w, phase, color="red", linewidth="1.1")
plt.xlabel ("Frequency")
plt.ylabel ("Amplitude")
plt.savefig ("c:\\phase.png", dpi=300, format="png")

[/code]

Output from Python script:

Magnitude plot:

Phase plot:

Bottlenecks and Slow Steps - again

October 28, 2013 5:55 pm

I've written a few articles on this blog about the irrationality of bottlenecks and slow steps in metabolism, hoping that one day my fellow scientists will feel the same. Sadly it is not true and maybe I should write more because I week or two ago I came from a talk about metabolic engineering which made some egregious statements that I wouldn't even expect my undergraduates to make. Apparently *the* bottleneck (note singular) in a metabolic pathway is the 'slowest' step and that's the one to engineer. I should add that this not quite right. Unfortunately I've no idea what they meant by the slowest step and I didn't have the heart to ask them given how lost they were. If I did ask, I suspect they'd become even more confused.

I think its time to write a blog on "What's the deal with slow steps"'

Linear Dynamics Application

May 16, 2013 9:22 am

There are a number of online apps that can show students the phase plane dynamics of a 2d system. Notable examples include:

Phase Portrait Viewer from the d’Arbeloff Interactive Math Project - requires Java installed
Wolfram Phase Portrait Viewer - required Mathematica CDF Installed
Programmable Interactive Simulator (DField and PPlane) - requires Java (Matlab version also available I believe)

Nevertheless, I decided, just for the hell of it to write my own, this time a desktop application for windows. Screenshot and download link below:

The download is below. Nothing to install, the application is a single exe that you just run. Works on Windows and Wine emulator.

Download linear dynamics simulator

Written using Delphi XE.

Choke points, Load points, Hots spots and Key steps

March 23, 2013 8:56 pm

Choke points, load points, hot spots, key steps and critical steps are some of the many phrases that have been coined to describe places in a biochemical network where perturbations are said to make a difference. Locating these places is important because they are sites where we would target a drug or reengineer to change a cell's phenotype. The question is what are these points, is a choke point something that can be identified as a physical thing, can a choke point be removed from a pathway and studied in isolation? To answer these questions let us look at a very simple two step pathway. Both steps in the pathway are governed by reversible Michaelis-Menten kinetics, so nothing out of the ordinary. Let's say that a perturbation to the enzyme activity at the second step results in a significant change in the steady-state flux through our little pathway, The first step has hardly any effect. We might therefore call the second step the choke point of the pathway, or the rate-limiting step, the hot spot etc.

Is there something special about the second step that makes it a 'choke point'? What may surprise many is that it is the first step that makes the second step a choke point. There is nothing intrinsically special about the second step that makes it the choke point, all the interesting action occurs at the first step.

The explanation is simple. The reaction rate through the first step is very sensitive to the product it makes. If the product of the first step goes up a bit the result is that the reaction rate through the first step goes down, this is simply due to strong product inhibition, that is the product is an effective competitor with respect to the substrate for the first step. So what? Consider this scenario, we add a drug that can inhibit the activity of the second step. Its reaction rate goes down. This results in the substrate for the second step to go up (its being consumed less so it must go up). Since the first step is sensitive to this concentration, its rate goes down. The net effect is that it looks as if the second step is the choke point. The name choke suggests a small diameter pipe so that making the pipe smaller has a marked effect on the flow. But this isn't what is happening here. What is happening is that there is negative feedback from the second step to the first step and it is the first step that slows down the flow. If it weren't for the product inhibition on the second step, the second step would have NO effect on the flow. The second step is actually NOT a choke point at all.

The words we use can be very misleading especially when dealing with complex systems. In this case the work choke is extremely misleading. What about the other phrases that people use, such as a hot spot. Hot spot is certainly a colorful phrase to use but is it a phrase that should be used by intelligent people? Its certainly a word that politicians or my 4 your old might understand, but is it a word that professional scientists should use? What about the word key step or critical step? These don't convey the point that the step itself is not the one responsible for it being influential, it was the first step that gave the second step its ability to control flux. If anything the key step is the first step. These words aren't therefore really appropriate either. What about the word rate-limiting step, still in use today in some circles? That word also suggests a pipe that is too small to carry the flow, it has the same problem as the word choke point.

Most of the words we use are either imprecise or just misleading. What do we use instead? As scientists we should define precisely what we mean, preferably operationally and quantitatively. Forty years ago such a definition was given, that definition is the flux control coefficient, it is simply a number that tells us how much of an effect a perturbation has on the system as a whole. It doesn't explain why a particular step has a given a degree of influence, it just tells us how much influence the step has.

Metabolic pathways such as the amino acid biosynthetic pathways which have allosteric negative feedback are an interesting case. The old literature (and even some new literature) will refer to the step that is regulated by the allosteric effector the rate-limiting step, the choke point or hot spot in more recent language. But modifying this step actually has little effect on the phenotype. It turns out that its the step after the feedback loop that has most of the influence. However it is the regulated step that allows influence to be on the last step. The allosterically regulated step is an important step but it certainly isn't the rate-limiting step, choke point or hot spot.

A real choke point, The Battle of Thermopylae.
Leonidas at Thermopylae, by Jacques-Louis David, 1814.
Image and data provided by Columbia University.

 

Jim Burns’ Ph.D Thesis

Originally Posted on March 27, 2012 by hsauro

Why on earth should anyone want to download a Ph.D thesis that is now almost 40 years old and presumably way out of date and completely irrelevant in today’s terabyte data world? Well for one, Jim’s thesis is an important historical document relating to the origin of Metabolic Control Theory (MCT). For those who don’t know, MCT (or BST as developed by M. Savageau) is the first, clear, practical and theoretically useful treatment of complex cellular enzyme networks. In a sense, it continues where enzyme kinetics becomes unmanageable though it uses a completely different approach. One of the nice aspects of MCT is that it is actually quite simple and is amenable to anyone with a basic understanding of differential calculus. Anyone who calls themselves a scientist with have this knowledge, those who don’t can’t. Those who dismiss MCT or pass over it without much thought have pretty much missed the point. What is the point? Well…..

• It’s a formal language to describe complex cellular networks.
• To those who are willing to spend time studying MCT, it forces clarity of thought.
• It connects genome and environment with phenotype.
• It helps to uncover the cause and effects in a complex system; who does what, how, and why.
• It helps to give direction in the search for novel drug targets and the rational manipulation of metabolism.

What I find most remarkable about Jim’s text is it’s modern feel which suggests that the content was decades ahead of his time. One finds in the thesis work that was anticipated by later workers, even results which were later presented, unknowingly of course, as novel by subsequent researchers. So I hope the thesis will be of interest, not only for historical reasons but also as a source of inspiration for future work.

Here is the thesis

I am grateful to Jim Burns for allowing me to upload the text of his thesis. Also thanks to Jannie Hofmeyr who assisted in the conversion process.

MAPK: Feedback Amplifier, Part 2

May 30, 2011 2:27 pm

In part 1 of this series, a brief history of negative feedback was given. Here we will look at the advantages and disadvantages of negative feedback, particularly in relation to signal transmission.

Amplifiers

Amplifiers used by electrical engineers are designed to magnify the current or voltage in an electrical circuit. A simple example is the voltage amplifier (i.e. voltage-controlled voltage source) which samples the voltage in one part of a circuit and produces a proportionally larger voltage in another. Of critical importance to an amplifier is not so much the amplification factor itself but how accurate the amplification is. There is no such thing however as the perfect amplifier and real amplifiers, electrical or biological will introduce errors or distortions into the signal. These distortions can be classed into three types:

1. Frequency distortion
2. Phase distortion
3. Harmonic distortion

Frequency distortion is do to the fact that amplifiers do not amplify signals at different frequencies to the same extend. The term bandwidth is used to indicate the range of frequencies over which a given amplifier can faithfully amplify a signal.

The second source of distortion, that is phase distortion, is do to the fact that as the amplifier operates it will add delays into the signal. The amount of delay will often be a function of the signal frequency.

Finally, harmonic distortion is due to the fact that amplifiers do not amplify a signal by a fixed amount. That is the amplifier will have some nonlinear behavior which will often be a function of signal frequency.

In the 1920s, such distortions were a huge problem to the new telecommunications industry and it was Harold Black's solution to use negative feedback that solved the problem.

Negative Feedback

In order to understand how negative feedback can improve the performance of a signal amplifier, we must consider a very simple example. The figure below comes from the paper: "MAPK Cascades as Feedback Amplifiers"

Let us consider only the steady-state behavior of the system. The input is given by $u$, the output by $y$, the error $e$, and the disturbance by $d$. The input is the signal we want to magnify and the magnified version of the input is the output, $y$. Some of the output we feed back via $F$ to the input, where we subtract it from the input, $u$. $A$ is the amplifier itself. We will ignore the disturbance $d$ for the moment. The way to look at this diagram mathematically is that any arrow coming out of a block is the product of the block and the arrow coming into the block. For example, consider the output, $y$. This output is a result of the amplifier, $A$, magnifying the error, $e$, that is $y = A e$. What about $e$? $e$ is the result of subtracting $F y$ from $u$. From these statements we can write the following two equations:

$$ y = A e $$ $$ e = u - F y $$ From these two equations we can eliminate $e$ to find: $$ y = \frac{A u}{1 + A F} $$

Calling $G = A/(1 + A F)$ the system gain, we have simply, $y = G u$. Comparing $G$ with $A$, it should be clear that the feedback reduces the gain of the amplifier. Further, if the loop gain $A F$ is large ($A F \ge 1$), then

$$ G \approx \frac{A}{A F} = \frac{1}{F} $$

That is, as the gain $A F$ increases, the system behavior becomes more dependent on the feedback loop and less dependent on the amplifier itself. But so what? Three things are apparent from this simple analysis, the first is that any variation is $A$ has no effect on the operation of the system, that is because $G$ is independent of $A$.From a practical point of view, the manufacturing tolerance of $A$ doesn't have to be so high which makes it possible to make cheap $A$s. Instead the designer need only provide a stable feedback mechanism, in electronics this is in the form of cheap but high tolerance resistors.

Secondly advantage of having feedback, if we introduce a disturbance, $d$, into the output we find that in the presence of feedback, the influence of the disturbance decreases. Finally, and this is the real magic, any nonlinearity present in the amplifier $A$, is eliminated (or at least greatly reduced), this means that our feedback amplifier is very good at faithfully magnifying the input signal, exactly what we want from an amplifier (Proofs of these assertions can be found in the original papers, eg see [3]). In summary a feedback amplifier provides the following desirable characteristics: 1. Increased robustness with respect to internal perturbations. 2. Insulation from external perturbation, resulting in functional modularization. 3. A linear graded response over an extended operating range.

A word about the source of the internal variations in $A$. There are two primary sources, the first is manufacturing variability, that is not every amplifier is exactly the same as it comes off the assembly line. Secondly, when the circuit operates, it heats up and this introduces thermal noise which feeds noise directly into the circuit. Also repeated heating and cooling can cause the amplifier components to slow change in behavior. All these sources of variation can be greatly reduced by adding a negative feedback loop.

What has this got to do with MAPK? The MAPK cascade has many features that are similar to a negative feedback amplifier. The input is from a small signal at the receptor, The amplifier part, that is the three phosphorylation cycles act as amplifiers with high gain. A negative feedback wraps the entire structure and finally, in the last stage, the protein ERK2, must diffuse to the nucleus to make any difference. This represents a disturbance in the output signal. The noise in the amplifier part of MAPK can comes from a number of sources. First, there may be natural allelic variation in the cascade proteins but perhaps more significant is the stochastic noise that occurs in transcription and translation which means that the mean concentration of the cascade proteins vary over time. Negative feedback will greatly reduce the effect the variations have on the performance of MAPK.

There is more to a negative feedback amplifier than described here but the most important points are described.

References

1. Quantitative analysis of signaling networks. Sauro HM, Kholodenko BN, Prog Biophys Mol Biol. 2004 Sep;86(1):5–43.

2. The Computational Versatility of Proteomic Signaling Networks, Sauro HM In: Current Proteomics, Vol. 1, Bentham Science Publishers Ltd. (2004) , p. 67-81.

3. MAPK Cascades as Feedback Amplifiers Sauro HM, Ingalls B

MAPK: Feedback Amplifier, Part I

 A recent paper by Sturm et al., report results that support the hypothesis that the MAPK cascade acts as a negative feedback amplifier.

The systems biology literature is full of reviews and articles about oscillators and bistable systems and very little else other than Uri Alon et al's refreshingly unique work on feedforward systems. An alien race, upon reading the literature, would most likely believe that the only thing biochemical networks can do is oscillate, show bistability and perhaps a little ultrasensitivity. This is probably because many of the modelers and theoreticians in systems biology are unaware of the possible signal processing capabilities offered by the engineering field. For example, an engineer looking at the MAPK cascade would probably immediately think of a negative feedback amplifier. Mention the word negative feedback amplifier to a systems biologist however and you're likely to get a blank stare. So what is a negative feedback amplifier? Let's start with some recent history.

ndustrial Revolution

Probably the most famous modern device that employed negative feedback was the governor. Thomas Mead in 1787 took out a patent on a device that could regulate the speed of windmill sails. His idea was to measure the speed of the mill by the centrifugal motion of a revolving pendulum and use this to regulate the position of the sail. Very shortly afterwards in early 1788, James Watt is told of this device in a letter from his partner, Matthew Boulton. Watt recognizes the utility of the governor as a device to regulate the new steam engines that were rapidly becoming an important source of new power for the industrial revolution. The image below illustrates an engraving of a governor from an early book entitled ”An Elementary Treatise on Stream and the Steam-engine by Clark and Sewell published in 1892.

The operation of the governor is simple (See Figure below), its purpose is to maintain the speed of a rotating engine at a constant predetermined value in spite of changes in load and steam pressure. The vertical axle of the governor is connected to the rotation of the steam engine. As the steam engine, for one reason or another, speeds up, the rotation increases, thereby causing the centrifugal pendulums to swing out. A linkage transmits this motion to the stream valve in such a manner that the flow of steam is reduced thus slowing down the engine. If the engine slows down too much, as a result of a sudden load, the flyweights will swing back and the steam value is opened so that the steam engine can accelerate. The governor was a highly successful device and it is estimated that by 1868, 75,000 governors were in operation (A History of Control Engineering, 1800-1930 By Stuart Bennett, 1979).

Figure: Illustration of a Governor from ”An Elementary Treatise on Stream and the Steam-engine by Clark and Sewell published in 1892.

This description of the governor illustrates one of the basic operational characteristics of negative feedback. The output of the device, in this case the steam engine speed, is ”fed back” to control the rate of steam entering the steam engine and thus influence the engine speed.

The Greeks

Although the governor is an example of one of the earliest negative feedback systems in modern times, the concept actually goes back much further in history. There is documentary evidence to show that the ancient Greeks were aware of the concept and used it in a wide variety of ways to control different mechanisms. Probably the most famous of these was the use of floats in water clocks to maintain a steady flow of water which could be used to measure time. The earliest recorded water clock that used negative feedback was described by Ktesibios who probably lived between 285 and 247 BC in Alexandria. Further work was done by Philon and particularly Heron (13 AD) who left us with an extensive book (Pneumatica) detailing many amusing water devices that employed negative feedback.

Figure: Ktesibios (270 BC) negative feeback value to regulate water flow. Modified from Stefano Penzier (http://www.dia.uniroma3.it/autom/FdAcomm/Lucidi)

Modern Times

In more recent times negative feedback has been used extensively in the electronics industry to confer, among other things, electrical stability to electronic devices and amplifiers. In fact, without negative feedback considerable swathes of modern technology would not be able to function. The application of negative feedback to modern devices is probably one of the most important innovations of the 20th century. Before continuing, let's make sure we understand what an amplifier is. The purpose of an amplifier is to faithfully scale up the power of a small time-varying electrical signal without adding distortion. This is an important task in both man-made and biological systems as we are often confronted with weak signals that need a boost to make them useful.

The story of how negative feedback came to be used in amplifiers begins in 1921 when Harold Black, who recently graduated from Worcester Polytechnic Institute in electrical engineering, took a job at Western Electric, the forerunner of Bell Labs. One of the challenges facing engineers in the 1920s in the US was how to design amplifiers that didn't distort the signal over long distances. In the early days, engineers would install what were called repeaters. Such repeaters would boost the signal but would also add their own distortions. By the time the signal had traveled 4000 miles across the country with repeaters less than 1000 miles apart, the signal at the end was barely intelligible.  These difficulties were ultimately overcome by the introduction of the feedback amplifier, designed in 1927 by Harold S. Black (Mindell, 2000). The basic idea was to introduce a negative feedback loop from the output of the amplifier to its input. At first sight, the addition of negative feedback to an amplifier might seem counterproductive. Indeed, Black had to contend with just such opinions when introducing the concept—his director at Western Electric dissuaded him from following up on the idea, and his patent applications were at first dismissed. In his own words ‘our patent application was treated in the same manner as one for a perpetual motion machine’ (Black, 1977). While Black’s detractors were correct in insisting that the negative feedback would reduce the gain of the amplifier, they failed to appreciate his key insight—that the reduction in gain is accompanied by increased robustness of the amplifier and improved fidelity of signal transfer. Since then, negative feedback has been widely used in the electrical industry (See opamps).

Biology

The reader may be wondering what on earth has this story got to do with MAPK cascades? Simple, we have a small hormonal signal coming in via receptors but need a larger signal inside the cell but without adding distortion and unwanted noise. You may be asking, where does the distortion and noise come from? The most obvious source of noise is the natural variability in protein levels due to stochastic events at the transcription and translation layers and source of the distortion is the nonlinear behavior of protein cascades. This combination is a recipe for disaster and to me at least, it isn't a surprise that evolution hit on the idea of wrapping the MAPK cascade in a negative feedback loop.

To understand the role of negative feedback is a system such as MAPK we need to examine more closely the advantages (and sometimes disadvantages) of negative feedback.

Go to Part II - under construction!

Black, H.S., 1977. Inventing the negative feedback amplifier. IEEE Spectrum 14, 55–60.

Mindell, D. (2000). Opening black’s box. Technology and Culture, 14, 405–434.

Sturm OE , Orton R , Grindlay J , Birtwistle M , Vyshemirsky V , Gilbert D , Calder M , Pitt A , Kholodenko B , Kolch W (2010) The mammalian MAPK/ERK pathway exhibits properties of a negative feedback amplifier. Sci Signal 3: ra90

 

Biochemical Control Analyis 101: Part 2

Originally Posted on June 16, 2011 by hsauro

What is a control coefficient?

First we should indicate what we mean by control. The term control has a special meaning in biochemical control analysis. Control refers to the ability of a system parameter to affect a system variable. For example, changes to the external glucose concentration in a microbial culture will most likely change the culture’s growth rate. The concentration of glucose therefore has ‘control’ over the growth rate. Engineering an enzyme in pathway so that its kcat is larger will result in changes to the pathways flux and metabolite concentrations. Changes to the promoter consensus sequence of a particular gene will result in changes to the concentration of the expressed protein and any other variables that depends on that protein. It is possible to quantify this concept of control by either measuring or computing control coefficients.

Control coefficients come in two flavors, flux control and concentration control. First consider flux control:

A flux control coefficient measures the relative steady state change in pathway flux (J)  in response to a relative change in enzyme activity, , often through changes in enzyme concentration.  This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. This assumes that a change on the level of the enzyme does not change the level of enzymes. This assumption will not always necessarily be true, in which case the control coefficient can be generalized to be independent of any particular parameter. For now however, with loss of generality, that we can change the enzyme concentration without affecting other enzymes. We define the flux control coefficients as follows:

  

The more generalized definition in terms of changes to the local reaction rate  of step  is given by:

  

so that the definition is now independent of the particular parameter, , used to perturb the reaction step. A very important property of all control coefficients is that they can only be measured in the intact system. It is not possible to isolate an enzyme and try to measure its control coefficient. The effect that a particular parameter, for example , has on a flux (or concentration) is a system property and depends on all the enzymes in the pathway. This is why it is not possible, or at least very difficult, to judge the importance of an enzyme by just looking at the enzyme alone.

A concentration control coefficient measures the relative steady state change in a species concentration (S)  in response to a relative change in enzyme activity, , often through changes in enzyme concentration.  This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. It is important to note that concentration control coefficients are properties of the intact system and cannot be measured from the isolated reaction step or enzyme. The same comments that were made with respect to the flux control coefficient applies here.

  

Or in generalized form:

  

What do the control coefficients actually mean, and why are they defined the way they are? The first thing to note about the control coefficient is that they are dimensionless. This is because we scale the derivative to eliminate units. The second important point is that as a result of the scaling, a control coefficients is a ratio of relative changes. This means that, roughly speaking, a control coefficient measures the effect a certain percentage change in enzyme concentration has on the percentage change in flux or metabolite concentration. For example a flux control coefficient of 0.4 means, that a 1% increase in the enzyme concentration, results in as 0.4% change in the steady state flux.

  

Metabolic Control Analyis 101: Part 1

Originally Posted on June 7, 2011 by hsauro

What is Metabolic Control Analysis?

Broadly speaking, Metabolic Control Analysis is a mathematical approach that allows us to understand and quantify how perturbations to a biochemical pathway propagate out from the disturbance to the rest of the system.

Metabolic Control Analysis (MCA) was born in an era when work on metabolism was in full swing. Since then, protein signaling and gene regulatory network have largely taken the limelight. Renewed interest in topics such biofuels, metabolism may be making a comeback. Since the development of MCA, we now know that it is much more general and applies to any kind of biochemical network be it metabolic, signal or gene regulatory. Therefore the first thing I’m going to do is stop called Metabolic Control Analysis, Metabolic Control Analysis. Instead, to indicate its generality, I will call it Biochemical Control Analysis (BCA).

BCA quantifies how variables, such as fluxes and species concentrations, depend on the system’s parameters. In particular it is able to describe how network dependent properties, called control coefficients, depend on local properties called elasticities.This will be the first in a series of short articles on the basic ideas embodied in BCA.

In this first article I will clarify the meaning of a couple of words used in BCA:

Variable

A variable, also called a dependent variable or state variable is a measurable characteristic of a system that can only be changed by an observer through changes to a suitable parameter. Variables are by definition determined by the system. Examples of possible variables include the pathway flux and species concentrations, such as metabolites or proteins.

Parameter

A parameter is a measurable characteristic of a system that can in principle be controlled by the observer. Parameters are also often called independent variables. By definition, a parameter cannot be changed by the system itself, if it can then it is called a variable. Examples of parameters include external concentrations such as glucose fed to a culture or externally added drug compounds, internal parameters such as kinetic constants, and depending on the system under study, enzyme concentrations.

Control

The term control has a special meaning in control analysis. Control refers to the ability of a system parameter to affect a system variable. For example, changes to the external glucose concentration in a microbial culture will most likely change the culture’s growth rate. The concentration of glucose therefore has ‘control’ over the growth rate. Engineering an enzyme in pathway so that its kcat is larger will result in changes to the pathways flux and metabolite concentrations. Changes to the promoter consensus sequence of a particular gene will result in changes to the concentration of the expressed protein and any other variables that depends on that protein. It is possible to quantify control by either measuring or computing control coefficients.

Regulation

Regulation may be defined as the capacity to achieve control. Such control may involve homeostasis or ability to move from one state to another in a particular manner.

Flux

The flux is the steady state flow of mass through a pathway.

MAPK: Feedback Amplifier, Part I

Originally Posted on May 26, 2011 by hsauro

A recent paper by Sturm et al., reports results that support the hypothesis that the MAPK cascade acts as a negative feedback amplifier.

The systems biology literature is full of reviews and articles about oscillators and bistable systems and very little else other than Uri Alon et al’s refreshingly unique work on feedforward systems. An alien race, upon reading the literature, would most likely believe that the only thing biochemical networks can do is oscillate, show bistability, and perhaps a little ultrasensitivity. This is probably because many of the modelers and theoreticians in systems biology are unaware of the possible signal processing capabilities offered by the engineering field. For example, an engineer looking at the MAPK cascade would probably immediately think of a negative feedback amplifier. Mention the word negative feedback amplifier to a systems biologist however and you’re likely to get a blank stare. So what is a negative feedback amplifier? Let’s start with some recent history.

Industrial Revolution

Probably the most famous modern device that employed negative feedback was the governor. Thomas Mead in 1787 took out a patent on a device that could regulate the speed of windmill sails. His idea was to measure the speed of the mill by the centrifugal motion of a revolving pendulum and use this to regulate the position of the sail. Very shortly afterward in early 1788, James Watt is told of this device in a letter from his partner, Matthew Boulton. Watt recognizes the utility of the governor as a device to regulate the new steam engines that were rapidly becoming an important source of new power for the industrial revolution. The image below illustrates an engraving of a governor from an early book entitled ”An Elementary Treatise on Stream and the Steam-engine by Clark and Sewell published in 1892.

The operation of the governor is simple (See Figure below), its purpose is to maintain the speed of a rotating engine at a constant predetermined value in spite of changes in load and steam pressure. The vertical axle of the governor is connected to the rotation of the steam engine. As the steam engine, for one reason or another, speeds up, the rotation increases, thereby causing the centrifugal pendulums to swing out. A linkage transmits this motion to the stream valve in such a manner that the flow of steam is reduced thus slowing down the engine. If the engine slows down too much, as a result of a sudden load, the flyweights will swing back and the steam value is opened so that the steam engine can accelerate. The governor was a highly successful device and it is estimated that by 1868, 75,000 governors where in operation (A History of Control Engineering, 1800-1930 By Stuart Bennett, 1979).