Wednesday, May 3, 2023

Optimal distribution of enzymes that maximizes flux

Here is an interesting metabolic question to ask. Consider a pathway with two steps: $$\large X_0 \stackrel{e_1}{\rightarrow} S \stackrel{e_2}{\rightarrow} $$ where the first step is catalyzed by enzyme $e_1$ and the second step by $e_2$. We can assume that a bacterial cell has a finite volume, meaning there is only so much space to fit all the proteins necessary for the bacterium to live. If the cell make more of one protein, it presumably needs to find the extra space by down expressing other proteins. If the two step pathway is our cell, this translates to there being a fixed amount of protein that can be distributed betwen the two, that is: $$ e_1 + e_2 = e_t $$ Presumably evolutionary pressure will maximize the flux through the pathway per unit protein. This means that given a fixed total amount of protein $e_t$ there must be a particular distribution of protein between $e_1$ and $e_2$ that maximizes flux. For example, if most of the protein were allocated to $e_1$ and very little to $e_2$ then the flux would be quite low. Likewise if most of the protein were allocated to $e_2$ and very little to $e_1$ then again the flux would be low. Between these two extremes there must be a maximum flux. To show this is the case we can do a simple simulation shown in the python code below. This code varies the levels of $e_1$ and $e_2$ in a loop such that their total is always 1 unit. Each time we change $e_1$ we must change $e_2$ by the same amount in the opposite direction in order to keep the constant fixed. As we do this we collect the steady-state flux and the current level of $e_1$. Finally we plot the flux versus $e_1$.

import tellurium as te
import roadrunner
import matplotlib.pyplot as plt

r = te.loada("""
     J1: $Xo -> S; e1*(k1*Xo - k2*S)
     J2: S ->; e2*k3*S
     Xo = 1
     k1 = 0.5; k2 = 0.2
     k3 = 0.45
     e1 = 0.01; e2 = 0.99


x = []; y = []; 
for i in range (49):
    r.e1 = r.e1 + 0.02
    r.e2 = 1 - r.e1  # Total assumed to be one
    x.append (r.e1)
    y.append (r.J1)
plt.grid(b=True, which='major', color='#666666', linestyle='-')
plt.grid(b=True, which='minor', color='#999999', linestyle='-', alpha=0.2)
plt.plot (x, y, linewidth = 2)
plt.xlabel('$e_1$', fontsize=36)
plt.ylabel('Flux', fontsize=36)

The following graph show the results from the simulation (the font sizes might be on the big size if you have a low res monitor):
You can see the flux reaching a maximum at around $e_1 = 0.6$, meaning also that $e_2 = 0.4$ in order to keep the total fixed. The actual position of the peak wil depend on the rate constants in the rate expressions. Let's assume we are at the maxium. The slope, $dJ/de_1$, at the maxium is obviously zero. That means if we were to move a small amount of protein from $e_1$ to $e_2$ the flux won't change. We can write this experiment in terms of the two flux control coefficients: $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{\delta e_1}{e_1} + C^J_{e_2} \frac{\delta e_2}{e_2} $$ However, we know that in this particlar experiment the change in $e_1$ is the same but oppisite to the change in $e_2$. That is $\delta e_1 + \delta e_2 = 0$ or $$ \delta e_1 = -\delta e_2$$ Replacing $e_2$ with $-\delta e_1$ gives: $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{\delta e_1}{e_1} - C^J_{e_2} \frac{\delta e_1}{e_2} $$ $$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{1}{e_1} - C^J_{e_2} \frac{1}{e_2} $$ $$ C^J_{e_1} \frac{1}{e_1} = C^J_{e_2} \frac{1}{e_2} $$ Giving the final result: $$ \frac{C^J_{e_1}}{C^J_{e_2}} = \frac{e_1}{e_2} $$ That is, when the protein distribution is optimized to maximize the flux, the flux control coefficients are in the same ratio as the ratio of enzyme amounts. This generalizes to any size pathway with multiple enzymes. This gives us a tantgilizing suggestion that we can obtain the flux control coeficients just by measuring the protein levels.

There is obviously a lot more one can write there and maybe I do that in future blogs but for now you can get further information:

S. Waley, “A note on the kinetics of multi-enzyme systems,” Biochemical Journal, vol. 91, no. 3, p. 514, 1964.

J Burns: “Studies on complex enzyme system.”, 1971 (page 141-) I have a LaTeX version at:

Guy Brown, Total cell protein concentration as an evolutionary constraint on the metabolic control distribution in cells,” Journal of theoretical biology, vol. 153, no. 2, pp. 195–203, 1991.

E. Klipp and R. Heinrich, “Competition for enzymes in metabolic pathways:: Implications for optimal distributions of enzyme concentrations and for the distribution of flux control,” Biosystems, vol. 54, no. 1-2, pp. 1–14, 1999

Sauro HM Systems Biology: An Introduction to Metabolic Control Analysis, 2018

Friday, April 21, 2023

Relationship of fluxes to enzyme levels in a metabolic pathway

To get hold of me it is best to contact me via my email address at:

Someone asked me the other day what the relationship was between the steady-state fluxe through a reaction and the coresponding level of enzyme. Someone else suggested that there would be a linear, or proportional relatinship between a flux and the enzyme level. However, this can’t be true, at least at steady. Considder a 10 step linear pathway. At steady-state each step in the pathway will, by defintion, carry the same flux. This is true even if each step has a different enzyme level. Hence the relationship is so simple. In fact the flux a given step carries is a systemic properties, dependent on all steps in the pathway. As an experiment I decided to do a simulation on some synthetic netowrks with random parameters and enzyme levels. For this exmaple I just used a simple rate law of the form: $$ v = e_i (k_1 A - k_2 B) $$ For a bibi reaction, A + B -> C + D, the coresponding rate law would be: $$ v = e_i (k_1 A B - k_2 C D) $$ a similar picture would be seen for the unibi and biuni reactions. Using our teUtils package I generated random networks with 60 species and 150 reactions. The reactions allowed are uiui-uinbi, biui or bibi. I then randomized the values for the enzymne levels $e_i$ and computed the steady-state flux. I used the following code to do the analysis. I have a small loop that generates 5 random models but obviously this number can be changed. I generate a random model, load the model into roadrunner, randomize the values for the $e_i$ parameters between 0 and 10, compute the steady-state (I do a presimulation to help things along) and collect the corresponding $e_i$ and flux values. Finally I plot each pair in a scatter plot.

import tellurium as te
import roadrunner
import teUtils as tu
import matplotlib.pyplot as plt
import random

for i in range (5):
      J = []; E = []
      antStr = tu.buildNetworks.getRandomNetwork(60, 150, isReversible=True)
      r = te.loada(antStr)

      n = r.getNumReactions()
      for i in range (n):      
          r.setValue ('E' + str (i), random.random()*10)    
      m = r.simulate(0, 200, 300)

      for i in range (n):
          J.append (abs (r.getValue ('J' + str (i)))) 
          E.append (r.getValue ('E' + str (i)))        
      plt.figure(figsize=(12, 8))
      plt.plot(E, J, '.')
        print ('Error: bad model')
The results for five random networks is shown below. Note the x axis is the enzyme level and the y axis the corresponding steady-state flux through that enzyme. It's intersting to see that there is a rough correlation between enzyme amount and the corresponding flux, but its not very strong. Many of the points are just scattered randomly with some showing a definite correlation. The short answer is the realtinship is not so simple.

Wednesday, March 1, 2023

How not to Comment Code

I recently came across one of the ten simple rules aricles in PLoS Comp Bio:

"Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students" by Korryn Bodner et al

One thing that struck me was Rule 5 on coding best practices with commenting being one of the discussion points. What struck me was their screen shot of a documented function shown below (in R):

My take on commenting is that it should be used to add human readable metadata on elements of a program that are not immediately obvious.

Most of the time, code should be sufficiently readable to indicate what it's doing. Obviously some languages are better than others when desribing an algorithm but it is also dependent on the programmer. I've seen code written in clear languages that are unintelliglbe, but I've also seen code written in poorly expressible languages that are easily readable. Although the programming language itself can influence code reability I think the programmer has much more influence.

But back to Rule 5. In the example you'll see something like:

# calculate the mean of the data

u <- mean (x)

This is completely redundant, as the coding states what it is going to do. In fact the authors comment every line like this. If anythng, I think the extent of comments actually hinders the reabilty of the code. The code itself is mostly clear as to what it is doing. There may be a justification to include a comment on next line that computes the standard error because the variables names are so badly chosen, e.g what does the following line do:

s <- sd(x)

sd might stand for standard deviation but the rest of the line offers no clue. If it had been written as:

standardDeviation <- sd(x)

It would have been much clearer, instead the authors add a comment to make up for poor choice of variable names. They also give the function itself a nondescriptive name, in this case ci. It would have been better to write the function using getConfidenceInterval or similar:

getConfidenceInterval <- f (data) {


Tuesday, January 24, 2023

Euclid's Elements

Nothing to do with cells and networks, but I have an interest in Euclid's Elements and decided to publish a new edition of Book I. The difference with previous editions is that this one is in color and also has a chapter on the history of the elements, together with commentaries on each proposition. For those unfamiliar with Euclid's Elements, it's a series of books (more like chapters in today's language) laying the foundation for geometry and number theory. Book I focuses on the foundation of geometry culminating in a proof for Pythagoras' theorem and other important but less known results. The key innovation is that it describes a deductive approach to geometry. It starts with definitions and axioms from which all results are derived using logical proofs.

It occurred to me that something similar could be done with deriving the properties of biochemical networks. For example, we might define the following three primitives:

I. Species

II. Reaction

III. Steady-state

We might then define the following axioms:

I. A species has associated with it a value called the concentration, x_i.

II. All concentrations are positive.

III. A reaction has a value associated with it called the reaction rate, v_i.

IV. Reaction rates can be negative, zero, or positive.

V. A reaction transforms one or more species (reactants) into one or more other species (products).

VI. The reaction rate is a continuous function of the reactants and products.

VII. The rate of change of a species can be described using a differential equation, dx/dt

VIII. All steps are reversible unless otherwise stated (may this can be derived?)


Given these axioms, we could build a series of propositions. This might be an interesting exercise to do. Some of the more obvious propositions would be the results from metabolic control analysis, such as the summation and connectivity theorems.

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