(c) 2019 Justin Bois and Michael Elowitz. With the exception of pasted graphics, where the source is noted, this work is licensed under a Creative Commons Attribution License CC-BY 4.0. All code contained herein is licensed under an MIT license.
This document was prepared at Caltech with financial support from the Donna and Benjamin M. Rosen Bioengineering Center.
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At the center of many of the genetic circuits we have discussed thus far are the processes of the Central Dogma, notably transcription and translation. The error rate in translation of mRNA to protein by ribosomes is about 10−4 errors per amino acid. In this lesson, we will work out whether or not this error rate is high or low, and then discuss mechanisms about how it is achieved.
The design principle we will explore is more general: Driven multistep pathways enable amplification of selectivity. By the word "driven," we mean energy consuming reactions are necessary for the multiple steps.
We can think of the process of translation as consisting of two successive steps. First, a tRNA (T) reversibly associates with the ribosome/mRNA complex (R). When associated, an amino acid can be irreversibly added to the nascent peptide. This is represented by the following reaction scheme.
Unfortunately, there can also be incorporation of erroneous tRNAs and we can also get erroneous amino acid incorporation. So, we should really write two sets of chemical reactions, one for correct incorporation of an amino acid and one for erroneous.
Here, we have denoted chemical rate constants for erroneous incorporation with a p. Note that we have assumed that the irreversible incorporation rate is the same for correct and incorrect incorporation.
Using mass action kinetics, we can write the dynamical equations for amino acid incorporation. Importantly, we have
rate of incorporation of correct AAs≡rc=vc,rate of incorporation of erroneous AAs≡re=ve,where c is the concentration of RC and e is the concentration of RE. We assume a separation of time scales, where incorporation of the amino acid is much slower than the other dynamics. Thus, the rate of incorporation of amino acids is proportional to the steady state concentration of RC complexes (or RE complexes for erroneous addition). We can write differential equations for c and e.
dcdt=k(1)+cRcC−k(1)−c,dedt=p(1)+cRcE−p(1)−e,where cR is the concentration of ribosomes, cC is the concentration of correct tRNA and cE is the concentration of erroneous tRNA. There are assumed to be constant, since they are in great excess and the cell is constantly producing them. So, in treating the dynamics, we only need the two above ODEs.
We are interesting in computing the error rate, f0, which is the number of erroneous amino acids incorporated per correct amino acid. We will compute this at steady state, so the steady state error rate is
f0=rate of incorporation of erroneous AAsrate of incorporation of correct AAs=ec.So, we just have to compute the steady state values for c and e. To do this, we take dc/dt=de/dt=0 and solve. We get
c=k(1)+k(1)−cRcCe=p(1)+p(1)−cRcE.Thus, the error rate is f0=ec=p(1)+p(1)−k(1)−k(1)+cEcC.
Note that the two ratios of rate constants are the dissociation constants for the interaction of the tRNA with the ribosome/mRNA complex; Kd,c=k(1)−/k(1)+ and Kd,e=p(1)−/p(1)+. Finally, to ease notation, we define the ratio of concentrations of erroneous and correct tRNAs to be q. Thus, we have
f0=qKd,cKd,e.From thermodynamic considerations, the dissociation constant is given by
Kd=exp[−ER+ET−ERTkBT],where T is either E or C. Since EC≈EE, the ratio of the dissociation constants is
Kd,cKd,e=exp[−ERE−ERCkBT].So, the error rate for this simple model is
f0=qexp[−ERE−ERCkBT].As mentioned before, about 10−4 erroneous amino acids per residue are incorporated. We will take q≈20, though this can vary significantly depending on the amino acid, since tRNA abundance is variable. So, to get an error rate of 10−4, we need
ERE−ERCkBT≈−ln(5×10−6)≈12.So, the difference in the interaction between an erroneous and correct tRNA and the ribosome/mRNA complex would have to be at least 12kBT (maybe more because some tRNAs are in higher abundance). So, the question is what is the energy difference between correctly and incorrectly bound tRNA-mRNA pairs and how does this compare to 12kBT? Being off by a single base pair could result in an erroneous amino acid being incorporated. The misalignment of the base pairs would result in disrupted hydrogen bonds. Base pairs have two or three hydrogen bonds. Presumably, at least one could form if the bases are misaligned. Thus, we could get an error with a single hydrogen bond being off. The stacking interactions are not as important here, since the bases are confined within the geometry of the ribosome. So, we will estimate that the energy difference is about one hydrogen bond, or 2kBT. So, there is just not enough energy in hydrogen bonding to give the necessary energy difference to give the low error rate.
Experimentally, it is known that there are intermediate steps in translation. In 1974, John Hopfield's big insight was that these steps require energy (e.g., GTP hydrolysis involving the elongation factor EF-Tu), and can therefore serve to decrease the error rate since the energy serves to overcome the limit set by thermodynamics. These hydrolysis events can cause a conformational change in the tRNA-codon complex that allow elongation and also for dissociation of tRNA from the codon. For this scheme, the chemical reactions are
We will now compute the error rate for this new scheme. To begin, we compute the steady rate of elongation with correct amino acids, which we will call rc. Our goal is to compute the steady state concentration of RC∗, c∗, since rc=vc∗. Some differential equations describing the dynamics of the system under mass action kinetics are
dcRdt=−k(1)+cRcC+k(1)−c+kdc∗dcdt=k(1)+cRcC−k(1)−c−k(2)+c+k(2)−c∗.Setting the time derivatives to zero, the first equation gives
c=1k(1)−(k(1)+cRcC−kdc∗).Substituting this expression into the second equation (with the time derivative set to zero) yields
kdc∗−k(2)+k(1)−(k(1)+cRcE−kdc∗)+k(2)−c∗=0This is readily solved for c∗ to give
c∗=k(1)+k(2)+cRcCk(2)+kd+k(1)−(k(2)−+kd).Similarly, we have for e∗,
e∗=p(1)+p(2)+cRcEp(2)+pd+p(1)−(p(2)−+pd),giving an error rate of
f=rerc=e∗c∗=q(p(1)+p(2)+k(1)+k(2)+)(k(2)+kd+k(1)−(k(2)−+kd)p(2)+pd+p(1)−(p(2)−+pd)).For ease of comparison and to build intuition, we will consider the following approximations.
Applying approximation (2), we have
k(2)+kd+k(1)−(k(2)−+kd)≈(k(1)−+k(2)+)kd.Applying approximation (3) gives
(k(1)−+k(2)+)kd≈k(1)−kd.Similar results hold for the terms describing erroneous binding. Thus, we have
f≈qp(2)+k(2)+k(1)−kdp(1)−pd.Applying approximation (1) allows us to write
f≈qp(2)+k(2)+Kd,ckdKc,epd,since Kd,e=p(1)−/p(1)+≈p(1)−/k(1)+. We use approximation (4), that the hydrolysis reaction is nonspecific such that k(2)+≈p(2)+, giving
f≈qkdpdKd,cKd,e=kdpdf0,which is a kd/pd-fold improvement over the error rate without proofreading. So, to get improved fidelity, we need kd<pd; incorrectly paired complexes dissociate faster than correctly paired ones. To bring the error rate down four more orders of magnitude to the observed error rate, we would need this ratio to be about 1/10000.
We have shown that adding the extra activation step allows for greater selectivity of correct amino acid incorporation in the context of translation. This extra step is a "proofreading" step that requires energy, hence the name kinetic proofreading. The kinetic proofreading principle is, in fact, more general. We have studied a recognition problem where there are two steps: bind a substrate and trigger a response. In this case, the substrate is tRNA and the response is addition of a new amino acid.
Kinetic proofreading has been applied extensively to recognition of major histocompatibility complex by T cell antigen receptors. The result of this recognition is T cell activation. This system exhibits extraordinary sensitivity, with the ability to detect a single ligand in a sea of thousands. This is better even than electronic circuits. In your homework, you will work out that this can be accomplished with many added steps of kinetic proofreading.