What are the constraints on an operon?

What are the constraints on an operon?

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From what I understand, the term operon is loosely defined to be a collection of genes that are found to be in the same proximity in a genome.

Some definitions enforce that an operon is only regulated by a single promoter and all of the genes are transcribed together.

If the above definition is strict, that would imply a couple of things

  1. Genes are opposite strands cannot be in the same operon (since they cannot be transcribed together)
  2. Overlapping genes cannot be in the same operon for the same reason.

I realize that part of the issue with these definitions stems from an ongoing investigation behind this phenomenon. But are there strict standards surrounding the definition an operon? What is the minimal definition of an operon?

Relevant links

From Henderson's Dictionary of Biology 14 ed. An operon is: a type of gene organization in bacteria, in which the genes coding for the enzymes of a metabolic pathway are clustered together in the DNA and transcribed together into a single mRNA. This mRNA is then translated to give the individual proteins. The expression of all the genes in an operon is controlled by a single promoter.

I'm not sure what an overlapping gene is, but according to this definition, yes genes on opposite directions of a promoter or different loci cannot be a part of an operon because they wouldn't be on the same mRNA.

Regulation of Enzyme Activity by Activation or Inhibition | Biochemistry

In connection with the tryptophan operon, that an excess of tryptophan can cause a repression of the genes of this operon, leading to an arrest of the synthesis of the enzymes required for the formation of tryptophan.

Beside this possibility of repression, there is very often feedback inhibition, i.e. the possibility for an essential metabolite (amino acid, nucleotide, etc.), which is the final product of a series of biosynthetic reactions, of inhibiting the activity of an enzyme catalyzing one of the first reactions of this series.

The inhibited enzyme is generally the one which catalyzes the first reaction leading specifically to the final product, and not an enzyme which catalyzes a reaction common to several metabolic pathways it is the enzyme situated at a strategic junction whose activity is inhibited by the final product.

This type of regulation is particularly characterized by the fact that the effector (the substance which activates or inhibits the enzyme) and the substrate of this enzyme are generally not isosteric., i.e. they have no structural analogy (contrary to the situation in competitive inhibition exerted by analogues of the substrate). That is why they are called allosteric effectors, while the term allosteric enzymes denotes the enzymes in which this type of control is observed.

1. General Properties of Allosteric Enzymes:

In connection with aspartate transcarbamylase, some important charac­teristics of the regulation at the level of allosteric enzymes, but it is of interest here to review the main properties.

A. Kinetics of Reactions Catalyzed by Allosteric Enzymes:

In general, allosteric enzymes have special kinetic properties, and when studying the variation of velocity as a function of substrate concentration one obtains, not a branch of an equilateral hyperbola as in the case of most enzymes, but a sigmoid curve. This S-shaped curve reflects a cooperative effect i.e. the fact that at least 2 substrate molecules interact with the enzyme and that the binding of the first molecule facilitates that of the second.

Very often such a cooperative effect is also manifest in the binding of allosteric effectors (see fig. 8-13), which suggests that the binding of the first allosteric activator (or inhibitor) molecule favours the binding of the second. These results already suggest that there are more than one catalytic site and more than one allosteric site per molecule of enzyme and imply the polymeric or oligomeric nature of allosteric enzymes.

We have not indicated where the allosteric effector binds, but our knowledge of the specificity of enzyme-substrate interaction and our observations on the absence of any structural similarity between substrate and effector suggest that the allosteric effectors do not bind to the active sites, but to different sites called allosteric sites.

Considering the sigmoid nature of the curves expressing enzymatic activity as a function of either the substrate, or the allosteric inhibitor (see fig. 8-13), one therefore has a threshold effect when inhibitor concentra­tion increases or when substrate concentration increases.

Below the threshold, an increase of [S] (see fig. 2-12) or [I] does not cause a significant change of velocity but beyond the threshold, velocity varies considerably for a relatively small increase of [S] or [I]. This enables the cell to adjust the enzymatic activity according to relatively small variations of [S] or [I], but occurring in a zone of critical concentration which corresponds to the intracel­lular concentrations of the metabolites involved.

B. Action of Allosteric Effectors:

There are various types of allosteric inhibitors and using the Lineweaver- Burk plot, it is observed that some allosteric inhibitors are of the competitive type and others of the non-competitive type. But contrary to what we have seen while studying the competitive inhibition of conventional enzymes, there is no competition — in the case of allosteric enzymes — between S and I for the active site of the enzyme (because they have no structural analogy).

The two types of inhibitors bind to allosteric sites, dif­ferent from the active sites, as shown by experiments of desensitization and fractionation of sub-units. In the case of a non-competitive al­losteric inhibitor, the binding of I to the allosteric site of an enzyme to give E — I can thus cause a change of conformation which still permits the binding of S to give E —S —I, but 1/Vmax is increased therefore Vmax is lower.

During the binding of a competitive inhibitor to the allosteric site, there is a change of conformation, an allosteric transition, which causes repercussions on the active site to which S can no longer bind. There is a decrease of -1/Km, i.e. an increase of Km, in other words a decrease of the affinity of the enzyme for S. There exists a type of mixed inhibition, charac­terized by an increase of 1/Vmax i.e. a decrease of Vmax, as well as an increase of Km, i.e. a decrease of the affinity of the enzyme for S.

We have seen how the allosteric transition caused by the binding of an activator favours the binding of the substrate. This change of conformation of the enzyme brings about a decrease of Km, i.e., an increase of affinity for S.

The curve representing the kinetics of the reaction can change from the sigmoid form to the hyperbolic form (see curve 2 of figure 2-12), but one must be very cautious because it has been shown in some cases that this change of order of the reaction was only apparent (it was due to an inaccuracy in the first part of the curve which did not show the sigmoid character).

C. Desensitization and Dissociation of Allosteric Enzymes:

Allosteric enzymes can be made insensitive to allosteric effectors, either after a mutation, or in vitro by a physical or chemical treatment: variation of pH, temperature, ionic strength action of urea, mercurial agents, proteolytic enzymes, etc.

This desensitization generally does not affect catalytic activity, which sup­ports the hypothesis of separate catalytic and allosteric sites. It is often reflected by a modification of kinetics which changes from the sigmoid form to the hyperbolic “Michaelian” form. The fact that the enzyme conserves a catalytic activity but is no longer sensitive to the allosteric site was first inter­preted as the sign of an alteration of the allosteric site (by the desensitizing agent) not affecting the catalytic site.

In reality, for the regulatory effects to manifest themselves not only the allosteric sites must be intact and the effectors able to bind to these sites but also, a conformation of the enzyme must be preserved which will enable the allosteric transition and especially the repercus­sion — at the catalytic site — of an event affecting the allosteric site.

In fact, it was observed in some cases, that after desensitization the effector can still bind to the allosteric site on the contrary, due to a modification of the spatial structure of the enzyme, there is disappearance of the cooperative interactions between the various catalytic sites of the same enzyme molecule, between its various allosteric sites and between its catalytic and allosteric sites, thus preventing the allosteric transition.

This allosteric transition consists of a modification of the bonds joining the promoters to one another, which permits the passage of the enzyme from a relaxed state to a constrained state, or vice-versa (see fig. 8-14).

The existence of separate sites for substrate and inhibitor, confirmed ex­perimentally in numerous cases, is particularly evident when it is possible to dissociate an allosteric enzyme in distinct sub-units, some carrying the catalytic sites and others carrying the allosteric sites.

This is the case for example with aspartate transcarbamylase, which is inhibited by CTP and activated by ATP, can be desensitized by heat or urea, but can also be dissociated by mercurial agents: the native enzyme (molecular weight = 310 000) consists of 6 polypeptide chains with catalytic activity (molecular weight = 33 000) and having each a site for the binding of the substrate, and 6 regulatory chains (molecular weight = 17 000) enabling the binding of 6 CTP.

On isolating the catalytic sub-units it is observed that their specific activity (quantity of substrate transformed per unit time, referred to the quantity of protein) is greater than that of the native enzyme, which is not surprising because the elimination of the regulatory sub-units — inactive in the catalysis process — is in a way a purification of the enzyme if one considers only the catalytic point of view.

D. Model of Monod, Wyman arid Changeux:

To explain the phenomena observed during the study of allosteric enzymes, these authors proposed a model whose important characteristics are as follows:

1. The allosteric enzymes are oligomers, whose protomers are associated so that the molecule comprises at least one axis of symmetry (the protomer is defined as the structure which has a binding site for each ligand, i.e. for each substance capable of binding — i.e. substrate, activator and inhibitor — and must not be mistaken for the sub-unit which results from the dissociation of the enzyme and can contain — as in the case of aspartate-transcarbamylase — only one site, catalytic or allosteric)

2. Each protomer possesses only one site permitting the formation of one specific complex with each category of ligand

3. The allosteric enzyme may have different but interconvertible conformations. One often speaks of relaxed state and constrained state. These states are in equilibrium and differ either by the distribution and energy of bonds between the protomers (which determine the constraints imposed oil protomers), or by the affinity of the various sites for the corresponding ligands.

Figure 8-14 shows a simple diagram — with only 2 protomers — to illustrate the model. At first, there is equilibrium between the relaxed form and the constrained form if one of the ligands (e.g., the substrate) has a greater affinity for one of these 2 forms, a relatively small concentration of this ligand will permit the binding of a substrate molecule to a protomer of the form con­sidered, which will shift the equilibrium in favour of this form and will facilitate the binding of the substrate.

But an increase of the concentration of an antagonistic ligand (here the inhibitor) is enough for the equilibrium to be shifted in the reverse direction. Allosteric phenomena are reversible and depend on the concentrations of the various ligands. Such a model explains the fact that a sigmoid curve is obtained when velocity is expressed as a function of [S] or [I].

The diagram of fig. 8-14 is valid for an allosteric enzyme of the K type. In this case, in the absence of substrate, the equilibrium is in favour of the form having a low affinity for the substrate. But as observed above, when [S] in­creases, the equilibrium is shifted in favour of the form having a greater affinity for S.

Conversely, the equilibrium is shifted by the inhibitor in favour of the form having a low affinity for S and the allosteric transition consists precisely in this change of equilibrium. Therefore, the inhibitor decreases the affinity of the enzyme for S (Ks increases), and conversely the substrate decreases the affinity of the enzyme for the inhibitor (K, increases), whence the name K type enzyme.

Other models were proposed to explain the kinetic properties of allosteric proteins. According to the model of induced fit proposed by Koshland, Nemethy and Filmer, there is only one configuration for the protein in the absence of ligand it appears that the binding of the ligand induces a conforma­tional modification of the protomer, which transforms the interactions between the sub-units and changes the catalytic properties.

It appears that the conformation of an enzymatic protein, which we called tertiary and quaternary structure, is not exclusively determined by the primary structure. Actually, it is observed that small molecules (substrates, activators, inhibitors), by binding to specific sites, are capable of causing slight modifications of the spatial structure of the protein.

2. Main Modes of Regulation:

1. Feedback inhibition consists in the inhibition of the first enzyme of a reaction series by the metabolite which is the terminal product of this series. The intracellular concentration of this metabolite therefore controls the rate of its own biosynthesis. In the following we are considering feedback inhibition in straight and branched reaction series.

2. Activation of an enzyme by a precursor of the substrate or by the substrate itself.

3. Activation by a degradation product of the terminal metabolite causing a new increase of the concentration of this metabolite (which may be a high energy potential substance for example).

4. Activation of an enzyme of a metabolic series leading to a metabolite A by a metabolite B, which is synthesized by an independent series, when A and B are both necessary for the synthesis of the same macromolecules, which permits a coordinated production of precursors (in the case of nucleotides).

The activity of an allosteric enzyme can be controlled by several of these regulatory modes. Thus, aspartate transcarbamylase, the first enzyme of the pathway leading to the synthesis of pyrimidine nucleotides, is feedback-in­hibited by a terminal product (CTP), activated by the substrate and also activated by ATP, a ribonucleoside triphosphate required – jointly with UTP and CTP – for the biosynthesis of RNAs.

3. Feedback Inhibition in Straight and Branched Reaction Chains:

A. Feedback Inhibition in Straight Reaction chains:

In straight metabolic sequences, it is generally the first enzyme (E1) which is the regulatory enzyme, i.e. the enzyme subjected to a control of allosteric type. By “first enzyme” one must generally understand the enzyme which catalyzes the first reaction specific of the metabolic pathways con­sidered.

For example, in the case of the biosynthesis of pyrimidine ribo­nucleotides, it is aspartate transcarbamylase which is subjected to feedback control and not an enzyme permitting the synthesis of carbamyl-phosphate or aspartate these two compounds can also enter other metabolic pathways, while their combination to give carbamyl aspartate is really the first reaction leading specifically to pyrimidine nucleotides. The first enzyme of the chain is generally the only one to be inhibited by the final product its activity therefore determines the functioning of the whole sequence of reactions.

The inhibition of this enzyme by the final product of the chain of reactions is of obvious interest. When this final product is in excess, the inhibiting effect it exerts on the first enzyme decreases the rate of this first reaction and consequently restricts its own biosynthesis. Since the series of biosynthetic reactions usually require energy, this regulation process enables the cell to save energy.

This economy is however smaller than the one made through the repression process: when a substance X is in excess, repression enables the cell to dispense with not only the biosynthesis reactions of X, but also the transcrip­tion of genes into mRNA and the translation of the polycistronic mRNA into the enzymes required for the biosynthesis of X, while in feedback inhibition the enzymes required are present but do not function.

On the contrary, feedback inhibition appears as a process more rapid than repression. An excess of a substance X can immediately inhibit the first enzyme of the chain of reactions leading to X, while the effects of repression are manifest only after the disappearance — through catabolism — of the molecules of enzymes and mRNAs existing in the cell (and which are not replaced because the expression of the corresponding genes is blocked).

Feed­back inhibition which is based — as mentioned above — on the phenomenon of allosteric transition, i.e. on the shift of a state of equilibrium in favour of one of the two conformations of the enzyme, is an easily reversible process, very sensitive to small variations of the concentrations of ligands beyond a particular threshold, and is therefore characterized by a great flexibility.

B. Feedback Inhibition in Branched Reaction Chains:

Feedback inhibition poses special problems in the case of branched reactions chains where one could a priori fear that the excess of the final product of one of the branchings would cause — if it inhibits the first enzyme of the chain — the arrest of the synthesis of substances produced by the other branchings, substances which are not necessarily in excess.

To study these problems we will take the example of the biosynthesis of amino acids deriving from aspartate, this will enable us to study their regulation with the help of a simplified diagram.

a) Feedback Inhibition Limited to Branchings:

As shown by figure 8-15, the amino acid which is the final product of a branching can inhibit the first step of the sequence of reactions leading only to its biosynthesis. The biosynthesis of other amino acids is therefore not affected. Lysine inhibits, dihydro-dipicolinate synthetase, threonine inhibits homoserine kinase (HK), methionine inhibits the succinylation of homoserine and isoleucine inhibits threonine deaminase (TD).

b) Iso-Enzymatic Control:

Three aspartokinases (AK) have been iden­tified in E.coli each of them is subjected to a regulation by a specific repression mechanism and two are subjected to a feedback inhibition which is also specific.

Moreover, there are also 2 homoserine dehydrogenases (HSDH) whose regulation is identical to that of the first two aspartokinases, as shown by the table below:

It has been shown that the two catalytic activities AK I and HSDH I are carried by one and the same polypeptide chain, the same is true of the activities AK II and HSDH II. It is evident that the existence, for example in the case of aspartokinase, of 3 isoenzymes whose synthesis and activity are controlled by different terminal products, enables the cell — in case of repression of the biosynthesis or inhibition of the activity of one of the aspartokinases due to a high concentration of one amino acid — to continue to synthesize the other amino acids deriving from aspartate thanks to the other two aspartokinases which are not affected. The existence of 3 isoenzymes to catalyze this first reaction permits an independent regulation by the various terminal products (fig. 8-16).

c) Concerted or Multivalent Feedback Inhibition:

In some organisms of the genus Rhodopseudomonas or Bacillus there is only one aspartokinase which is not affected by the excess of only one of the terminal products (Lys, Thr, Ile), but which is feedback inhibited when there is excess of both lysine and threonine. However this concerted feedback inhibition is not total, which permits the synthesis of methionine.

Other possibilities of control exist in some organisms in this branched chain of biosynthesis of amino acids deriving from aspartate. We cannot study all of them but the types of regulation we examined do show that varied mechanisms were selected by the living organisms in the course of evolution to solve the special problems posed by regulation in the metabolic pathways presenting branchings.

Teaching the Big Ideas of Biology with Operon Models

This paper presents an activity that engages students in model-based reasoning, requiring them to predict the behavior of the trp and lac operons under different environmental conditions. Students are presented six scenarios for the trp operon and five for the lac operon. In most of the scenarios, specific mutations have occurred in genetic elements of the system that alter the behavior from the norm. Students are also challenged to relate their understanding of operon behavior to the &ldquoBig Ideas&rdquo of homeostasis, evolution, information, interactions, and emergent properties. By using operons to teach students to reason with models of complex systems and understand broad themes, we equip them with powerful skills and ideas that form a solid foundation for their future learning in biology.


The American Biology Teacher &ndash University of California Press

Published: Jan 1, 2015

Keywords: Key Words: Biochemistry biological themes core concepts disciplinary core ideas model-based reasoning molecular biology operons problem solving

Non-specific DNA Binding of Genome Regulating Proteins as a Biological Control Mechanism: 1. The lac Operon: Equilibrium Aspects

The regulatory system of the lactose operon has been “modeled” by a set of mass action equations and conservation constraints which describe the system at equilibrium. A �se-set” of values of binding constants and total component concentrations has been assembled from the available experimental data, and the simultaneous equations solved by computer procedures, to yield equilibrium concentrations of all the relevant molecular species. Considering the operator-repressor-inducer system alone, it is shown that the in vivo basal and induced (derepressed) levels of lac enzyme synthesis in both wild-type and certain mutant Escherichia coli can be accounted for only if binding of repressor and repressor-inducer complexes to non-specific DNA sites is included in the calculations as an integral component of the ovrall control system. A similar approach was applied to the RNA polymerase-promoter system to show that sigma factor may modulate the general level of transcription in the cell by “inducing” polymerase off non-specific DNA binding sites, thus making it available to promoters. Competitive and non-competitive models for the interaction of repressor and polymerase at the lac operon can, in principle, be distinguished by these computational procedures, though data sufficient to permit unambiguous differentiation between the models are not available at this time. However, for any competitive binding model the results show that repression in the entire (operator-repressor-RNA polymerase-lac promoter) system can occur only because non-specific binding of the regulatory proteins reduces the concentration of free polymerase, relative to that of repressor, to appropriate levels.


Recapitulation Edit

A recapitulation theory of evolutionary development was proposed by Étienne Serres in 1824–26, echoing the 1808 ideas of Johann Friedrich Meckel. They argued that the embryos of 'higher' animals went through or recapitulated a series of stages, each of which resembled an animal lower down the great chain of being. For example, the brain of a human embryo looked first like that of a fish, then in turn like that of a reptile, bird, and mammal before becoming clearly human. The embryologist Karl Ernst von Baer opposed this, arguing in 1828 that there was no linear sequence as in the great chain of being, based on a single body plan, but a process of epigenesis in which structures differentiate. Von Baer instead recognised four distinct animal body plans: radiate, like starfish molluscan, like clams articulate, like lobsters and vertebrate, like fish. Zoologists then largely abandoned recapitulation, though Ernst Haeckel revived it in 1866. [2] [3] [4] [5] [6]

Evolutionary morphology Edit

From the early 19th century through most of the 20th century, embryology faced a mystery. Animals were seen to develop into adults of widely differing body plan, often through similar stages, from the egg, but zoologists knew almost nothing about how embryonic development was controlled at the molecular level, and therefore equally little about how developmental processes had evolved. [7] Charles Darwin argued that a shared embryonic structure implied a common ancestor. As an example of this, Darwin cited in his 1859 book On the Origin of Species the shrimp-like larva of the barnacle, whose sessile adults looked nothing like other arthropods Linnaeus and Cuvier had classified them as molluscs. [8] [9] Darwin also noted Alexander Kowalevsky's finding that the tunicate, too, was not a mollusc, but in its larval stage had a notochord and pharyngeal slits which developed from the same germ layers as the equivalent structures in vertebrates, and should therefore be grouped with them as chordates. [8] [10] 19th century zoology thus converted embryology into an evolutionary science, connecting phylogeny with homologies between the germ layers of embryos. Zoologists including Fritz Müller proposed the use of embryology to discover phylogenetic relationships between taxa. Müller demonstrated that crustaceans shared the Nauplius larva, identifying several parasitic species that had not been recognised as crustaceans. Müller also recognised that natural selection must act on larvae, just as it does on adults, giving the lie to recapitulation, which would require larval forms to be shielded from natural selection. [8] Two of Haeckel's other ideas about the evolution of development have fared better than recapitulation: he argued in the 1870s that changes in the timing (heterochrony) and changes in the positioning within the body (heterotopy) of aspects of embryonic development would drive evolution by changing the shape of a descendant's body compared to an ancestor's. It took a century before these ideas were shown to be correct. [11] [12] [13] In 1917, D'Arcy Thompson wrote a book on the shapes of animals, showing with simple mathematics how small changes to parameters, such as the angles of a gastropod's spiral shell, can radically alter an animal's form, though he preferred mechanical to evolutionary explanation. [14] [15] But for the next century, without molecular evidence, progress stalled. [8]

The modern synthesis of the early 20th century Edit

In the so-called modern synthesis of the early 20th century, Ronald Fisher brought together Darwin's theory of evolution, with its insistence on natural selection, heredity, and variation, and Gregor Mendel's laws of genetics into a coherent structure for evolutionary biology. Biologists assumed that an organism was a straightforward reflection of its component genes: the genes coded for proteins, which built the organism's body. Biochemical pathways (and, they supposed, new species) evolved through mutations in these genes. It was a simple, clear and nearly comprehensive picture: but it did not explain embryology. [8] [16]

The evolutionary embryologist Gavin de Beer anticipated evolutionary developmental biology in his 1930 book Embryos and Ancestors, [17] by showing that evolution could occur by heterochrony, [18] such as in the retention of juvenile features in the adult. [11] This, de Beer argued, could cause apparently sudden changes in the fossil record, since embryos fossilise poorly. As the gaps in the fossil record had been used as an argument against Darwin's gradualist evolution, de Beer's explanation supported the Darwinian position. [19] However, despite de Beer, the modern synthesis largely ignored embryonic development to explain the form of organisms, since population genetics appeared to be an adequate explanation of how forms evolved. [20] [21] [a]

The lac operon Edit

In 1961, Jacques Monod, Jean-Pierre Changeux and François Jacob discovered the lac operon in the bacterium Escherichia coli. It was a cluster of genes, arranged in a feedback control loop so that its products would only be made when "switched on" by an environmental stimulus. One of these products was an enzyme that splits a sugar, lactose and lactose itself was the stimulus that switched the genes on. This was a revelation, as it showed for the first time that genes, even in an organism as small as a bacterium, were subject to fine-grained control. The implication was that many other genes were also elaborately regulated. [23]

The birth of evo-devo and a second synthesis Edit

In 1977, a revolution in thinking about evolution and developmental biology began, with the arrival of recombinant DNA technology in genetics, and the works Ontogeny and Phylogeny by Stephen J. Gould and Evolution by Tinkering by François Jacob. Gould laid to rest Haeckel's interpretation of evolutionary embryology, while Jacob set out an alternative theory. [8] This led to a second synthesis, [24] [25] at last including embryology as well as molecular genetics, phylogeny, and evolutionary biology to form evo-devo. [26] [27] In 1978, Edward B. Lewis discovered homeotic genes that regulate embryonic development in Drosophila fruit flies, which like all insects are arthropods, one of the major phyla of invertebrate animals. [28] Bill McGinnis quickly discovered homeotic gene sequences, homeoboxes, in animals in other phyla, in vertebrates such as frogs, birds, and mammals they were later also found in fungi such as yeasts, and in plants. [29] [30] There were evidently strong similarities in the genes that controlled development across all the eukaryotes. [31] In 1980, Christiane Nüsslein-Volhard and Eric Wieschaus described gap genes which help to create the segmentation pattern in fruit fly embryos [32] [33] they and Lewis won a Nobel Prize for their work in 1995. [29] [34]

Later, more specific similarities were discovered: for example, the Distal-less gene was found in 1989 to be involved in the development of appendages or limbs in fruit flies, [35] the fins of fish, the wings of chickens, the parapodia of marine annelid worms, the ampullae and siphons of tunicates, and the tube feet of sea urchins. It was evident that the gene must be ancient, dating back to the last common ancestor of bilateral animals (before the Ediacaran Period, which began some 635 million years ago). Evo-devo had started to uncover the ways that all animal bodies were built during development. [36] [37]

Deep homology Edit

Roughly spherical eggs of different animals give rise to extremely different bodies, from jellyfish to lobsters, butterflies to elephants. Many of these organisms share the same structural genes for body-building proteins like collagen and enzymes, but biologists had expected that each group of animals would have its own rules of development. The surprise of evo-devo is that the shaping of bodies is controlled by a rather small percentage of genes, and that these regulatory genes are ancient, shared by all animals. The giraffe does not have a gene for a long neck, any more than the elephant has a gene for a big body. Their bodies are patterned by a system of switching which causes development of different features to begin earlier or later, to occur in this or that part of the embryo, and to continue for more or less time. [7]

The puzzle of how embryonic development was controlled began to be solved using the fruit fly Drosophila melanogaster as a model organism. The step-by-step control of its embryogenesis was visualized by attaching fluorescent dyes of different colours to specific types of protein made by genes expressed in the embryo. [7] A dye such as green fluorescent protein, originally from a jellyfish, was typically attached to an antibody specific to a fruit fly protein, forming a precise indicator of where and when that protein appeared in the living embryo. [38]

Using such a technique, in 1994 Walter Gehring found that the pax-6 gene, vital for forming the eyes of fruit flies, exactly matches an eye-forming gene in mice and humans. The same gene was quickly found in many other groups of animals, such as squid, a cephalopod mollusc. Biologists including Ernst Mayr had believed that eyes had arisen in the animal kingdom at least 40 times, as the anatomy of different types of eye varies widely. [7] For example, the fruit fly's compound eye is made of hundreds of small lensed structures (ommatidia) the human eye has a blind spot where the optic nerve enters the eye, and the nerve fibres run over the surface of the retina, so light has to pass through a layer of nerve fibres before reaching the detector cells in the retina, so the structure is effectively "upside-down" in contrast, the cephalopod eye has the retina, then a layer of nerve fibres, then the wall of the eye "the right way around". [39] The evidence of pax-6, however, was that the same genes controlled the development of the eyes of all these animals, suggesting that they all evolved from a common ancestor. [7] Ancient genes had been conserved through millions of years of evolution to create dissimilar structures for similar functions, demonstrating deep homology between structures once thought to be purely analogous. [40] [41] This notion was later extended to the evolution of embryogenesis [42] and has caused a radical revision of the meaning of homology in evolutionary biology. [40] [41] [43]

Gene toolkit Edit

A small fraction of the genes in an organism's genome control the organism's development. These genes are called the developmental-genetic toolkit. They are highly conserved among phyla, meaning that they are ancient and very similar in widely separated groups of animals. Differences in deployment of toolkit genes affect the body plan and the number, identity, and pattern of body parts. Most toolkit genes are parts of signalling pathways: they encode transcription factors, cell adhesion proteins, cell surface receptor proteins and signalling ligands that bind to them, and secreted morphogens that diffuse through the embryo. All of these help to define the fate of undifferentiated cells in the embryo. Together, they generate the patterns in time and space which shape the embryo, and ultimately form the body plan of the organism. Among the most important toolkit genes are the Hox genes. These transcription factors contain the homeobox protein-binding DNA motif, also found in other toolkit genes, and create the basic pattern of the body along its front-to-back axis. [43] Hox genes determine where repeating parts, such as the many vertebrae of snakes, will grow in a developing embryo or larva. [7] Pax-6, already mentioned, is a classic toolkit gene. [44] Although other toolkit genes are involved in establishing the plant bodyplan, [45] homeobox genes are also found in plants, implying they are common to all eukaryotes. [46] [47] [48]

The embryo's regulatory networks Edit

The protein products of the regulatory toolkit are reused not by duplication and modification, but by a complex mosaic of pleiotropy, being applied unchanged in many independent developmental processes, giving pattern to many dissimilar body structures. [43] The loci of these pleiotropic toolkit genes have large, complicated and modular cis-regulatory elements. For example, while a non-pleiotropic rhodopsin gene in the fruit fly has a cis-regulatory element just a few hundred base pairs long, the pleiotropic eyeless cis-regulatory region contains 6 cis-regulatory elements in over 7000 base pairs. [43] The regulatory networks involved are often very large. Each regulatory protein controls "scores to hundreds" of cis-regulatory elements. For instance, 67 fruit fly transcription factors controlled on average 124 target genes each. [43] All this complexity enables genes involved in the development of the embryo to be switched on and off at exactly the right times and in exactly the right places. Some of these genes are structural, directly forming enzymes, tissues and organs of the embryo. But many others are themselves regulatory genes, so what is switched on is often a precisely-timed cascade of switching, involving turning on one developmental process after another in the developing embryo. [43]

Such a cascading regulatory network has been studied in detail in the development of the fruit fly embryo. The young embryo is oval in shape, like a rugby ball. A small number of genes produce messenger RNAs that set up concentration gradients along the long axis of the embryo. In the early embryo, the bicoid and hunchback genes are at high concentration near the anterior end, and give pattern to the future head and thorax the caudal and nanos genes are at high concentration near the posterior end, and give pattern to the hindmost abdominal segments. The effects of these genes interact for instance, the Bicoid protein blocks the translation of caudal's messenger RNA, so the Caudal protein concentration becomes low at the anterior end. Caudal later switches on genes which create the fly's hindmost segments, but only at the posterior end where it is most concentrated. [49] [50]

The Bicoid, Hunchback and Caudal proteins in turn regulate the transcription of gap genes such as giant, knirps, Krüppel, and tailless in a striped pattern, creating the first level of structures that will become segments. [32] The proteins from these in turn control the pair-rule genes, which in the next stage set up 7 bands across the embryo's long axis. Finally, the segment polarity genes such as engrailed split each of the 7 bands into two, creating 14 future segments. [49] [50]

This process explains the accurate conservation of toolkit gene sequences, which has resulted in deep homology and functional equivalence of toolkit proteins in dissimilar animals (seen, for example, when a mouse protein controls fruit fly development). The interactions of transcription factors and cis-regulatory elements, or of signalling proteins and receptors, become locked in through multiple usages, making almost any mutation deleterious and hence eliminated by natural selection. [43]

Among the more surprising and, perhaps, counterintuitive (from a neo-Darwinian viewpoint) results of recent research in evolutionary developmental biology is that the diversity of body plans and morphology in organisms across many phyla are not necessarily reflected in diversity at the level of the sequences of genes, including those of the developmental genetic toolkit and other genes involved in development. Indeed, as John Gerhart and Marc Kirschner have noted, there is an apparent paradox: "where we most expect to find variation, we find conservation, a lack of change". [51] So, if the observed morphological novelty between different clades does not come from changes in gene sequences (such as by mutation), where does it come from? Novelty may arise by mutation-driven changes in gene regulation. [43] [52] [53] [54]

Variations in the toolkit Edit

Variations in the toolkit may have produced a large part of the morphological evolution of animals. The toolkit can drive evolution in two ways. A toolkit gene can be expressed in a different pattern, as when the beak of Darwin's large ground-finch was enlarged by the BMP gene, [55] or when snakes lost their legs as distal-less became under-expressed or not expressed at all in the places where other reptiles continued to form their limbs. [56] Or, a toolkit gene can acquire a new function, as seen in the many functions of that same gene, distal-less, which controls such diverse structures as the mandible in vertebrates, [57] [58] legs and antennae in the fruit fly, [59] and eyespot pattern in butterfly wings. [60] Given that small changes in toolbox genes can cause significant changes in body structures, they have often enabled the same function convergently or in parallel. distal-less generates wing patterns in the butterflies Heliconius erato and Heliconius melpomene, which are Müllerian mimics. In so-called facilitated variation, [61] their wing patterns arose in different evolutionary events, but are controlled by the same genes. [62] Developmental changes can contribute directly to speciation. [63]

Consolidation of epigenetic changes Edit

Evolutionary innovation may sometimes begin in Lamarckian style with epigenetic alterations of gene regulation or phenotype generation, subsequently consolidated by changes at the gene level. Epigenetic changes include modification of DNA by reversible methylation, [64] as well as nonprogrammed remoulding of the organism by physical and other environmental effects due to the inherent plasticity of developmental mechanisms. [65] The biologists Stuart A. Newman and Gerd B. Müller have suggested that organisms early in the history of multicellular life were more susceptible to this second category of epigenetic determination than are modern organisms, providing a basis for early macroevolutionary changes. [66]

Developmental bias Edit

Development in specific lineages can be biased either positively, towards a given trajectory or phenotype, [b] or negatively, away from producing certain types of change either may be absolute (the change is always or never produced) or relative. Evidence for any such direction in evolution is however hard to acquire and can also result from developmental constraints that limit diversification. [68] For example, in the gastropods, the snail-type shell is always built as a tube that grows both in length and in diameter selection has created a wide variety of shell shapes such as flat spirals, cowries and tall turret spirals within these constraints. Among the centipedes, the Lithobiomorpha always have 15 trunk segments as adults, probably the result of a developmental bias towards an odd number of trunk segments. Another centipede order, the Geophilomorpha, the number of segments varies in different species between 27 and 191, but the number is always odd, making this an absolute constraint almost all the odd numbers in that range are occupied by one or another species. [67] [69] [70]

Ecological evolutionary developmental biology (eco-evo-devo) integrates research from developmental biology and ecology to examine their relationship with evolutionary theory. [71] Researchers study concepts and mechanisms such as developmental plasticity, epigenetic inheritance, genetic assimilation, niche construction and symbiosis. [72] [73]


By using operons to teach students to reason with models of complex systems and understand broad themes, we equip them with powerful skills and ideas that form a solid foundation for their future learning in biology. These skills and ideas have broad application in biology, but they also potentially have application in other areas. Constructs, such as negative feedback and natural selection, that are used to explain changing and self-organizing systems constitute what Ohlsson (1993) refers to as “abstract schemas,” which encode the structure of discourse rather than its content. Their abstract nature allows for the possibility of cross-domain transfer. For example, the schema for negative feedback was developed originally in the context of specific engineering problems and was later found to have application in biology. Similarly, natural selection was developed to explain the origin of adaptations in organisms but has subsequently been applied to the development of the immune and nervous systems, computer programming, and artificial intelligence. Mastering abstract schemas enables students to develop into mature thinkers with powerful minds who can imagine solutions to the world’s future problems.

Synthetic gene circuits for metabolic control: design trade-offs and constraints

A grand challenge in synthetic biology is to push the design of biomolecular circuits from purely genetic constructs towards systems that interface different levels of the cellular machinery, including signalling networks and metabolic pathways. In this paper, we focus on a genetic circuit for feedback regulation of unbranched metabolic pathways. The objective of this feedback system is to dampen the effect of flux perturbations caused by changes in cellular demands or by engineered pathways consuming metabolic intermediates. We consider a mathematical model for a control circuit with an operon architecture, whereby the expression of all pathway enzymes is transcriptionally repressed by the metabolic product. We address the existence and stability of the steady state, the dynamic response of the network under perturbations, and their dependence on common tuneable knobs such as the promoter characteristic and ribosome binding site (RBS) strengths. Our analysis reveals trade-offs between the steady state of the enzymes and the intermediates, together with a separation principle between promoter and RBS design. We show that enzymatic saturation imposes limits on the parameter design space, which must be satisfied to prevent metabolite accumulation and guarantee the stability of the network. The use of promoters with a broad dynamic range and a small leaky expression enlarges the design space. Simulation results with realistic parameter values also suggest that the control circuit can effectively upregulate enzyme production to compensate flux perturbations.

1. Introduction

Synthetic biology aims at engineering cellular systems to perform customized and programmable biological functions. The seminal works published in 2000 [1,2] kick-started the development of a wide range of gene circuits with prescribed functions, including bacterial logic gates [3], mechanisms for programmed cell-to-cell communication [4] and light-responsive modules [5]. This progress has recently been followed by the so-called ‘second wave’ of synthetic biology [6], which aims at scaling up the designs from individual genetic modules to whole cellular systems that operate across different layers of cellular regulation, including signalling networks and metabolic pathways [7,8].

One of the most prominent applications of synthetic biology is the manipulation of bacterial metabolism for chemical production in sectors such as energy, biomedicine and food technology [6]. Effective control of metabolism hinges on the ability to upregulate or downregulate pathways in response to changes in the intracellular conditions, cell requirements or environmental perturbations [9]. These requirements call for dynamic control strategies that can modulate enzyme expression in a metabolite-dependent fashion [10,11]. One of the key bottlenecks in this respect is our limited understanding of how genetic design knobs modulate the metabolic responses.

The goal of this paper is to reveal new insights into the design limitations and trade-offs arising from the interplay between gene circuits and metabolic pathways. To that end, we analyse a dynamic model for a feedback system comprising nonlinear kinetic equations for the metabolic species, together with product-dependent enzyme expression controlled by a synthetic gene circuit. We focus on the existence and stability of the steady state, the dynamic response of the network under perturbations and the dependence of these on the design knobs of the synthetic gene circuit.

Two landmark implementations of engineered genetic–metabolic circuits are the genetic control of lycopene production [12] and the metabolic oscillator described in Fung et al. [13]. These works were followed up by the recent study by Zhang et al. [14], whereby the authors reported the first successful implementation of a genetic control circuit to increase biofuel production. In a way akin to man-made technological systems, the use of feedback control plays a pivotal role in ‘robustifying’ pathway dynamics under changing environmental conditions, cell-to-cell variability and biochemical noise. Despite the ubiquity of control engineering methods [15], only a few works have rigorously addressed the problem of genetic feedback design on the basis of mathematical models. Notably, Anesiadis et al. [16] demonstrated the use of a genetic toggle switch [2] as an ON–OFF controller for metabolism, whereas Dunlop et al. [17] explored different genetic control architectures for biofuel production.

From a control engineering standpoint, catalytic enzymes act as inputs to a metabolic pathway in order to drive the metabolite dynamics (i.e. the outputs). The pathway outputs are then sensed by metabolite-responsive molecules that can modulate enzyme expression levels (e.g. transcription factors (TFs) or riboswitches [18]). In the control engineering jargon, this feedback system can be seen as a ‘plant’ (i.e. the pathway to be controlled), and a ‘controller’ (i.e. the gene regulatory circuit controlling the expression of the catalytic enzymes) see figure 1a. The design of the genetic controller must then account for two complementary control objectives: firstly, it must dynamically adjust pathway activity to match the cellular demand for product and sustain the homeostatic balance of native cellular processes. Secondly, a common strategy in metabolic engineering is to modify host microbes by expressing heterologous enzymes that convert metabolic intermediates into a chemical of interest [19]. The consumption of intermediates diverts part of the flux allocated to the host native processes (figure 1b), and, therefore, the controller must also alleviate the impact of these engineered pathways on the native flux.

Figure 1. Control design for metabolic pathways. (a) Transcriptional regulation of metabolic pathways seen as a feedback control system: effector molecules (such as transcription factors) sense metabolite concentrations and modulate the expression of catalytic enzymes, which act as inputs to the pathway. (b) Engineered pathways can divert part of the native metabolic flux to the production of foreign compounds. (Online version in colour.)

In this paper, we study an unbranched metabolic pathway under transcriptional repression from the product. The synthetic circuit consists of an operon encoding all the catalytic enzymes that is repressed by a product-responsive TF (§2). The operon feedback architecture mimics natural circuits enabling cellular adaptations to environmental perturbations (e.g. in bacterial amino acid metabolism [20] and nutrient uptake [21]). To maintain a general analysis, we do not specify the kinetics of the metabolic model, but rather work with a generic class of enzyme turnover rates satisfying mild assumptions. These are satisfied by a wide range of saturable enzyme kinetics, including Michaelis–Menten kinetics and cooperative behaviour described by sigmoidal kinetics [22]. We parameterize the genetic model in terms of the promoter characteristic and the ribosome binding site (RBS) strengths, which are typical design elements used as tuneable knobs in synthetic biology applications. As with the enzyme kinetics, we do not fix the shape of the promoter characteristic, but rather consider a generic class of repressive functions that account, in particular, for the standard Hill equation model for transcriptional repression [23].

Model analysis revealed that enzymatic saturation and promoter leaky expression limit the RBS strength design space (§3.1). These constraints must be satisfied to guarantee the existence of an equilibrium point, to prevent the accumulation of metabolites and to ensure the stability of the network under small perturbations. The feasible set for the RBS strengths depends critically on the promoter leakiness and substrate availability. Within the feasible set, RBS strengths may be used to fine-tune the balance between the intermediate metabolite levels and the gene expression burden imposed on the host cell. We also obtained analytical formulae for the modes of the feedback system these showed that the operon architecture leads to slow fixed modes, and suggests a separation principle between the effect of RBS strengths and the promoter characteristic (§3.2).

We also show that engineered pathways consuming an intermediate add further constraints to the RBS strengths design space, which can be relaxed by using promoters with a high dynamic range and small leakiness (§4.1). We performed numerical simulations of the model with physiologically realistic parameters in Escherichia coli (§§3.2 and 4.2). The simulations show that the control circuit can effectively upregulate enzyme production to compensate an increase in the cell's native demand for product and the impact of engineered pathways. These also suggest that, in terms of both flux and product homeostasis, the synthetic circuit always outperforms an uncontrolled pathway (i.e. with constant enzyme levels), thus highlighting the advantages of using a dynamic feedback control strategy.

2. Unbranched pathway under transcriptional feedback regulation

We consider an unbranched metabolic pathway as in figure 2b, where s0 denotes the concentration of substrate, s1 and s2 are intermediate metabolites and s3 is the metabolic product. The metabolic reactions occur at a rate vi (each one catalysed by an enzyme with concentration ei) and d denotes the rate of product consumption by the cell. The metabolic genes are encoded in a single operon controlled by a product-responsive TF that represses enzyme expression. This kind of transcriptional feedback is common, for example, in bacterial nutrient uptake systems (e.g. the lactose operon [21]) and amino acid metabolism (e.g. the tryptophan operon [20]).

Figure 2. Generic model for an unbranched metabolic pathway under transcriptional repression from the product. The enzymes ei catalyse the reactions at a rate vi, and the cell consumes the product at a rate d. The enzymes are encoded in an operon under the control of a single promoter that is repressed by a TF. (Online version in colour.)

2.1. Metabolic pathway

The network in figure 2 exchanges mass with the environment and/or other networks in the cell. The model accounts for this interaction via the input substrate s0 and the product consumption rate d. We are interested in biologically meaningful phenotypes, and, therefore, we assume that s0 is constant to ensure that, the network can reach a non-zero steady state [24]. Note that, if the substrate decays in time, the network eventually reaches a zero equilibrium, whereby the substrate, intermediate metabolites and product are fully depleted. The constant substrate assumption is also suitable for scenarios where s0 is an extracellular substrate pool shared by a low-density cell population (so that the effects of cell-to-cell competition are negligible).

In a pathway with n reactions and n metabolites, the rate of change of metabolite concentrations can be described by

The rate of product consumption d(sn) is typically modelled as a saturable function of Michaelis–Menten type [20], but, for the sake of generality, we will consider a generic saturable function d(sn) satisfying d(0) = 0, and (cf. assumptions (2.3) and (2.4) for the turnover rates). The cellular demand for product depends on the concentration of a product-catalysing enzyme (which is not explicitly modelled in (2.1)). For typical consumption kinetics such as the Michaelis–Menten or Hill equation, the maximal consumption rate dmax is proportional to the enzyme concentration, and therefore in our model we can describe changes in cellular demand as changes in the parameter dmax.

2.2. Synthetic gene circuit

In an operon architecture all the enzymes are under the control of a single promoter (for the multi-promoter case see [26]), and therefore we model the expression of catalytic enzymes as

Promoter characteristic. It describes the regulatory effect of the TF on gene transcription. The function depends on the specific molecular mechanisms underlying the product–TF and TF–promoter interactions. In order to keep a generic description of the regulatory effect, and to parameterize the model in terms of experimentally accessible design parameters, we opt for a phenomenological description of the promoter characteristic. We therefore consider a function that depends directly on the product concentration and represents the net effect of the product on the transcription rates. Gene transcription under the action of a repressible promoter is typically modelled using Hill functions, but to keep the analysis general we consider generic transcription repression functions satisfying σ(0) = 1, and .

Figure 3. Tuneable knobs in a synthetic operon control circuit. (a) The promoter characteristic and RBS strengths modulate gene transcription and translation rates, respectively (the symbols are described in the legend of figure 2). (b) Sigmoidal characteristic of a repressible promoter. (Online version in colour.)

Promoters are typically described in terms of their tightness (κ 0 ) and strength (κ 1 ) see figure 3b. The tightness refers to the level of baseline transcription (i.e. under full repression by the product), whereas the strength is the gap between the ON and OFF transcription levels. The promoter strength is quantified in terms of the dynamic rangeμ,

Ribosome binding site strengths. RBSs are mRNA sequences that are bound by the ribosomes to initiate translation [10]. The translation rate of the enzymes can then be modified by choosing RBS sequences with different affinities to ribosome binding [28]. We model the effect of the RBS strengths on the enzyme expression rate via the parameters bi.

With the above assumptions and definitions, we can write the complete model for the feedback system as

3. Circuit design for cellular demands

3.1. Trade-offs and constraints in the design of ribosome binding site strengths

The operon circuit must be able to sustain a metabolic flux that feeds the product into the downstream native processes of the host. In this section, we show how this essential requirement translates into constraints on the RBS strength design space.

We will denote the steady-state metabolite concentrations, enzyme concentrations and reaction rates as , and , respectively. We first note that the steady-state enzyme concentrations can be obtained by setting in (2.7), leading to

The equilibrium concentrations of the intermediates can be obtained by setting for i ≥ 2,

Effect of the promoter characteristic. From the steady-state equation in (3.2), we can calculate (see appendix A.1 for a detailed derivation) the sensitivity of the product concentration to changes in promoter tightness κ 0 , promoter strength κ 1 or RBS strength b1,

Effect of the RBS strengths. Using (3.1), we can write the steady state of the downstream enzymes as

In the above discussion, we have implicitly assumed that a solution to equations (3.2) and (3.3) exists. However, because of the saturable characteristic of the product consumption rate (d) and enzyme kinetics (gi), both equations may lack a solution. Firstly, the solution of (3.2) can be computed as the intersection of the two curves shown in figure 4. From these plots, we can see that an intersection exists only when dmax/g1(s0 ) > b1κ 0 /γ, or equivalently

which defines a constraint on the RBS strength of the first enzyme. Since both sides of (3.2) are monotonic in , the solution is unique. By equation (3.1), the existence of also guarantees the existence of the steady-state enzyme concentrations.

Figure 4. Existence of the metabolic flux, product and first enzyme concentration. (a) The solution of steady-state equation (3.2) can be seen as the intersection of two curves, h1(x) = d(x)/g1(s0) and h2(x) = b1(κ 0 + κ 1 σ(x))/γ. (b) The intersection does not exist when condition (3.9) fails.

Secondly, since the enzyme turnover rates saturate at , equation (3.3) has a finite solution provided that

which defines a constraint on the bi/b1 ratio. Taken together, conditions (3.9) and (3.10) define a feasible region in the RBS strengths design space that prevents the accumulation of intermediates and product (figure 5b). If the condition in (3.9) is not satisfied, the substrate will be consumed at a higher rate than the maximal product consumption, and therefore the design will lead to an infinite accumulation of the product. Likewise, violation of at least one of the bounds in (3.10) will cause enzymatic saturation and lead to infinite accumulation of an intermediate.

Figure 5. Design trade-offs and constraints for the RBS strengths. (a) Design trade-off: a low bi/b1 ratio yields low steady-state enzyme levels and high steady-state intermediate levels, whereas high ratios increase enzyme concentrations in favour of lower concentrations for the intermediates. (b) Design constraint: the feasible RBS region prevents the accumulation of intermediates and product the region is defined by conditions (3.9) and (3.10). (c) Tighter promoters enlarge the feasible region, whereas (d) leakier promoters or higher substrate concentrations tighten the constraints. (Online version in colour.)

Conditions (3.9) and (3.10) link together genetic and metabolic parameters (the RBS strengths bi and promoter tightness κ 0 , together with the substrate availability s0 and the enzyme saturation ), and therefore they shed light on how the design constraints appear due to the interplay between metabolic and enzyme expression dynamics. In figure 5c,d, we illustrate the effect of promoter tightness and substrate availability on the feasible region for the RBS strengths. Tighter promoters relax condition (3.9) and therefore enlarge the feasible region (figure 5c). In the limit case of a perfect leak-less promoter (i.e. κ 0 = 0), condition (3.9) does not limit the RBS strength of the first enzyme. Conversely, by conditions (3.9) and (3.10), a higher substrate tends to tighten the feasible region (figure 5d).

3.2. Adaptation to changes in cellular demand

One of the purposes of the genetic feedback circuit is to sustain pathway operation under changes in the cellular demand for product. From a control engineering standpoint, a change in cellular demand can be seen as a perturbation signal acting on the network. A useful approach to study dynamical systems under perturbations consists in examining their linear approximation around their equilibrium points. If we write the model (2.7) as and compute its Jacobian matrix ( ), then trajectories starting in a small vicinity of the steady state can be approximated as

In the case of the feedback system in (2.7), we can exploit the structure of the Jacobian matrix to obtain analytic expressions for its eigenvalues in terms of the design knobs of the gene circuit (see appendix A.2 for details). We found that the 2n eigenvalues can be classified into three categories λfixed, λRBSi and λprom. The system has the following:

— (n−1) stable eigenvalues at λfixed = −γ < 0. These eigenvalues are independent of the circuit design parameters, and therefore they lead to fixed modes, which can be adjusted only by changing the degradation rate (e.g. with various degradation tags). They cannot be suppressed or changed by tuning the circuit design knobs, and, from (3.11), we see that they translate into (n − 1) modes of the form e −t/γ , t e −t/γ , … , t n−2 e −t/γ . The enzyme degradation rates γ are inversely proportional to their half-lives, which are in turn much longer than metabolic time scales (enzymatic half-lives are of the order of minutes to hours, whereas metabolic time scales are typically milliseconds to seconds [22]). Therefore, depending on the initial conditions the network can potentially display very slow transients, and this appears to be aggravated in long pathways.

— (n−1) stable eigenvalues at

— Two stable eigenvalues at

with and . Unlike λfixed and λRBSi, these two eigenvalues depend on the steady-state product concentration, and therefore they can be fine-tuned through the promoter characteristic (see equation (3.2)). To study the dependence of λprom on the promoter design parameters, we computed them for a pathway with realistic parameter values. In figure 6a, we show the steady-state values of the product, flux and first enzyme level for a wide span of promoter dynamic range μ. We observe that strong promoters tend to increase pathway flux, in agreement with the sensitivity equation previously derived in (3.5). We also see that, as shown by the steady-state relation , the flux corresponds to a scaled version of the concentration of the first enzyme. In figure 6b, we plot the location of the promoter-dependent eigenvalues λprom in the complex plane. These indicate that, in the case of weak promoters, the eigenvalues λprom lie on the real axis, becoming complex only for a sufficiently broad dynamic range μ. For strong promoters, the real part becomes closer to the imaginary axis, potentially leading to slow transients. Moreover, since stronger promoters lead to a higher flux, the eigenvalues in figure 6b suggest that flux maximization may entail a reduction in the response speed.

Figure 6. Impact of the promoter dynamic range on the metabolic steady state and the modes of the feedback system. (a) Steady-state value of the metabolic flux, and the concentrations of the product and the first enzyme. (b) Location of the promoter-dependent eigenvalues λprom. The plots were generated by solving the steady-state equation (3.2) (a) and computing λprom from (3.13) (b) for a different dynamic range μ spanning five orders of magnitude and a fixed tightness κ 0 . The real and imaginary parts in (b) are normalized to the degradation rate γ. We considered a pathway with two metabolites and two enzymes with Michaelis–Menten kinetics (gi(si−1) = kcat isi−1/(KM i + si−1)) with kcat i = 32s −1 and KM i = 4.7 µM. These are representative values for PRA isomerase (extracted from the BRENDA database [29], EC number, a transcriptionally regulated enzyme in the tryptophan pathway of E. coli. We took the enzyme degradation rate as 2 × 10 −4 s −1 (half-life h), and used a substrate concentration s0 = KM1 (so that g1(s0) is at half-saturation). We modelled the product consumption as a Michaelis–Menten function d(sn) = dmaxsn/(Kd + sn) with dmax = 24.96 µM min −1 and Kd = 0.2 µM, both taken from an experimentally validated model for the tryptophan pathway [20]. Promoter tightness was fixed to κ 0 = 0.03 nM min −1 and the RBS strength to b1 = 1, whereas the repression function was described by a Hill function σ(sn) = θ h /(θ h + sn h ) with Hill coefficient h = 2 and repression threshold θ = Kd. (Online version in colour.)

To illustrate the dynamic response of a pathway under the control of the transcriptional control circuit, we simulated the network under a change in the cell demand for product (see figure 7a). We modelled a change in the cell demand as a slow S-shaped temporal increase in the maximal product consumption rate dmax (see the inset in figure 7a). This describes, for example, cases in which the demand increase is due to native processes upregulating the enzyme that metabolizes the product. Note that, from the steady-state equation in (3.2), a higher dmax inevitably leads to a higher flux and a lower steady-state product concentration (see also figure 4a). Before the perturbation, the network is in steady state with a pre-stimulus flux d pre = 19.5 µM min −1 . We considered a Michaelis–Menten consumption rate of the form d(sn) = dmaxsn/(Kd + sn), with Kd being the product concentration needed for half-maximal consumption. Upon the increase in dmax at t = 50 min, we observe that the promoter responds to the drop in product concentration and upregulates enzyme expression so as to drive the pathway to a new post-stimulus flux d post that is approximately 40 per cent higher than d pre , and a product concentration that is approximately 20 per cent lower than its pre-stimulus value. Using equations (3.1) and (3.2), we can compute the enzyme upregulation factor as

which is equivalent to the relative change in pathway flux. The dynamic upregulation of enzyme expression can be seen in the lower panel of figure 7a, where we can also verify that the upregulation factor is approximately per cent as predicted in (3.14). Note that as a consequence of the operon architecture, all the enzymes are upregulated by the same fold-factor. This factor depends on the pre- and post-stimulus fluxes, which by (3.2) depend only on the promoter design and first RBS strength.

Figure 7. Response of a synthetic operon to changes in the cellular demand for product. (a) Comparison between the dynamic responses of the synthetic control circuit and the uncontrolled case the change in cell demand was induced at t = 50 min and modelled as an S-shaped increase in the maximal consumption rate dmax, reaching 50% of the pre-stimulus dmax in h (see inset). (b) Drop in the steady-state product concentration relative to its pre-stimulus value as a function of the change in metabolic flux and the promoter dynamic range. In (a), the promoter dynamic range was set to μ = 100 and the RBS strengths to bi = <1,4> for the pathway parameters used in figure 6, the chosen bi comply with the feasible region of figure 5 and lead to steady-state enzyme concentrations that are within the physiological range of E. coli [30]. In (b), we swept the dynamic range μ between 3 and 200 by changing the promoter strength. All the remaining network parameters are the same as in figure 6. (Online version in colour.)

As a way of comparison, in figure 7a, we have also simulated the response of a pathway without feedback regulation (i.e. with constant enzyme levels chosen to match the flux of the controlled case dpre). The uncontrolled pathway is unable to increase the flux, and we observe that this leads to a considerable decrease in product concentration (approx. 70% reduction). In the uncontrolled case, the rate of substrate uptake is fixed to v1 unc so that the equilibrium product satisfies

We study the performance of the control circuit in more detail in figure 7b, where we show the drop in product concentration relative to the pre-stimulus level as a function of the change in flux and dynamic range of the promoter. We observe that stronger promoters can significantly improve the compensation of the drop in product concentration (a perfect compensation would correspond to a flat curve at 0% in figure 7b). For example, under a 50 per cent increase in the pathway flux, a mild promoter (μ = 10) leads to a drop in product of approximately 47 per cent, whereas a strong promoter (μ = 100) can bring down the drop in product to approximately 20 per cent (the latter corresponds to the design simulated in figure 7a). As predicted by the upper bound in (3.7), the flux is limited by the promoter strength, and therefore weak promoters do not allow for large increases in flux (as a consequence, the domain of the curves in figure 7a decreases with decreasing promoter strength) for example, for the weakest promoter tested, the flux could not be increased beyond approximately 10 per cent.

4. Circuit design for compensation of flux perturbations

A common strategy in metabolic engineering is to modify bacteria by expressing heterologous enzymes that convert natural metabolic intermediates into a compound of interest [19]. The target compound is synthesized by ‘branching out’ a specific intermediate from a natural pathway, and therefore part of the metabolic flux needed to sustain the host native processes is redirected to the production of the foreign chemical. The choice of a good branching point (i.e. one that does not lead to lethal metabolic imbalances for the host) is a major problem typically addressed with the aid of optimization-based computational tools [31,32]. In this section, we turn our attention to the effect of a perturbation in the native flux as a consequence of branching out from an intermediate metabolite.

4.1. Trade-offs and constraints in the design of the RBS strengths

To account for an engineered pathway consuming the intermediate at a constant rate dext, we include dext as a consumption rate in the ODE for

with . From these steady-state equations, we observe similar properties as in the case without a branch. The promoter characteristic and the first RBS strength determine the metabolic flux, whereas the RBS ratio bi/b1 can be used to fine-tune the balance between enzyme expression and the concentrations of the intermediates. In addition, in this case, we see that the intermediates after the branch point also depend on the promoter characteristic.

Figure 8. Design constraints of the RBS strengths with a branch consuming an intermediate. (a) Metabolic pathway under transcriptional regulation with a branch consuming an intermediate. Model equations are shown in (4.1) and symbols are described in the legend of figure 2. (b) In addition to the constraints in figure 5b, the branch introduces a further constraint to avoid product depletion the region is defined by conditions (4.5)–(4.7). (c) Stronger and tighter promoters enlarge the gap Δ and the RBS design space according to equation (4.8). (Online version in colour.)

Using similar arguments to those in figure 4, we find that a solution to (4.2) exists if

4.2. Adaptation to a flux perturbation

To illustrate the effect of an engineered branch on the dynamic response of the feedback system, we simulated a network with two metabolites and two enzymes under a flux perturbation that consumes the intermediate s1 (see figure 9a). Before the perturbation, the network is in steady state with a native flux d pre = 19.5 µM min −1 . We modelled the engineered branch as an S-shaped increasing rate d ext (t) (see the inset in figure 9a). Upon the activation of the branch, induced at t = 50 min, the synthetic operon circuit upregulates enzyme expression by approximately 45 per cent to drive the pathway to a new native flux d post . Using equation (3.1), together with the pre- and post-stimulus steady-state equations ((3.2) and (4.2)), we find that the enzymes are upregulated by the factor

Figure 9. Response of a synthetic operon to flux perturbations induced by engineered pathways. (a) Comparison between the dynamic responses of the synthetic control circuit and the uncontrolled case the flux perturbation was induced at t = 50 min, and modelled as an S-shaped increasing rate consuming the intermediate s1, reaching 50% of the pre-stimulus flux in h (see the inset). (b) Flux reduction and enzyme upregulation relative to the pre-stimulus levels as a function of the consumption rate in the branch and the promoter dynamic range. The uncontrolled case (μ = 1) is represented by the dashed lines. As in figure 7, in (a) we chose a promoter with dynamic range μ = 100 and RBS strengths bi = <1,4>, whereas in (b) we swept μ between 3 and 200 by changing the promoter strength. All the remaining network parameters are the same as in figure 6. (Online version in colour.)

As in the case without the branch, the expression in (4.9) indicates that all enzymes are upregulated by an identical fold-factor that depends on the promoter design and the first RBS strength.

In figure 9a, we have also simulated the response of a pathway without feedback regulation (i.e. with constant enzyme levels chosen to match the flux d pre of the controlled case). In terms of both flux and product concentration, we observe that the feedback-controlled network displays a dramatic improvement compared with the uncontrolled case: the operon circuit reduces the loss in native flux from 50 per cent to approximately 5 per cent, whereas the decrease in steady-state product concentration is brought down from approximately 82 per cent to 20 per cent. In the uncontrolled case, the rate of substrate uptake is fixed to v1 unc and therefore the post-stimulus flux is given by

5. Discussion and outlook

In this paper we have presented a detailed analysis of a synthetic gene circuit designed to dynamically control metabolic pathways. The goal of this feedback control system is twofold: to adjust pathway activity so as to match the cell demand for product, and to dampen flux perturbations that divert the native flux to the synthesis of foreign molecules. The control strategy relies on encoding the metabolic genes in a single operon repressed by a product-responsive TF. The TF can sense a drop in product concentration and upregulate enzyme expression to bring the pathway close to its homeostatic levels.

Since the seminal operon paper [33], the interaction between the genetic machinery and metabolism has been extensively studied in the context of natural systems. These studies typically focus on understanding how observed phenotypes emerge from the genetic–metabolic cross talk [34–38], and a number of detailed mechanistic models for operon regulation have been developed (e.g. [20,21]). The goal in synthetic biology, however, is to design regulatory circuits for controlling metabolism in a customized fashion. Model-based design therefore requires mathematical descriptions that are explicitly parameterized in terms of the design knobs that can be manipulated in synthetic biology applications. Consequently, we have used a gene expression model that is deliberately not mechanistic, and instead describes the genetic feedback in terms of tuneable parameters such as the promoter's dynamic range, RBS strengths and protein half-lives. This approach has proved to be adequate to explore the genetic design space and to quantify the impact of the promoter characteristic and RBS strengths on the system response.

A typical complication in engineered pathways is that enzymatic saturation may cause intermediates to accumulate in prohibitively large concentrations, thus affecting the viability of the host due to toxic effects [11]. Metabolite accumulation arises when the steady state lies beyond the saturation limit of a catalytic step, and available models for pathways under transcriptional regulation [20,39–41] have generally overlooked the impact of enzyme saturation on the existence of a metabolic steady state. In our aim to carry out a general analysis, we have used a metabolic model that accounts for a whole class of saturable enzyme kinetics under mild assumptions. By explicitly accounting for enzyme saturation, we characterized a feasible set for the design parameters which ensures that the steady state lies within the saturation limits. The feasible set also guarantees the local stability of the network, and we found that the constraints on the RBS strengths can be relaxed with the use of promoters with a high dynamic range and small leakiness. The geometry of the feasible set depends on a combination of genetic and kinetic parameters, thus highlighting the emergence of design constraints as a consequence of the interplay between the genetic and metabolic subsystems.

The steady-state equations reveal a trade-off between the steady-state enzyme expression levels and the concentration of intermediates: the enzyme concentrations are inversely proportional to the concentration of the intermediate they catalyse. We found that a critical parameter is the RBS ratio, i.e. the relative strength of an RBS with respect to the strength of the first one in the operon, which can be used to fine-tune the circuit between high-enzyme/low-intermediate or low-enzyme/high-intermediate designs.

The two considered design knobs, promoter characteristic and RBS strengths, seem to have decoupled roles in the steady state and transient behaviour of the network. The promoter characteristic together with the first RBS determine the steady state of the product and the first enzyme. A strong promoter and a strong RBS for the first enzyme can be used to increase the pathway flux, but this may come at the expense of slow modes in the transient response. In the absence of an engineered pathway consuming an intermediate, the remaining RBS strengths can be used to independently adjust the concentrations of the intermediates and the remaining enzymes. In the case of consumption of an intermediate, however, this design rule applies only to the metabolites upstream of the consumed intermediate, i.e. the steady state of the downstream metabolites depends on a combination of the RBS strength, promoter characteristic and the size of the perturbation.

The closed-form expressions for the transient modes of the feedback system show further evidence of the separation principle between promoter and RBS design. From the 2n modes of an n-step pathway, we found that only two depend on the promoter characteristic, whereas further (n − 1) modes depend exclusively on the RBS ratio. The remaining (n − 1) modes correspond to the enzyme half-lives and are independent of the promoter characteristic and RBS strengths. Since enzyme half-lives are considerably slower than metabolic time constants (even with the use of protein degradation tags), the system dynamics can be dominated by slow transients.

We ran numerical simulations that demonstrate the potential of the proposed control strategy. Using physiologically realistic parameter values for E. coli, the synthetic operon control circuit can dramatically compensate the loss in flux by sensing the drop in product concentration and subsequently upregulating the enzyme concentrations. The feedback-controlled pathway outperforms the uncontrolled one even when weak promoters are used, thus underscoring the tremendous advantage of taking a feedback approach to metabolic control.

In this work we focused on a control circuit with an operon architecture, a choice inspired by the fact that operons are one of the building blocks in genome-wide bacterial networks [42]. The ubiquity of natural metabolic pathways under operon regulation [43] makes them a reasonable choice as template architectures for engineered circuits. In addition, the main difficulty in building genetic–metabolic systems is to find suitable regulatory molecules to interface a metabolite of interest with the genetic machinery. Some of the available alternatives are engineered promoters [44,45], metabolite-responsive riboswitches [18,46,47] and natural TFs (for a comprehensive catalogue of natural metabolite-responsive TFs see Zhang et al. [14, supp. table 5]). In this respect, an operon architecture stands as a simple yet effective topology, as it requires only one metabolite-responsive TF. More complex architectures can certainly add more flexibility to the design, but this will probably come at the expense of more intricate relationships between the design parameters and the metabolic response. For example, the use of multi-promoter circuits allows for independent tuning of the enzyme upregulation factor, but at the same time the pathway may display sustained oscillations if the characteristics of the different promoters are not carefully designed [26].

We should point out that the derived design constraints guarantee the existence and stability of the metabolic steady state, and thus they are only baselines for the correct functioning of the genetic control circuit. In most applications, the design must also account for more demanding objectives such as maximization of flux, minimization of energy expenditure, or a combination of these. Since these objectives may conflict with each other, selecting an appropriate combination of circuit parameters requires the use of multi-objective optimization methods within the feasibility sets derived here (see, for example, figures 5 and 8). Optimization routines can therefore be used to single out the parameter values that lead to an acceptable compromise between mutually colliding objectives see Banga [48] for a review of a number optimization methods available.

As a consequence of a compromise between model complexity and the generality of the analytic results, our results have two main limitations. Firstly, we have restricted the analysis to pathways with irreversible reactions, and, secondly, our results are limited to unbranched pathways operating in isolation of the remaining metabolism of the host cell. Enzymatic reactions are inherently reversible processes and, although many biosynthetic reactions operate in a regime where the forward reaction is much more likely to occur than its backward counterpart [22], their reversibility cannot always be neglected [49]. In our case, the use of irreversible reactions is an important simplification that allowed the derivation of intuitive and easy to interpret relations between the network parameters and its steady state. Other instances where the analysis of irreversible pathways led to new insights into biological design principles include, for example, the works in [38,43]. Our derivation of the design constraints on the promoter parameters and RBS strengths relies on the structure of the steady-state equations and the fact that most of them are decoupled from each other. However, in the case of an n-step pathway with reversible reactions, the steady-state equations form a system of 2n coupled algebraic equations. These equations may admit an analytic solution for specific enzyme kinetics (see Heinrich & Klipp [50] for the solution in the case of linear and Michaelis–Menten kinetics with constant enzyme concentrations), but its extension to transcriptionally controlled enzymes and general reversible kinetics is cumbersome and lies outside the scope of our paper.

A possible workaround to deal with reversible kinetics is to exploit the natural timescale separation between enzyme expression and metabolic reactions. In this approach, the metabolite trajectories are assumed to evolve in a much faster time scale than the enzyme concentrations. This allows us to approximate the metabolite concentrations as algebraic functions of the enzymes, leading to an enzyme-only ODE model subject to the algebraic relations between metabolites and enzymes. We have previously used such an approach in the case of ON–OFF promoters [36] (i.e. promoters that are either fully active or inactive, without intermediate levels of gene expression), and future work will focus on its use with graded promoters such as those considered here. Another advantage of the time scale separation is that it may allow for the analysis of pathways with more complex stoichiometries. This is of enormous relevance in practical applications, as the cross talk between the controlled pathway and the rest of the host metabolism is likely to have a detrimental impact on the performance of the feedback control system.

We are exploring a number of extensions to this work, aiming primarily at the use of alternative feedback topologies and at quantifying the impact of biochemical noise on the pathway performance. The implementation of genetic–metabolic circuits, let alone parameter fine-tuning, can be costly and time-consuming. Our work provides a first step towards understanding the fundamental limitations and trade-offs that must be addressed at the design stage, potentially facilitating the implementation using a model-guided rationale.

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Y.C. and J.N. acknowledge funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 686070. Y.C. and J.N. also acknowledge funding from the Novo Nordisk Foundation (grant no. NNF10CC1016517). E.v.P.-K., B.v.O., S.B., H.B., D.M., and B.T. acknowledge funding from the Netherlands Organisation for Scientific Research (grant no. ALWTF.2015.4). S.D., H.B., D.M., and B.T. acknowledge funding from the Top-sector Agri&Food (grant no. AF-15503). The computations were partially enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N partially funded by the Swedish Research Council through grant agreement no. 2018-05973.

Study conception: BT, HB, and JN Modeling, simulations, and data analysis: YC and EvP-K Experiments: BvO and SD Data analysis: DM and SB Manuscript writing: YC, EvP-K, and BT Manuscript editing: All authors.

Watch the video: Jacob-Monod: The Lac operon. Biomolecules. MCAT. Khan Academy (August 2022).