Allostery and conformational changes upon binding as generic features of proteins: a high-dimension geometrical approach
AAllostery and conformational changes upon binding as genericfeatures of proteins: a high-dimension geometrical approach.
A.S. Zadorin
Chimie Biologie Innovation, ESPCI Paris, CNRS, PSL University, 75005 Paris, France.Center for Interdisciplinary Research in Biology (CIRB), Coll`ege de France, CNRS, INSERM, PSL Research University,Paris, France.
Abstract
A growing number of experimental evidence shows that it is general for a ligand binding protein to havea potential for allosteric regulation and for further evolution. In addition, such proteins generically changetheir conformation upon binding. O. Rivoire has recently proposed an evolutionary scenario that explainsthese properties as a generic byproduct of selection for exquisite discrimination between very similar ligands.The initial claim was supported by two classes of basic examples: continuous protein models with smallnumbers of degrees of freedom, on which the development of a conformational switch was established, and a 2-dimensional spin glass model supporting the rest of the statement. This work aimed to clarify the implication ofthe exquisite discrimination for smooth models with large number of degrees of freedom, the situation closer toreal biological systems. With the help of di ff erential geometry, jet-space analysis, and transversality theorems,it is shown that the claim holds true for any generic flexible system that can be described in terms of smoothmanifolds. The result suggests that, indeed, evolutionary solutions to the exquisite discrimination problem, ifexist, are located near a codimension-1 subspace of the appropriate genotypical space. This constraint, in turn,gives rise to a potential for the allosteric regulation of the discrimination via generic conformational changesupon binding. Introduction
Significant conformational changes in proteins upon their binding to specific ligands are ubiquitous in nature.Their occurrence spans from signaling proteins and transcription factors to enzymes. They are observed evenoutside the realm of proteins: in aptamers and ribozymes. Allosteric regulation, the modulation of the primaryfunction by binding of another molecule at a distant site, is often associated with such biopolymers either in anactual or in a potential form. There are two common features in all these systems. First, they are flexible. Second,the primary task that all of them solve includes a fine discrimination between di ff erent but close ligands vs.solvent. Namely, the desirable ligand must be more preferably than the solvent with the contrary for undesirableligands. For signaling proteins it is the ability to distinguish between the specific signal and similar molecules.For transcription factors it is the recognition site among similar DNA motifs. For enzymes and ribozymes it isthe ability to su ffi ciently strongly bind the substrate and / or the transition state and to release the (often similar)product [1].The main focus of researchers since the discovery of conformational changes has been on the nature andmechanisms of these changes. Their evolutionary origin is usually seen either as a requirement for the functionof the protein in question or as a subsequent development of regulation of this function. For example, for asignal transduction receptor, a conformational change upon binding is necessary to initiate the signal responsepathway (be it binding to a DNA site or a transmembrane activation of a signaling cascade). For enzymes,such explanations include the correct positioning of aminoacids in the reaction center and creation of the correctenvironment around the substrate, as well as kinetic control of the reaction rate. For the enzymes that aremolecular motors, conformational changes are the essence of their function. Finally, the conformational changeis seen as an adaptation to allosteric regulation of protein function [2]. It is important to emphasize that in thisview allostery is actively selected for and a conformational change serves as a means to fulfill this demand.1 a r X i v : . [ q - b i o . B M ] M a y uch explanations assume some adaptive value of the conformational change in light of selection for a complexproperty of the protein. The conformational changes themselves are understood as highly orchestrated events.The main two hypothetical mechanisms of the conformational change itself, however, assume it to be generic.These two mechanisms are the hypothesis of induced fit and the hypothesis of conformational selection . In theinduced fit scenario, the ligand provokes a conformational change in the su ffi ciently flexible binding proteinafter an initial weak binding. This results in a strongly bound complex [3, 4]. In the conformational selectionparadigm, the native state and the conformation of the strongly bound complex exist as possible conformationsin the population of the free protein under normal conditions. The native state is assumed to reflect the globalminimum of the free energy while the other state is assumed to be metastable with lower probability in the popu-lation. The binding of the ligand change their roles and the other state becomes predominant. The conformationsthemselves, in this scenario, do not significantly change—only their free energy levels change [5, 6].Recently, a di ff erent view on the problem of allostery and conformational changes in proteins was introducedby O. Rivoire in [7]. Rivoire noticed that allostery can be a consequence of an existing conformational changein a discriminating protein. Indeed, if a conformational change is invelved in an exquisite discrimination, theability to discriminate can be turned o ff by blocking the movement itself and thus changing the energies ofbound states. If the conformation changes far from the initial binding site, as it takes place in su ffi ciently largeconformational change, the regulating binding can happen far from the initial binding pocket. This situation isinterpreted as allosteric regulation. The conformational change in the first place, in this scenario, comes as abyproduct of the selection towards the exquisite discrimination. Thus, a potential for allostery emerges from aselection for a much simpler property.The validity of this scenario was demonstrated in [7] on two types of models: 1) extremely simple elasticnetwork models and 2) a spin glass model of proteins. The elastic models treated the protein as a single masson one or two springs with only one degree of freedom. Possible ligands were treated as numerical values on anaxis of environmental variables that served as an additional force constantly acting on the system. In addition,the evolutionary degree of freedom also was considered to be a single continuous variable that imposes anotherforce of the same sort. The spin glass model, from the other hand, had multiple configurational and evolutionarydegrees of freedom, but a configuration of each “aminoacid” was described only by either “up” or “down” state.Both these models are strongly simplified descriptions of proteins.Both models gave the same qualitative result, formulated in [7] as follows. Under the assumption of asystem’s flexibility, the discrimination of particular similar ligands requires the system to be evolutionary finelytuned to respect requirements on free energies of the complexes. This constraint causes a generic conformationalchange upon binding. A hypothesis, tested only on the spin glass model of proteins, was formulated that connectsa large enough conformational change with the potential for allosteric regulation by involvement of distant partsof the protein in the movement. Being involved in keeping a delicate free energy balance, such sites becomea potential target for further regulation by another ligand, for example. Furthermore, it was shown (on thecontinuous elastic model) that the conformational change may have a form of continuous deformation of theinitial state to the final one or these states may be di ff erent ones and may even coexist as a global and a localminimum. Thus the distinction between the induced fit and the conformational selection becomes moot fromthis point of view.However, the simplicity of the models used for illustration of this powerful principle comes with stronglimitations that may prevent a direct generalization. The drawback of the spin glass model is in its intrinsicdiscontinuity, and it is di ffi cult to say if the observed e ff ects are related to general properties of protein-likesystems or to this particularity of the model. This is especially true for the claim about a conformational switch,since the behaviour of the system is switch-like at the level of each element from the beginning.The continuous elastic model su ff ers from its low dimensionality. It is not clear how the conclusion can bedrawn from an example with one physical degree of freedom, one scalar phenotypic trait, and a ligand spacedescribed by a single number. Furthermore, the particularly simple relation between the ligand and the system’spotential may turn out to be a very particular case.In the current work, it is proven that the conclusions of [7] are, indeed, valid, under a certain interpretation,for a much wider class of models: continuous systems with any number of degrees of physical freedom, anydimensionality of the phenotypical trait space, and any number of parameters describing ligands. In addition,an estimate on the abundance of evolutionary solutions to the exquisite discrimination of particular ligands(equivalently, on the required fine tuning of the protein sequence) is derived in terms of the dimensionality ofthe set in the trait space around which the solutions are concentrated. It is also shown that the proposed scenario2or the origin of allosteric regulation is plausible in these settings, too.The work is organized in the following way. In Section 2, the problem is formulated in terms of physicalchemistry. In Section 3, it is translated to a mathematical model. In Section 4, the problem is rigorouslyformalized in the language of di ff erential geometry and three main theorems are stated, constituting the mainresult about the exquisite discrimination problem, the conformational changes, and allostery: Theorem 1, 2,and 3. A biological interpretation and some implications of theses results are outlined in Section 5. The finalSection 6 is completely devoted to formal mathematical proofs of the main theorems. Physical formulation of the problem
Following [7], we assume that evolution of a protein involves three types of variables: 1) physical (conforma-tional) degrees of freedom x , 2) environmental degrees of freedom (cid:96) that define the surrounding medium, and 3)evolutionary degrees of freedom a associated with the protein sequence.Typically, the variables in x include positions of single atoms or distances between pairs of them and com-pletely describes the shape (conformation) of the molecule. Depending on the coarse graining of the model,it may describe mutual orientation of larger portions of the molecule, like individual aminoacids. In the lattercase, x may also involve angles on top of distances. The variables in (cid:96) contain information about the environ-ment around the molecule. In this particular case they are restricted to the identification of a ligand bound to aparticular site of the protein or to the absence of any such ligand (a molecule of the solvent can be taken as theligand for this case). Finally, a describes the genetic information involved in building the molecule. In the mostdirect case it is its aminoacid sequence. Alternatively, it can reflect some higher level aggregated phenotypicalproperties of the molecule or its parts that define its behaviour in the selected level of abstraction.These three types of variables are linked via the parametrized potential energy U ( x , (cid:96), a ) of the protein, where (cid:96) and a are parameters. Thus, a function U ( x , (cid:96), a ) describes a family of potential energies U (cid:96), a ( x ) with a constantparameter a (it defines the protein) and an environment-dependent parameter (cid:96) (it describes how the energychanges with the binding of the ligand (cid:96) ). At given (cid:96) and a , the distribution of conformations of the protein isgiven by the Boltzmann distribution with that potential energy such that the probability of conformation x in anensemble of molecules is (cid:144) ( x | (cid:96), a ) = exp (cid:16) − β (cid:0) U ( x , (cid:96), a ) − F ( (cid:96), a ) (cid:1)(cid:17) , where F ( (cid:96), a ) is the free energy of thesystem and β is the inverse temperature measured in energy units. Following the treatment of the continuouscase in [7], we will only consider the zero temperature limit β → ∞ . In this case (cid:144) ( x | (cid:96), a ) degenerates to a δ -function at the point (or points) of the global minimum of U (cid:96), a and F ( a , (cid:96) ) is equal to this minimum.For example, in [7], variables x , (cid:96) , and a were natural numbers and the considered potentials had the follow-ing forms U ( x , (cid:96), a ) = k ( | x | − r ) − ( (cid:96) − a ) x , U ( x , (cid:96), a ) = k ( x − r ) − ( (cid:96) − a ) x , (1) U ( x , (cid:96), a ) = k (cid:16) √ x + d − r (cid:17) − ( (cid:96) − a ) x , where k , d , and r are positive constants.The exquisite discrimination problem is formulated in the following way. Given a desirable ligand (cid:96) r andan undesirable ligand (cid:96) (cid:52) such that (cid:96) r ≈ (cid:96) (cid:52) , and assuming that the environment defined by the solvent alone isrepresented by (cid:96) ∅ , find a such that F ( (cid:96) r , a ) < F ( (cid:96) ∅ , a ) < F ( (cid:96) (cid:52) , a ) , (2)or, in the zero temperature limit, min U (cid:96) r , a < min U (cid:96) ∅ , a < min U (cid:96) (cid:52) , a . (3)This condition is schematically shown on the left part of Figure 1. Mathematical formulation of the problem
For the sake of brevity, we will use the term “protein” for “a system that needs to do an exquisite discrimination”,although, of course, the general argument is not restricted only to proteins. We will assume that a protein is3haracterized by three types of variables: conformational variables x that take value in the configuration space X , environmental variables (cid:96) taking value in L (the space of possible ligands), and evolutionary variables a takingvalue in A (the space of protein sequences, the phenotypical trait space, etc.).The main claim of the current work can informally be expressed in the following statement. Under generalassumptions, a su ffi ciently flexible system that solves the exquisite discrimination problem generically experi-ences a large conformational change upon binding to its substrate. The ability to discriminate requires evolution-ary fine tuning and possible solutions are concentrated near a codimension-1 hypersurface of the phenotypicaltrait space A. The combination of the fine tuning and the conformational change makes the discrimination abilitysensitive to binding of other ligands to distant sites. To make these statement precise we will fix the following assumptions. Spaces X , L , and A are assumedto be smooth ( C ∞ ) compact manifolds. The physical behaviour of a protein is defined by an energy function U : M → (cid:146) , where M = X × L × A . (4)The product is understood in the category of smooth manifolds so M is assumed to be endowed with the structureof a C ∞ manifold. U is assumed to be smooth, as well. U is understood as a family of potentials on X withparameters from L and A . In the zero temperature approximation, the configuration of a protein with sequence a that corresponds to a ligand (cid:96) is considered to be x that minimizes U ( x , (cid:96), a ) with constant (cid:96) and a globallyin X . We also assume that X represents only the shape of the protein and the degrees of freedom of the wholemolecule (translational and rotational) are already excluded as well as that the dimensionality of X correspondsto the number of the leftover independent degrees of freedom.Let us discuss these assumptions. The representation of the configuration space of a physical system bya manifold is very natural and does not require any special explanation. The compactness of X is a technicalrequirement, which is not very restrictive. Indeed, if the configurations are given by the collection of pairwisedistances between elements (aminoacids, nucleotides) with hard links between neighbours in the primary se-quence, as it is commonly assumed in physical models of macromolecules, the configuration space naturally hasa form of closed (without border) compact multiply connected submanifold of some Euclidean space. If, instead,no restrictions are applied, but the interatomic interactions are described by some pairwise potentials, the actualconfiguration space is some Euclidean space, which is not compact. However, as the interaction potential eitherincreases with acceleration (as in elastic network models) or monotonously increase to some finite limit (as inreal molecules) as some atom approaches an infinite distance from the rest of the structure, we are not interestedat the behaviour of U in the neighbourhood of infinity. In this case, it is enough to consider some smaller, com-pact, subspace of the initial configuration space. Finally, when there is a finite potential energy in the bound stateof the protein(as in real molecules), the initial space (cid:146) n can be compactified to the projective space P (cid:146) n . Thecompactness of L , as well as its smooth nature, is just a convenient hypothesis as there are no good models forthis space. The sequence space A is not a smooth manifold in nature. It is rather a nondirected graph with highsymmetry. However, we assume that it can be well approximated by some smooth manifold with a smooth func-tion U ( x , (cid:96), a ). For example, for binary sequences of length n the sequence space is an n -dimensional hypercube.It can be approximated by an ( n − (cid:96) r , stronger than the solvent, (cid:96) ∅ , and the incorrect ligand, (cid:96) (cid:52) , must be bound weaker than thesolvent. (cid:96) r and (cid:96) (cid:52) are assumed to be close in L ( L , being described by some physical parameters, is usuallymetrizable, so one may assume that the distance between the states given by some metrics on this space is muchsmaller than between either of them and (cid:96) ∅ ).As the first step, we will solve a simpler problem. Given two ligands (cid:96) , (cid:96) ∈ L , we want to find suchphenotypes a and such configurations x that the protein bound to either of the ligands has the same minimalenergy U . This point of view ignores the inevitably small di ff erences between the minimal energy levels ofthe complexes with (cid:96) r and (cid:96) (cid:52) . Moreover, the values of (cid:96) (cid:52) and (cid:96) r are considered to be indistinguishable, too.Therefore, we assume (cid:96) (cid:52) = (cid:96) r = (cid:96) . As a consequence, the minimal energy that corresponds to the binding to (cid:96) ∅ = (cid:96) is equal to that of (cid:96) , as we assume it to be between (cid:96) (cid:52) and (cid:96) r (see Figure 1).This simplification can be regarded as a coarse grained view on the problem, where any slightly di ff erentpoints in any of the spaces are seen as equal. In this picture, a significant di ff erence between real configurations x and x means simply x (cid:44) x . Such abstraction allows a rigorous mathematical treatment as it can be recastto questions about intersections of submanifolds of appropriate manifolds. Such questions, taking into account4 • (cid:96) ∅ • (cid:96) (cid:52) • (cid:96) r min x UU ∅ U r U (cid:52) coarse graining( (cid:96) (cid:52) → (cid:96) r ) L • (cid:96) • (cid:96) min x UU = U (cid:51) Figure 1: A schematic representation of the coarse graining approach to build the mathematical formalization ofthe problem. Here L is an abstract ligand space, (cid:96) r is the correct ligand, (cid:96) (cid:52) is the incorrect lignad, (cid:96) ∅ symbolisesthe solvent or the empty binding site, U is the potential energy of the system.the notion of general position (generic case), result in exact answers in qualitative terms. The backinterpretationis, however, much less rigorous. We will address the issues related to it in the end of the article.The initial problem sought a phenotype a ∈ A that brings minimal energies for the ligand (cid:96) ∅ to that of the lig-ands (cid:96) (cid:52) and (cid:96) r (with the correct ordering, but we will consider this issue separately). The corresponding proteinwould be considered to take a large conformational change upon binding if the corresponding configurations x ∅ and x (cid:52) ≈ x r are very di ff erent. In the simplified problem, the initial discrimination problem reduces to finding a such that the global minima of U in X corresponding to (cid:96) and (cid:96) have the same energy level. We will call this a reduced discrimination problem . Then the initial question is whether or not the corresponding minimum points x and x coincide in X .We will also consider an infinitesimal discrimination problem , where (cid:96) r ≈ (cid:96) (cid:52) and we replace the di ff erencebetween (cid:96) (cid:52) and (cid:96) r by a vector (cid:51) in L at (cid:96) that shows the direction from (cid:96) r to (cid:96) (cid:52) . We will assume that a given a , which solves the reduced discrimination problem, also solves the infinitesimal discrimination problem if thedisplacement along (cid:51) in L at (cid:96) corresponds to a positive change in the energy value at the global minimum (seeFigure 1). Main results4.1
Exquisite discrimination, conformational changes, and fine tuning
Let us recall the following notion.
Definition.
A subset of a topological space is called a residual set if it can be represented by a countableintersection of open dense subsets. A typical element of the topological space is an element that belongs to someresidual set. A situation is generic if it can be represented as a typical element of some space relevant to theproblem. A complement to a residual set is called meager set .We must consider that the naturally defined system, given by U , is typical. Indeed, the meaning of residualsets is that their complements, meager sets, can be considered as negligible and points that belong to them asspecial. An assumption that naturally occurring systems do not belong to some negligible sets is a kind of anextension of the Copernican principle. It is in this sense U is typical.The first main result now can be formulated in the following theorem. Theorem 1.
For a typical family of potentials U ∈ C ∞ ( M ) , solutions to the reduced discrimination problem forligands (cid:96) and (cid:96) either do not exist, or a typical solution is located on a (dim A − -dimensional submanifold ˆ Υ of A and, if dim X > , its minimum points x (for (cid:96) ) and x (for (cid:96) ) are di ff erent. It should be noted that this mathematical result is intuitively expected from the beginning. Indeed, one for-mally has to find points of minimum for U (cid:96) , a and U (cid:96) , a . Let them be ˆ x ( a ) and ˆ x ( a ), respectively, where the de-pendence on a is explicitly indicated. The solution is given by traits a such that U ( ˆ x ( a ) , (cid:96) , a ) = U ( ˆ x ( a ) , (cid:96) , a ).This constitutes one condition on a , which is intuitively expected to be satisfied on a codimension-1 hypersurfaceof A . In the same way, additional constraint of no conformational change is written in the form ˆ x ( a ) = ˆ x ( a ) andis equivalent to dim X additional conditions. One would intuitively expect that the set of solution to discrimina-tion without conformational changes occupies a submanifold of codimension dim X +
1. However, the intuition5lone is not suitable to treat multidimensional problems. In particular, the condition on a is not a simple equationbut depends on solutions ˆ x and ˆ x of the energy minimization problem. These solutions themselves depend on a in a complex manner, which may involve discontinuities of rearrangements. The purpose of Theorem 1 andthe following Theorem 2 is to justify the intuitive conclusion and to clarify in which sense it is true.The evolutionary solutions delivered by Theorem 1 only guarantee that, after going back from the coarsegraining picture, the minimal energies for (cid:96) ∅ , (cid:96) r , and (cid:96) (cid:52) will be close. However, for a discriminating proteinto work correctly, it is important to have the right order of these energies: U r < U ∅ < U (cid:52) . Therefore, we willconsider the infinitesimal discrimination problem that probes the validity of this constraint by infinitesimallysmall deformation of solutions for the reduced discrimination problem at (cid:96) .More specifically, consider a nonzero vector (cid:51) on L emanating from (cid:96) ( (cid:51) ∈ T (cid:96) L ). This vector can beregarded as showing the direction from (cid:96) r to (cid:96) (cid:52) . These points are considered to be infinitesimally close to (cid:96) , and (cid:96) has the same energy minimum as (cid:96) . Therefore, for the phenotype a to be a solution to the full discriminationproblem, the displacement along (cid:51) with the fixed a must increase the minimal energy. The boundary betweensolutions that respect this requirement and those that do not is made of a such that there is no change in theminimal energy level in the direction spanned by (cid:51) .The second main result concerns this additional infinitesimal constraint on the order of the minima and isformulated as the following theorem. Theorem 2.
For a typical family of potentials U, solutions to the infinitesimal discrimination problem for ligands (cid:96) and (cid:96) and for a separating vector (cid:51) either do not exist, or a typical solution is located on a (dim A − -dimensional submanifold ˜ Υ of A and, if dim X > , its minimum points x and x are di ff erent. In other words, the additional requirement of a correct order in energy minima does not qualitatively changethe situation. It can make the set ˆ Υ smaller, though ( ˜ Υ ⊂ ˆ Υ in general). Conformational changes and allosteric regulation
Let us now look at how the development of a conformational change as a byproduct of a solution to the exquisitediscrimination problem can help a development of allosteric regulation. By allosteric regulation we will under-stand the disruption of the initial ability to discriminate two ligands by binding of another ligand to a distant siteof the molecule.Let us assume now that the protein in question can bind two di ff erent ligands: λ and ρ . Therefore, theenvironmental variable takes the form (cid:96) = ( λ , ρ ) and it belongs to the space L = Λ × P, λ ∈ Λ , ρ ∈ P. Let usdenote, as before, the situation when λ is bound by λ and when it is not bound by λ . Likewise, we have ρ and ρ for the bound and free state of the ligand ρ . Note, that we assume, as before, that λ in fact representstwo ligands: λ r and λ (cid:52) . The protein discriminates these ligands. In contrast, ρ is assumed to be a single ligand,which is bound by the protein without discrimination. Based on the theorems of the previous section we canexpect a conformational change of the protein upon binding of λ , upon binding of ρ , upond binding of λ , when ρ is already bound, and vice versa .Let us now assume in addition that the binding of λ and ρ is localized on the molecule in question and thatit happens at di ff erent sites. Let us also assume that the sites are not to directly coupled. This can be expressedin the following way. Let x be the degrees of freedom involved in the interaction with the ligand λ (coordinatesof atoms interacting with λ , for example), x be the degrees of freedom involved in binding ρ , and x be theresidual degrees of freedom. We assume thus that X = X × X × X with x ∈ X , x ∈ X , and x ∈ X . Thenthe potential decomposes in this case as U ( x , (cid:96), a ) = U ( x , x , x , a ) + U ( x , λ , a ) + U ( x , ρ , a ) . (5)Let a be a solution to the reduced exquisite discrimination problem (the reasoning is analogous for theinfinitesimal problem) for (cid:96) = ( λ , ρ ) and (cid:96) = ( λ , ρ ), and define (cid:96) = ( λ , ρ ) with (cid:96) = ( λ , ρ ). Then thefollowing result holds. Theorem 3.
Suppose that the protein changes its conformation during the switch from (cid:96) to (cid:96) (upon bindingof λ on the background of bound ρ ). Then the situation described by min U (cid:96) , a = min U (cid:96) , a and min U (cid:96) , a = min U (cid:96) , a (6)6 (a) U r < U ∅ U ∅ < U (cid:52) ˆ Υ r ˆ Υ (cid:52) solutions • U r < U ∅ ˆ Υ r solutions • A (b) Figure 2: Solutions to the zero-temperature exquisite discrimination problem with finite di ff erence between (cid:96) r and (cid:96) (cid:52) (a) and to the simple binding problem for a single ligand (cid:96) r (b) and their sensitivity to mutations (see thetext). A bundle of arrows illustrates mutations away from a given solution. is not structurally stable in the sense that it can be turned by an arbitrarily small perturbation of U into situationdescribed by min U (cid:96) , a = min U (cid:96) , a but min U (cid:96) , a (cid:44) min U (cid:96) , a . (7) In the contrary, situation (7) is structurally stable in the sense that for any small enough perturbation of U itcannot be turned into situation (6).
In other words, situation (7) means that when ρ is not bound, the protein performs the exquisite discrimi-nation for λ , while when ρ is bound, this ability is broken. Therefore, ρ acts as an allosteric regulator for theexquisite discrimination of λ . The theorem asserts that such behaviour is typical. The condition of the theoremsubstantiantly uses the genericity of the conformational change upon binding provided by Theorem 1. Discussion and biological interpretation
Theorems 1–3 provide rigorous results in the limit of indistinguishable ligands ( (cid:96) (cid:52) → (cid:96) r ) and for the zero tem-perature approximation. In real systems, the di ff erence between the right and the wrong ligands is finite, freeenergies of di ff erent states are allowed to be di ff erent provided that the correct order is preserved, and the temper-ature is positive. Going back from the mathematical idealization adopted above to physically meaningful modelswith finite di ff erences and nonzero temperature blurs the rigor of the statements. An exclusion from a genericsituation must be understood as not something impossible for practical observation but rather as something lessprobable than the generic case. The more the di ff erence and the temperature the less strong the statement. Thiscan be graphically demonstrated for the case of a nonzero di ff erence between (cid:96) r and (cid:96) (cid:52) (and between the energylevels U r , U (cid:52) , and U ∅ ) still assuming zero temperature. A solution to the exquisite discrimination problem in thiscase corresponds to a phenotype a such that (3) holds. If we denote (cid:96) = (cid:96) ∅ and (cid:96) = (cid:96) r , the corresponding set ˆ Υ provided by Theorem 1 (we will denote ˆ Υ r ) defines the border of phenotypes that respect U r < U ∅ . In the sameway, the analogous set ˆ Υ (cid:52) defined for (cid:96) = (cid:96) ∅ and (cid:96) = (cid:96) (cid:52) marks the border of phenotypes that respect U ∅ < U (cid:52) .When (cid:96) r becomes close to (cid:96) (cid:52) , ˆ Υ r becomes close to ˆ Υ (cid:52) . From this it is clear that for (cid:96) r (cid:44) (cid:96) (cid:52) , phenotypes thatsolve the exquisite discrimination problem are situated between ˆ Υ r and ˆ Υ (cid:52) . This is schematically shown by theshaded region on Figure 2. In fact, this regions is a “thick” version of the codimension-1 submanifold ˜ Υ givenby Theorem 2 with an appropriately chosen direction (cid:51) . Indeed, when (cid:96) r approaches (cid:96) (cid:52) by some trajectory, theshaded region collapses to the submanifold that corresponds to (cid:51) such that (cid:51) is tangent to the trajectory. We seethat a nonzero di ff erence between the ligands makes the possible evolutionary solutions to their discriminationproblem to occupy a spatial domain in the trait space A rather than its infinitely thin codimension-1 submanifold.Yet, with su ffi ciently similar ligands (which is supposed by the exquisite discrimination problem) they stay nearsuch manifold.Another notion that is blurred in real systems is that of a large conformational change. In the idealized coarsegrained mathematical model, any conformational change was interpreted as being large. The proven theoremsdo not provide any means to determine how large the conformational change is or what large means in general.Such problems are typical for topological but not metric theorems. Addition of real physics on top of the bare7opology in this problem (such as the limits on the stifness of the chemical bonds, assumption of a nonzerotemperature, the value of the mutational e ff ects of individual aminoacids, and so on) might help to destinguishbetween essential and nonessential changes in the discrimination ability of a protein and in its conformation.Although the first part of the main result (Theorems 1 and 2) can be shortly stated as discrimination requiresa conformational change , the statement would not be entirely correct. First, the requirement must not be un-derstood as direct causality. The correct interpretation is that most solutions to the discrimination problem willinvolve a conformational change. It implies that if a system performs discrimination and changes its conforma-tion, it should not be surprising and no special explanation is required to this fact. In contrary, if a discriminatingsystem does not show a conformational change, it is an indication on a special additional circumstances that maybe of interest. Second, an application of the same theoretical approach to a protein that just binds to a ligandbut not necessarily discriminates between similar ligands results in a conclusion that a conformational change isexpected to accompany any binding in general.Let us elaborate the latter statement. The part of the reasoning (in a simplified form) in the proof of Theo-rem 1 that involves a conformational change still holds in the case of a simple binding without discrimination.This means that we should expect a conformational change upon binding in general , not only when a discrim-ination is performed. This general statement is very close to the classical induced fit scenario. The competingconformational selection hypothesis in its strict form, instead, represents a very special case (very special formof energy landscape), as it requires the conformation of the global minimum of free energy for the unbound stateto be also a conformation of a local minimum for the bound state and vice versa . This situation is not typical forsmooth potentials. However, if the relevant conformations are themselves allowed to change upon binding, thenthe situation becomes as typical as the pure induced fit situation. In fact, the distinction between these two casesbecomes irrelevant, as was already demonstrated in [7] on a simple model. A similar conclusion was formulatedin [1] based on biochemical arguments and experimental observations, and in a new vision of the protein bindingproposed in [8].What discrimination does require is a an evolutionary fine tuning expressed in the dimension of the set ofpossible solutions. It is this fine tuning that brings about the potential for allosteric regulation (in the samesense as a discrimination causes a conformational change). If we combine the conclusion of Theorem 3 withthe above understanding of how such rigorous statements should be interpreted in application to real systems,we may conclude the following. Proteins that discriminate ligands are prone to allosteric regulation by anotherligand at a di ff erent binding site. This sensitivity of the discrimination to the distant binding is associated withthe conformational change during the primary binding. Although this question was not studied in this work, wemay also expect a wide sensitivity of such protein to mutations. Indeed, a mutation of an aminoacid is in somesense analogous to a local binding in its e ff ect on the potential energy. Repeating for this case the reasoningabout allosteric e ff ects, we conclude that the ability to discriminate is broken by mutations in many sites. Onecan justify this assertion from a di ff erent perspective. Since the exquisite discrimination requires an evolutionaryfine tuning, we can expect a mutation to break this tuning in a generic case. This is graphically represented onFigure 2. As a consequence, we expect a wide (in the spreading on the level of the primary sequence) mutationale ff ect, when many mutations, however distant from the binding site, destroy the discrimination.Note that the ability to bind a ligand without an imposed discrimination problem is generically robust tomost mutations. Indeed, the solutions to the binding problem for a single ligand lay in a half space of A to oneside of ˆ Υ given by Theorem 1 for that ligand and the solvent. If a solution is situated deep in this region, it isexpected to survive mutations in the sense that the resulting protein retains the binding ability (perhaps, with aweaker a ffi nity, see Figure 2).The modelling approach taken in this work is in the family of folding landscape models [9]. Looking at aprotein through its (free) energy landscape is very natural from the point of view of physics and deserves moreattention. The fact that such model supports the conclusion of [7] is very important. It shows that an emergenceof sophisticated properties of proteins and other biological heteropolymers, upon which substantial part of thecomplexity of life is built, can be attributed to a very simple evolutionary process: selection for a local property,that is the ability to discriminate between similar ligands. It is not di ffi cult to imagine a selection process thatoptimizes this task. Furthermore, such ability very probably was required even back at the earliest times ofabiogenesis or very early life. 8 Proofs of Theorem 1, Theorem 2, and Theorem 3
We imply in the following that all manifolds and functions (maps) are smooth. The main tool of the proof is thejet-bundle and the multijet transversality theorem that is a consequence of the Thom’s transversality theorem.We will first recall some definitions and fix some notations.
Definition.
Let M and N be two smooth manifolds and S ⊂ N be a submanifold. Let p be a point in M . Asmooth function f : M → N is said to be transverse to S at p , if d f p T p M + T f ( p ) S = T f ( p ) N , where T p M meansthe tangent space to M at p and d f p is the di ff erential of f at p . f is said to be transverse to S , if it is transverseto S at each point of M . This situation will be denoted by f (cid:116) S . Let P ⊂ N be another submanifold. S and P are said to intersect transversely (or simply to be transverse ), if T q S + T q P = T q N for each q ∈ S ∩ P . Thissituation will be denoted by S (cid:116) P . Definition.
The codimension of a submanifold N of a manifold M is the number codim N = dim M − dim N .If S and N are submanifolds of the same manifold and N (cid:116) S , then codim N ∩ S = codim N + codim S (assuming N ∩ S (cid:44) ∅ ). If this number is negative, then N ∩ S = ∅ . Definition.
Let M be a smooth manifold. Two smooth functions f and g from M to (cid:146) are said to have k-th ordercontact at p ∈ M , if in some coordinate chart around p their values and all their partial derivatives up to order k are equal at p . The relation of k -th order contact is independent of the coordinate chart and defines equivalentclasses. The equivalent class of function f by k -th order contact at p , denoted [ f ] kp , is called k-jet of f at p . Let J k ( M , (cid:146) ) p be the set of all k -jets at p . The bundle of k-jets of functions on M is the set J k ( M , (cid:146) ) = (cid:96) p ∈ M J k ( M , (cid:146) ) p with the projection π k : J k ( M , (cid:146) ) → M , [ f ] kp (cid:55)→ p endowed with the di ff erential structure lifted from M by π k .Every function f : M → (cid:146) generates a special section of the k-jet bundle j k f : p (cid:55)→ [ f ] kp .Note that jet bundles can be generalized to maps between arbitrary manifolds. Essentially, k -jets of functionsrepresent an invariant notion of their Taylor polynomials truncated to order k . In the special case k =
1, theonly one we will be interested in the following, the 1-jet bundle J ( M , (cid:146) ) is naturally isomorphic to the product (cid:146) × T ∗ M (we denote this by J ( M , (cid:146) ) (cid:39) (cid:146) × T ∗ M ), where T ∗ M is the cotangent bundle of M . Definition. An s-fold multijet bundle J ks ( M , (cid:146) ) is defined as follows. We denote M ( s ) = { x ∈ M s : ∀ i , j , (cid:54) i < j (cid:54) s ⇒ x i (cid:44) x j } . (8)It is a submanifold of M s . Let π k be the bundle projection J k ( M , (cid:146) ) → M . Then J ks ( M , (cid:146) ) = ( π × sk ) − ( M ( s ) ). It isa submanifold of J k ( M , (cid:146) ) s and is a fibre bundle over M ( s ) . Every function f on M generates its special section j ks f by the rule j ks f ( x ) = ( j k f ( x ) , . . . , j k f ( x s )).Let us denote the diagonal of the direct product M as ∆ M . In the special case s =
2, the only one we willbe interested in the following, M (2) has a simple representation: M (2) = M \ ∆ M . Definition.
Map f : X → Y is regular at p ∈ X , if the rank of d f p is maximal. If f is not regular at p , it is called singular at p and p is called its critical point . Map f : X → Y is an immersion , if d f p is injective at every p ∈ X .We also need some known theorems. Theorem 4 ([10], page 52, Theorem 4.4) . Let X and Y be manifolds, W ⊂ Y be a submanifold, and f : X → Ybe a function and let f (cid:116)
W. Then f − ( W ) is a submanifold of X. If in addition f ( X ) ∩ W (cid:44) ∅ , then codim f − ( W ) = codim W. Theorem 5 (A special case of Mather’s multijet transversality theorem, [10], page 57, Theorem 4.13) . Let Xbe a manifold and W be a submanifold of J ks ( X , (cid:146) ) . The subset of C ∞ ( X ) constructed of functions f that verifyj ks f (cid:116) W is a residual set of C ∞ ( X ) in the Whitney C ∞ topology (for the definition, see [10, p. 42]). Moreover,if W is compact, this subset is open. This theorem is called Thom’s transversality theorem for s = and thusJ ks ( M , (cid:146) ) = J k ( M , (cid:146) ) , j ks f = j k f . We will first prove some lemmas. 9 emma 1.
Let X be a compact manifold and Y be a manifold, let π : X × Y → Y be the projection to the secondfactor. Then for a typical function f : X × Y → (cid:146) , the set of critical points of f | π − ( y ) is finite for any y ∈ Y.Proof.
It is known that the subspace of functions that have only isolated points is of so called infinite codimension (see [11] and [12] for definition and explanation and [11] and [13] for the proof), a notion that is stronger thanbeing typical (the latter implies the former). This property implies that for any k , the set of k -parameter familiesof functions with only isolated points is residual in the set of all k -parameter families. Function f on X × Y is a(dim Y )-parameter family of functions on X (cid:39) π − ( y ). Thus, for any y , f | π − ( y ) has only isolated critical points.The finiteness of the number of critical points on every layer follows from the compactness of X . (cid:3) Lemma 2.
Let X and Y be manifolds and S ⊂ X × Y be a compact submanifold, let π : X × Y → Y be theprojection to the second factor, let dim S = dim Y − . Suppose that for each y ∈ Y, π − ( y ) ∩ S is finite. Then π | S is regular at a typical point of S .Proof. The conclusion of the lemma is trivially true for dim Y = S = Y > M (dim M = n ), a smooth association of a point p ∈ M with a k -dimensionalsubspace in T p M is called a k -dimensional distribution on M [14, §
3] (not to be confused with probabilitydistributions). In other words, a distribution associates a tangent hyperplane with each point of a manifold. Adistribution can be viewed as a subbundle of the tangent bundle. Another way to define a distribution is bydefining a collection of (at least k ) vector fields V i on M that span the corresponding subspace of the distributionat each point. Finally, the same distribution can be defined by (at least n − k ) di ff erential 1-forms ω j that annulate V i : ω j ( V i ) = i and j . Any distribution can be defined in such way at least locally (in a neighbourhoodof each point of M ).Now, the regularity of π | S at point p ∈ S means that T p S ∩ T p π − ( π ( p )) =
0. The association p (cid:55)→ T p π − ( π ( p ))defines a distribution D on X × Y , which is vertical towards the projection π (it is mapped to the trivial distribu-tion on Y that associates 0 ∈ T y Y with each point of Y ) and has its layers as integral manifolds (the layers aretangent to the distribution at each point). In a local chart around p ∈ X × Y with coordinates ( x µ , y ν ), the layers ofthe bundle π are defined by the conditions y ν = const, and thus the distribution is defined by 1-forms α ν = dy ν ,where d is the exterior derivative.The distribution D on X × Y induces a distribution D S on S in the following way. Let ι : S → X × Y bethe inclusion of S . Then the collection { ω ν } , ω ν = ι ∗ α ν , defines a distribution on S , where ι ∗ is the pullback ofdi ff erential forms induced by ι . The dimension of D S , however, can change from point to point depending onthe degeneracy of { ω ν } .Choose a coordinate neighbourhood O around s ∈ S in S with local coordinates s λ . Then locally we have ω ν = (cid:80) λ ω νλ ds λ , where ω νλ ∈ C ∞ ( S ). In these terms, the regularity of π | S at s means that the rank of matrix ω νλ is maximal at s (rank ω νλ ( s ) = dim S , as dim S < dim Y ).Consider the sets Ω = { σ ∈ O : ∀ ν , ω ν = } = { σ ∈ O : rank ω νλ (cid:54) } , Ω = { σ ∈ O : ∀ ν , ν , ω ν ∧ ω ν = } = { σ ∈ O : rank ω νλ (cid:54) } ,. . . Ω k = { σ ∈ O : ∀ ν , . . . , ν k , ω ν ∧ . . . ∧ ω ν k = } = { σ ∈ O : rank ω νλ (cid:54) k } , (9) . . . where ∧ is the exterior product of di ff erential forms. Note that for each k (cid:54) l , Ω k ⊂ Ω l . Note also that trivially Ω k = O for all k (cid:62) dim S . Consider also sets Θ k , where Θ = Ω and Θ k = Ω k \ Ω k − for k >
0. These setsdefine the points where rank of ω νλ is equal to k .Let us prove that Ω k are closed and nowhere dense in O for k < dim S , and thus Θ dim S is open and densein O . The closeness of all Ω k follows from the fact that the defining equations in (9) are equivalent to a finiteset { F m = } of functional equations on ω νλ , where F m are homogeneous polynomials of ω νλ of k -th order withcoe ffi cients from {− , } . Indeed, the equations in (9) reflect nothing else but setting to zero all k -th minors of ω νλ . As F m are smooth, the set Ω k = { σ ∈ O : F m = } is closed.10ow suppose that ˚ Ω (cid:44) ∅ , where ˚ A is the interior of A . Then D S has constant dimension dim S in ˚ Ω , andthe set of equations ω ν ( V ) = S linearly independent solutions. Choose one such V , a point σ ∈ ˚ Ω ,where V σ (cid:44)
0, a neighbourhood O σ of σ , where V (cid:44) V ),the integral curve γ of this vector field in O σ that passes through σ , any σ ∈ γ di ff erent from σ , and denote ˜ γ ⊂ γ the interval of γ that connects σ and σ . By necessity, for an arbitrary such σ we have ∀ ν (cid:82) ˜ γ ω ν = , and thus (cid:82) ι (˜ γ ) dy ν = y ν (cid:0) ι ( σ ) (cid:1) − y ν (cid:0) ι ( σ ) (cid:1) = . (10)The equality y ν (cid:0) ι ( σ ) (cid:1) = y ν (cid:0) ι ( σ ) (cid:1) for all ν and σ means that ι ( γ ) ⊂ π − ( π ( σ )) and thus S ∩ π − ( π ( σ )) is uncount-able. This contradicts the premise, therefore ˚ Ω = ∅ , which means that Ω = Θ is nowhere dense.Repeat this reasoning in the inductive manner for Θ k , 0 < k < dim S . The only di ff erence at each stepis the number of independent vector fields that solve ω ν ( V ) =
0, which is equal to dim S − k . In the end ofeach step ˚ Θ k = ∅ (the proved expression) and ˚ Θ k − = ∅ (the expression from the previous step) together imply˚ Ω k = ∅ . The induction chain breaks at Θ dim S , since in this case the aforementioned equations have no nontrivialsolutions, and thus D S is 0-dimensional in Θ dim S .Now select a chart around every point of S and then subselect a finite covering from this collection (whichis possible by the compactness of S ). Repeat the reasoning for all of them to get the conclusion of the lemma. (cid:3) Note that the requirements of compactness of manifolds and finiteness of π − ( y ) ∩ S are not essential. Itis only essential for π − ( y ) ∩ S to be at most countable. But this level of generality brings about unneededcomplications that are not relevant for the following. Proof of Theorem 1.
Let us call a presolution to the reduced discrimination problem a phenotype a such that theprotein has equal in the energy level critical points of energy for (cid:96) and (cid:96) , and not necessarily minima. It isclear that the proper solutions make a subset of the presolutions.Consider the following diagram, associated with energy functions on M : (cid:146) (cid:39) ∆ (cid:146) ι E (cid:47) (cid:47) (cid:146) L { ( (cid:96) , (cid:96) ) } ι L (cid:111) (cid:111) J ( M , (cid:146) ) π E (cid:101) (cid:101) π T ∗ M (cid:121) (cid:121) π (cid:47) (cid:47) M (2) j U (cid:97) (cid:97) π L (cid:62) (cid:62) π A (cid:47) (cid:47) π X (cid:32) (cid:32) A ∆ A (cid:39) A ι A (cid:111) (cid:111) P X ι PX (cid:47) (cid:47) ( T ∗ M ) X ∆ X (cid:39) X ι X (cid:111) (cid:111) (11)Here π is the projection of J ( M , (cid:146) ) as a bundle over M (2) . π X is ( p X × p X ) | M (2) , where p X is the natural projectionof M on X , analogously for π L and π A . π E is the projection to pairs of energy values, associated with a multijet(it can be seen as ( p (cid:146) × p (cid:146) ) | J ( M , (cid:146) ) , where p (cid:146) is the projection to the first factor of (cid:146) × T ∗ M (cid:39) J ( M , (cid:146) )). ι X , ι L , ι A , and ι E are the obvious natural inclusions (embeddings).Finally, P X and π T ∗ M are defined as follows. Let us consider again J ( M , (cid:146) ) as (cid:146) × T ∗ M and let p T ∗ M be thenatural projection on the second factor. In turn, T ∗ M (cid:39) T ∗ X ⊕ T ∗ ( L × A ) (12)where V ⊕ W is the Whitney sum of two vector bundles π V : V → B , π W : W → B over the same base B , i. e.the pullback from the following commutative diagram ( ι ∆ is the diagonal inclusion map) V ⊕ W (cid:47) (cid:47) (cid:15) (cid:15) V × W π V × π W (cid:15) (cid:15) B ι ∆ (cid:47) (cid:47) B × B (13)11et 0 X be the image of the 0-th section of T ∗ X in T ∗ X . Then we define P X = X ⊕ T ∗ ( L × A ) , π T ∗ M = ( p T ∗ M × p T ∗ M ) | J ( M , (cid:146) ) , (14)and ι P X as the natural inclusion P X → ( T ∗ M ) Let us define W = ( π E ) − (Im ι E ) ∩ ( π T ∗ M ) − (Im ι P X ) ∩ ( π L ◦ π ) − (Im ι L ) ∩ ( π A ◦ π ) − (Im ι A ) ⊂ J ( M , (cid:146) ) , W = W ∩ ( π X ◦ π ) − (Im ι X ) ⊂ J ( M , (cid:146) ) , Y = Im j U ∩ W ⊂ J ( M , (cid:146) ) , Y = Im j U ∩ W ⊂ J ( M , (cid:146) ) , V = π ( Y ) ⊂ M (2) , (15) V = π ( Y ) ⊂ M (2) , Υ = ι − A ( π A ( V )) ⊂ A , Υ = ι − A ( π A ( V )) ⊂ A . The meaning of these sets is the following. Space J ( M , (cid:146) ) consists of pairs of jets over two at least somehowdistinct points of M . The preimage of ∆ (cid:146) defines pairs of jets that have the same energy values. The preimageof P X defines pairs of jets both of which have zero partial derivatives in X (so, the corresponding points in X arecritical points for any representatives of these jets). The preimage of ( (cid:96) , (cid:96) ) defines pairs of jets one of which isover (cid:96) and the other one is over (cid:96) . The preimages of ∆ A and ∆ X define pairs of jets that have the same valuesof a and x , correspondingly.Therefore, V corresponds to pairs of tuples ( x , (cid:96) , a ) and ( x , (cid:96) , a ) ( (cid:96) i are fixed to the values of the prob-lem) such that functions U ( · , (cid:96) i , a ) have corresponding x i as critical points and U ( x , (cid:96) , a ) = U ( x , (cid:96) , a ). V corresponds to the same tuples but with the additional constraint x = x . Accordingly, Υ corresponds to evo-lutionary presolutions of the reduced discrimination problem, while Υ corresponds to such presolutions wherethe critical points coincide.By Theorem 5, for a typical U (from a residual set of all U ∈ C ∞ ( M )), we have both j U (cid:116) W and j U (cid:116) W . Indeed, the theorem guaranties that each of the condition is verified on a residual set. Therefore,they both are verified on a residual set, as an intersection of two residual sets is residual.Sets Y i are compact submanifolds of J ( M , (cid:146) ). Indeed, consider W i and Y i as subsets of J ( M , (cid:146) ) ⊃ J ( M , (cid:146) ).If on the diagram above we replace J ( M , (cid:146) ) by J ( M , (cid:146) ) , j U by j U × j U , π by its nonrestricted version,all other projections of the form π • by nonrestricted p • × p • , and then repeat the construction of Y i in the sameway, they will coincide with Y i constructed in the old way. Indeed, the only di ff erence could be some addi-tional points on the preimage of the diagonal π − ( ∆ M ), but since ( p L × p L ) − (Im ι L ) ∩ ∆ M = ∅ , we have Y i ∩ π − ( ∆ M ) = ∅ , hence the equality. As j U × j U ( M ) is compact due to the compactness of M , Y i arecompact, too. This property conserves upon restriction to J ( M , (cid:146) ).From the transversality properties and from π ◦ j U = Id M , by Theorem 4, V i are submanifolds of M (2) .These submanifolds are compact, too.Note that codim( π E ) − (Im ι E ) = , codim( π T ∗ M ) − (Im ι P X ) = X , codim( π L ◦ π ) − (Im ι L ) = L , codim( π A ◦ π ) − (Im ι A ) = dim A , (16)codim( π X ◦ π ) − (Im ι X ) = dim X , dim M (2) = X + dim L + dim A ) . Therefore, by Theorem 4 and assuming Y i (cid:44) ∅ ,codim V = codim W = + X + L + dim A , codim V = codim W = + X + L + dim A , (17)dim V = dim A − , dim V = dim A − dim X − . We already see that if dim X > V > dim V . As V ⊂ V , wecan conclude that points of V that correspond to coincident critical values are not typical. More specifically,12hey form a submanifold of codimension dim X . For instance, if dim A < dim X +
1, such points do not exist atall. Otherwise, if dim X > V is a negligible subset in V in the sense that it is nowhere dense and closed.Note that by construction, V i ⊂ π − A (Im ι A ) ⊂ M (2) and Υ i can be understood as projections of V i on A from M (2) , for example as Υ i = π ( V i ), where π = p ◦ π A : M (2) → A and p : A × A → A is the projection onthe first factor. Unfortunately Υ and Υ are not in general submanifolds of A due to generic singularities ofthe corresponding projection and generic self-crossings of the images of Υ i . We do not expect them to be evenimmersed manifolds. In fact, in a typical case, they form so called stratified sets of A . We will show, however,that typical points of Υ do form (dim A − M → L × A , where the projection is the natural projection of M to the correspondingfactor. Then U can be viewed as a family of potentials U (cid:96), a on X parametrized by (cid:96) and a . By Lemma 1, fortypical U , all U (cid:96), a have only finite number of critical points for each pair ( (cid:96), a ). Using the diagram P X (cid:47) (cid:47) T ∗ M J ( M , (cid:146) ) (cid:47) (cid:47) (cid:111) (cid:111) M j U (cid:94) (cid:94) (cid:47) (cid:47) L { (cid:96) i } (cid:111) (cid:111) (18)and the same reasoning as in the beginning of the proof, we conclude that for a typical U , the sets V (cid:96) i of all criticalpoints in X of U (cid:96) i at di ff erent a (we will call V (cid:96) i the critical set of U (cid:96) i ) constitute manifolds in X × { (cid:96) i } × A ⊂ M and can be regarded as subsets of X × A (cid:39) X × { (cid:96) i } × A .Consider a fibre bundle with the natural projection ξ A : X × A → A with two functions U (cid:96) , U (cid:96) on X × A thatcorrespond to U at di ff erent values of (cid:96) . They too have finite number of critical point over every a ∈ A , sincethese functions are restrictions of U . If we consider the direct product of these fibre bundles with projection ξ A × ξ A : ( X × A ) → A , function U (cid:96) × U (cid:96) has V (cid:96) × V (cid:96) as its critical set. It is finite over any ( a , a ) ∈ A ,too. Therefore, V regarded as a submanifold of X × A (cid:39) X × ∆ A , is a submanifold of V (cid:96) × V (cid:96) , and thusfinite over any point a ∈ A . In other words, only finite number of points from V are projected to any a ∈ Υ .By Lemma 2, a typical point of V (from an open dense subset) is projected regularly. Therefore, π A | V is alocal immersion in a neighbourhood of a typical point with finite preimage. Due to compactness of V , the setof points of change of the number of preimages and the intersection locus of the immersion in regular pointsare closed nowhere dense sets of Υ . Therefore, typical points of Υ form open submanifold of A of dimensiondim A − V projects to this submanifold as a (dim A − dim X − Υ ), we are left with an open (dim A − Υ ofthe phenotype space A , which is dense in Υ .Finally, let us return to the proper solution of the reduced discrimination problem, that is, when we consideronly the parts of V that correspond to the global minima and only the corresponding parts of Υ . This willreduce Υ to a smaller subset ˆ Υ , but all the conclusions will hold for it, too. Typical points of ˆ Υ form an open(dim A − A and the corresponding minimum points in X over (cid:96) and (cid:96) do notcoincide.The only possible complication can come from situations when at least at one of (cid:96) i , U (cid:96) i , a has multiple globalminima. Each part of V that is projected to the corresponding a guarantees only that each pair of a minimumpoint at (cid:96) and a minimum point at (cid:96) do not coincide, but it does not preclude a situation, when to the samepoint a , for example, two parts of V are projected that correspond to pairs of minimum points at (cid:96) and (cid:96) ofthe form ( x , x ) and ( x , x ). In this case we would have two coinciding global minimum points for (cid:96) and (cid:96) .However, as two members of the same pair correspond to the same value of the global minima, we have in thiscase U ( x , (cid:96) , a ) = U ( x , (cid:96) , a ) and U ( x , (cid:96) , a ) = U ( x , (cid:96) , a ) . (19)But all the minima must have the same value of energy, as they are global minima. Therefore, we must have U ( x , (cid:96) , a ) = U ( x , (cid:96) , a ) , and thus, U ( x , (cid:96) , a ) = U ( x , (cid:96) , a ) . (20)It follows that such a belongs to Υ and is not in the described submanifold of typical solutions to the reduceddiscrimination problem. This concludes the proof. (cid:3) roof of Theorem 2. Consider the following diagram P X (cid:47) (cid:47) T ∗ M J ( M , (cid:146) ) π , (cid:47) (cid:47) (cid:111) (cid:111) J ( M , (cid:146) ) π (cid:47) (cid:47) M j U (cid:104) (cid:104) j U (cid:127) (cid:127) (21)where π , is the natural projection [ f ] p (cid:55)→ [ f ] p (it forgets about tangency and keeps information only aboutintersections of function graphs) and P X is as in the proof of Theorem 1. Consider the manifold (for generic U ) Y ⊂ J ( M , (cid:146) ) defined by the intersection of the preimage of P X and the image of j U . Consider also, as before,the set V of critical points of U , which is a manifold in M , V = ( j U ) − ( Y ). Consider now the projection V of Y to J ( M , (cid:146) ). It is a manifold of J ( M , (cid:146) ). Indeed, it is just j U ( V ) and Im j U is an embedding of M (it isjust the graph of U ).Note that J ( M , (cid:146) ) (cid:39) (cid:146) × M . As before, we can assume that for every (cid:96) and a , U (cid:96), a has only finitely manycritical points. Therefore, V projected to (cid:146) × L × A by the natural projection from (cid:146) × L × A locally over atypical point ( (cid:96), a ) looks like a finite set of sections (graphs) of the bundle (cid:146) × L × A → L × A . Nontypical pointsof L × A constitute a codimension-1 subset, which consists of either intersections of these manifolds or of thesingularities of their projections. The set W that corresponds to global minima of U is a subset of this union of(dim L + dim A )-dimensional manifolds that locally in a typical point looks like a single such manifold that isregularly projected to L × A .For a typical U and (cid:96) , U ( · , (cid:96), · ) : X × A → (cid:146) is a typical family of functions on X parametrized by A . Thus,for a typical a , U (cid:96) , a is Morse function. It has only separate nondegenerate critical points that are all mapped todi ff erent values. A codimension-1 bifurcations of a typical family (bifurcations that happen on a codimension-1subset of A ) include only fold bifurcations and equality of the function value for some two critical points. Allother bifurcations have codimension greater than 2 and thus those of them that happen to be in ˆ Υ form a meagerset of ˆ Υ (due to compactness of A , the complement to this set is open and dense in A ). Therefore, we can considerthat typical points of ˆ Υ that possess the properties stated in Theorem 1 do not include these higher codimensionbifurcations. Furthermore, possible codimension-1 bifurcations of the global minimum (and maximum) excludefold bifurcations. Indeed, during a fold bifurcation, a critical point disappears in a collision with a nondegeneratecritical point with di ff erent Morse index but for the global minimum it is impossible unless a third point becomesthe new global minimum at the same time (this makes the bifurcation of codimension at least 2). Therefore, theonly codimension-1 bifurcations of the global minimum are switches of the minimal points (coincidence ofminimal values).Let ˜ A be the open and dense subset of A formed by the complement to the set of bifurcations of codimensiongreater than 2 for global minima of the family U (cid:96) , a . Let us consider the intersection W (cid:96) of the submanifold (cid:146) × { (cid:96) } × ˜ A (cid:39) (cid:146) × ˜ A of (cid:146) × L × A and W . From the previous paragraph we conclude that in some neighbourhoodof each point of W (cid:96) , W looks either like a graph of some smooth function ϕ : L × A → (cid:146) or like a graphof a continuous function ( (cid:96), a ) (cid:55)→ min( ϕ ( (cid:96), a ) , ϕ ( (cid:96), a )), where ϕ and ϕ are two smooth functions that areequal at the point in question. Let, for the first case, s = j ϕ be the corresponding section of the bundle π : (cid:146) × L × A → L × A , where π is the natural projection. Let s and s be the sections corresponding to ϕ and ϕ of the second case. Let I = { } and I = { , } for the first and the second cases, correspondingly. Let the localcoordinates in (cid:146) × L × A be ( E , (cid:96), a ).Let us locally define,for each ( (cid:96) , a (cid:48) ) ∈ W (cid:96) , a (cid:48) ∈ ˜ A , and a corresponding neighbourhood O ( (cid:96) , a (cid:48) ) that admitsthe aforementioned representation, functions on U a (cid:48) = O ( (cid:96) , a (cid:48) ) ∩ ˜ Af i : a (cid:55)→ dE s i ( (cid:96) , a ) d ( s i ) ( (cid:96) , a ) ( (cid:51) , , i ∈ I , (22)where by ( (cid:51) ,
0) we understand the vector in T (cid:96) L ⊕ T a A (cid:39) T ( (cid:96) , a ) ( L × A ) that corresponds to (cid:51) . These functionsare smooth. Let us also define, for the same point, neighbourhood, and representation, the following piecewisesmooth function f on U a (cid:48) . If the representation corresponds to I = { } , then we define f ( a ) = f ( a ) . (23)14 × X X A ••• O ( ˆ x , a )( ˆ x , a ) g ε ( ˆ x , a )ˆ x , ˆ x Figure 3: A possible deformation of U to recover ˆ x (cid:44) ˆ x (see the text).If the representation corresponds to I = { , } , then we define f ( a ) = f ( a ) , ϕ ( a ) < ϕ ( a ) , f ( a ) , ϕ ( a ) < ϕ ( a ) , min( f ( a ) , f ( a )) , ϕ ( a ) = ϕ ( a ) . (24)It is not di ffi cult to see that, by construction, the values of thus defined functions for any a and any pair itsintersecting neighbourhoods U (cid:48) a and U (cid:48)(cid:48) a (induced from any two O (cid:48) (cid:96) , a and O (cid:48)(cid:48) (cid:96) , a ) agree in U (cid:48) a ∩ U (cid:48)(cid:48) a . Thus, function f is globally defined on ˜ A . Moreover, we can use any its possible representation to study its local behaviour.By construction, the sign of this function defines the sign of di ff erence between U (cid:52) and U r under an infinites-imal separation of (cid:96) to (cid:96) r = (cid:96) and (cid:96) (cid:52) , when the latter moves along (cid:51) . Let us extend f on the whole A by setting f ( a ) = − a ∈ A \ ˜ A . Then the set Ω = { a ∈ A : f ( a ) > } is an open subset of A . Indeed, it means that thesubset K = { a ∈ A : f ( a ) (cid:54) } is closed. To show this, let us choose some convergent sequence { a n } in A suchthat a n ∈ K and a n → a . First note that A \ ˜ A is trivially in K. Let a ∈ ˜ A . Starting from some number, all pointsof a n lay in some of U a with one of the considered representations of f . If I = { } for this representation, by thesmoothness of functions ϕ and f and thus of f in U a , we have f ( a n ) → f ( a ) and f ( a ) (cid:54)
0, thus a ∈ K. If,on the other hand, I = { , } for the selected representation, the situation f , ( a ) > f , ( a ) (cid:54)
0, and thus a is automatically in K, or one of f i ( a ) must be greater than 0 and the other onebe smaller or equal to 0. Let us for concreteness assume f ( a ) (cid:54) < f ( a ). But then, starting from su ffi cientlylarge number, we must have f ( a n ) = f ( a ) and thus f ( a ) = f ( a ) (cid:54)
0, so again a ∈ K. Therefore, K contains allits limit points and thus is closed, from which follows that Ω is open.As an open subset of A , Ω is its (dim A )-dimensional submanifold. Let ˇ Υ be the open (dim A − A made of typical points granted by Theorem 1 ( ˇ Υ is an open dense subset of ˆ Υ ). Then, trivially,ˇ Υ (cid:116) Ω and thus ˜ Υ = ˇ Υ ∩ Ω is either empty or a (dim A − A . By construction, ˜ Υ consists of typical solutions to the infinitesimal discrimination problem with di ff erent minimal points. (cid:3) Proof of Theorem 3.
Let ˆ x i j denote the minimal points of, respectively, U (cid:96) ij , a (for a typical value of a theminimal points are unique and nondegenerate [10, Propositions 6.3 and 6.13]). They can be represented asˆ x i j = ( ˆ x i j , ˆ x i j , ˆ x i j ), ˆ x i jk ∈ X k . Let us also suppose that ˆ x (cid:44) ˆ x . If not, this can be recovered by an arbitrarilysmall perturbation of U as follows. First recall that if W is a manifold, O is any its open subset, C and C areany its closed subsets such that C ∩ C = ∅ , than there are smooth functions ϕ and ψ on W such that ϕ ( p ) > p ∈ O and ϕ ( p ) = p (cid:60) O ; (25) ψ ( p ) = p ∈ C , ψ ( p ) = p ∈ C , and 0 < ψ ( p ) < p (cid:60) C ∪ C . (26)An explicit construction of such functions can be found in [15, Part 2, p. 13].15y the premise, we have ˆ x (cid:44) ˆ x . Consider now a neighbourhood O of point ( ˆ x , a ) in X × A such that( ˆ x , a ) (cid:60) O . Choose an associated with O by (25) function ϕ and a nonzero in O rectifiable vector field (cid:51) tangent to T p X at each point p ∈ X × A (such vector field always exists in a small enough O ). Consider nowthe vector field (cid:52) = ϕ (cid:51) and the phase flow g t generated by (cid:52) (so q ( t ) = g t ( p ) is the solution to the di ff erentialequation dq / dt = (cid:52) q with q (0) = p ). The function U ( ε )0 = U ◦ g ε provides the needed deformation of U (seeFigure 3). The corresponding deformation U ( ε ) = U ( ε )0 + U + U of U tends to U (in the Whitney C ∞ topology)as ε →
0. Note also that U ( ε ) still verifies (6), but for any ε (cid:44)
0, ˆ x (cid:44) ˆ x . From the other hand, if these pointsare di ff erent, this cannot be changed by an arbitrarily small perturbation of U , as the position of nondegeneratecritical points of a function smoothly depends on this perturbation to a certain extent. Therefore, the inequalityof ˆ x and ˆ x is typical for a typical a that solves the reduced discrimination problem for (cid:96) and (cid:96) .Now let (6) hold for U and ˆ x (cid:44) ˆ x . Let us denote for brevity N = X × P × A . Let us also denote p i j = ( ˆ x i j , ρ , a ), p i j ∈ N . Let C and C be two closed subset of N such that p ∈ ˚ C (the interior of C ), X × { ρ } × A ⊂ ˚ C , p ∈ ˚ C , and C ∩ C = ∅ (such subsets always exist as ˆ x and ˆ x are di ff erent). Choose afunction ψ defined by ψ ≡ C , ψ ≡ C , as in (26). Consider a deformation U ( ε )2 of U in the followingform U ( ε )2 = U + ε ψ and the corresponding deformation of U given by U ( ε ) = U + U + U ( ε )2 . For any number ε it has the same structure as in (5) and U ( ε ) → U (in the Whitney C ∞ topology) as ε →
0. However, for anarbitrary small enough ε (cid:44)
0, it violates (6) but verifies (7). The robustness of situation (7), in turn, again followsfrom the properties of nondegenerate critical points of functions. (cid:3)
Acknowledgments
The author thanks Olivier Rivoire for the formulation of the problem and for fruitful discussions, and Cl´ementNizak for carefully reading the manuscript and for his advices on making it more accessible to readers withoutstrong mathematical background. This work was supported by ProTheoMics grant from Paris-Sciences-LettresUniversity and by ANR-17-CE30-0021-02 RBMPro grant from Agence Nationale de la Recherche.
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