A Computational Study of Alternate SELEX

Abstract

Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure for identifying nucleic acid (NA) molecules with affinities for specific target species, such as proteins, peptides, or small organic molecules. Here, we extend the work in Seo et al. (Bull Math Biol 72:1623–1665, 2010) (multiple-target SELEX or positive SELEX) and examine an alternate SELEX process with multiple targets by incorporating negative selection into a positive SELEX protocol. The alternate SELEX process is done iteratively by alternating several positive selection rounds with several negative selection rounds. At the end of each positive selection round, NAs are eluted from the bound product and amplified by polymerase chain reaction (PCR) to increase the size of the pool of NA species that bind preferentially to the given positive target vector. The enriched population of NAs is then exposed to the negative targets (undesired targets). The free NA species (instead of the bound NA species being eluted) are retained and amplified by PCR (negative selection). The goal is to minimize an enrichment of nonspecifically binding NAs against multiple targets. While positive selection alone results in a pool of NAs that bind tightly to a given target vector, negative selection results in the subset of the NAs that bind best to the nontarget vectors that are also present. By alternating the two processes, we eventually obtain a refined population of nucleic acids that bind to the desired target(s) with high “selectivity” and “specificity.” In the present paper, we give formulations of the negative and alternate selection processes and define their efficiencies in a meaningful way. We study the asymptotic behavior of alternate SELEX system as a discrete-time dynamical system. To do this, we use the chemical potential to examine how alternate SELEX leads to the selection of NAs with more specific interactions when the ratio of the number of positive selection rounds to the number of negative selection rounds is fixed. Alternate SELEX is said to be globally asymptotically stable if, given the initial target vector and a fixed ratio, the distribution of the limiting NA fractions does not depend on the relative concentrations of the NAs in the initial pool (provided that all of the NA species are initially present in the initial pool). We state conditions on the matrix of NA—target affinities that determine when the alternate SELEX process is globally asymptotically stable in this sense and illustrate these results computationally.

Appendices

Appendix A: The Contamination Effect—Analysis of the Specificity Matrix $C_\lambda$

When $\lambda$ is near to, but larger than, $1/2$, NAs with indices that are not in $\mathcal {L}_{1}$ can be selected by alternate selection, i.e., NAs other than those that would ordinarily be obtained by positive SELEX alone. (Figure 10a, $\lambda =0.52$, $\mathcal {L}_{\lambda }(m)=\{8,17\}$, $\mathcal {M}_1=\{1,2,3,4\}$.)

Definition 7

The ability of poorer binders to survive a small excess of positive selection over negative selection or the ability of better binders to survive a small excess of negative selection over positive selection is called the contamination effect.

We take $[T_{\nu }]$ large and fixed and replace the matrix $C_{\lambda }$ by $[T_{\nu }]^{(1-\lambda )/\lambda }C_{\lambda } \approx C_{\lambda ,\infty }$ where $\overrightarrow{C}_{\lambda ,\infty }^j= \overrightarrow{A}^j/(\overrightarrow{A}^j\cdot \widehat{\Omega }_{\nu })^{(1-\lambda )/\lambda }$ in order to focus on the contribution of the affinities to the contamination effect.

If $\overrightarrow{A}^j\cdot \widehat{\Omega }_{\nu }<\overrightarrow{A^{j'}}\cdot \widehat{\Omega }_{\nu }$, we say that the removed target component leaves a larger target affinity component for NA $j$ than for NA $j'$ because if $A_{ij}=A_{ij'}$ the effective affinity for target $i$ for NA $j$ (namely, $C_{\lambda ,\infty ,ij}$) will be larger than the effective affinity for target $i$ for NA $j'$ ($C_{\lambda ,\infty ,ij'}$).

In the following discussion, we assume that only one sub-target is removed during negative selection. Call it sub-target $T_{i_0}$, and recall that $\Omega _{\nu ,i_0}=0.$

The entries have the form

$$\begin{aligned} C_{\lambda ,\infty ,ij}= \frac{A_{ij}}{(\Omega _{\nu ,i}A_{ij}+\sum _{i'\in \mathcal {M}_1, i\ne i'}\Omega _{\nu ,i'}A_{i'j} )^{(1-\lambda )/\lambda }}. \end{aligned}$$

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Suppose first that $\lambda >1/2$ and $i\ne i_{0}$ corresponds to one of the remaining target components. Examining the formula for $C_{\lambda ,\infty ,ij}$ more closely, we find

$$\begin{aligned} C_{\lambda ,\infty ,ij}\approx \frac{A_{ij}^{(2\lambda -1)/\lambda }}{(\Omega _{\nu ,i}+ \sum _{i'\in \mathcal {M}_1,i'\ne i}\Omega _{\nu ,i'}(A_{i'j}/A_{ij} ))^{(1-\lambda )/\lambda }}. \end{aligned}$$

Suppose $j$ is fixed and $A_{ij}>>A_{ij'}$ for all $j'\ne j$. The matrix entry $C_{\lambda ,\infty ,ij}$ will behave like $A_{ij}^{(2\lambda -1)/\lambda }/\Omega _{\nu ,i}$. The other entries in this row, namely the $C_{\lambda ,\infty ,ij'} =A_{ij'}/(\overrightarrow{A^{j'}}\cdot \widehat{\Omega }_{\nu })^{(1-\lambda )/\lambda }$, do not depend upon $A_{ij}$. These entries will be bounded above for $0<\lambda <1$ while $C_{\lambda ,\infty ,ij}$ will be very large if $1/2<\lambda <1.$ Therefore, we expect that, under many circumstances, that for $\lambda >1/2$, $C_{\lambda ,\infty ,ij}>>C_{\lambda ,\infty ,ij'}.$ (This inequality could fail if $A_{ij}$ is not too much larger than the other entries in this row as one can infer from the following discussion.)

Suppose $i=i_{0}$. Suppose that for positive SELEX, $A_{i_{0}j}$ is the affinity of NA $NA_j$ that is selected by target $T_{i_{0}}$. Then, $C_{\lambda ,\infty ,i_{0}j}<C_{\lambda ,\infty ,i_{0}j'}$ will hold if and only if

$$\begin{aligned} \frac{A_{i_{0}j}}{A_{i_{0}j'}}<\biggl (\frac{\sum _{i'\in \mathcal {M}_1}\Omega _{\nu ,i'}A_{i'j}}{\sum _{i'\in \mathcal {M}_1}\Omega _{\nu ,i'}A_{i'j'}}\biggr )^{(1-\lambda )/\lambda }. \end{aligned}$$

Such an inequality is achievable for some $j'\ne j$ if the affinities $A_{i'j'}$ are relatively small compared with the affinities $A_{i'j}$. Suppose, for example, that

$$\begin{aligned} \frac{\sum _{i'\in \mathcal {M}_1}\Omega _{\nu ,i'}A_{i'j}}{\sum _{i'\in \mathcal {M}_1}\Omega _{\nu ,i'}A_{i'j'}}=1+\delta . \end{aligned}$$

Then, for all sufficiently small $\lambda$, $A_{i_{0}j}/A_{i_{0}j'}<(1+\delta )^{(1-\lambda )/\lambda }.$ When this occurs, the NA with index $j'$ will be preferred over the NA with index $j$ in the $i_{0}^{th}$ row.

If $\lambda =1/2$, $i\ne i_{0}$, then $C_{1/2,\infty ,ij}\approx 1/\Omega _{\nu ,i}$ provided the affinity $A_{i)ij}$ is sufficiently large relative to the other entries in this row. Then, there will be a preference for $NA_{j}$ over $NA_{j'}$ at $\lambda =1/2$ if $C_{1/2,\infty ,i,j}>C_{1/2,\infty ,ij'}$. This last inequality holds if

$$\begin{aligned} \frac{1}{\Omega _{\nu ,i}}>\frac{A_{ij'}}{\Omega _{\nu ,i}A_{ij'}+\sum _{i'\in \mathcal {M}_1, i'\ne i}\Omega _{\nu ,i'}A_{i'j'}} \end{aligned}$$

which is always true. Thus, for $1/2\le \lambda \le 1$, negative SELEX cannot influence positive SELEX for the nonremoved target component (provided $A_{ij}>>A_{ij'}$ for $j'\ne j$).

If, at $\lambda =1/2$ and $A_{i_{0}j}/A_{i_{0}j'}<(1+\delta )$, this will also be true in a small interval $(1/2-\epsilon ,1/2+\epsilon )$ about $\lambda =1/2$. Therefore, for such an interval, the nucleic acid $j'$ contaminates the pool. See Fig. 13 for the relative affinities $C_{1/2,\infty ,ij}/\max _{i'j'}C_{1/2,\infty ,i'j'}$. If one thinks of $C_{1/2,\infty }$ as an “affinity” matrix, the figure indicates that $NA_{17}$ is the “best binder.” The NA fractions $\{F_8(\lambda ,m), F_{17}(\lambda ,m)\}$ for a large number of grand rounds and $m=1$ are $\{ 0.0113, 0.9887 \}$, $\{ 0.0001, 0.9999 \}$, $\{ 0.0004, 0.9996 \}$, $\{ 0.9302, 0 \}$ for $\lambda =0.50, 0.52, 0.54, 0.56$, respectively. (Note that the sum of the two fractions, $F_8$ and $F_{17}$, is not one since for $\lambda =0.56$ the selected NA indices are $\{8,10,12,16\}$.) When $m=30$, the limiting NA fractions, $\{F_8, F_{17}\}$, are $\{ 0, 1 \}$, $\{ 0, 1 \}$, $\{1, 0\}$, $\{ 1, 0 \}$ for $\lambda =0.50, 0.52, 0.54, 0.56$, respectively. We see that small differences in the specificities can lead to relatively large fractions of contaminant even when the number of grand rounds is large but the multiplier, $m$, is small.

The problem is further aggravated if one restricts oneself to a single grand round as illustrated in Fig. 12. For $\lambda =0.4, 0.5, 0.6,$ we calculated $\lambda R$ rounds of positive SELEX followed by $(1-\lambda )R$ rounds of negative SELEX with $R=80$. We see that for $\lambda =0.4, 0.5, 0.6$, there is penetration of poorer binders into a pool that should consist only of the ultimately selected NA $NA_8$ but does not if $[T_\nu ]$ is of moderate size. For $\lambda =0.6$, the ultimately selected acid pool consists only of $NA_8$ when $[T_\nu ]$ is large.

The problem goes away in the limit of infinitely large grand round number. The reason is that when $m>>>1$, so that we have done $mR_{s,0}>>1$ rounds of positive SELEX before starting the negative rounds, we have very nearly achieved a pure pool of NAs whose indices are in $\mathcal {L}_\lambda (1)$ before we have even begun the negative SELEX rounds. If we are lucky, there may not be any NAs other than the best binders to the full target before we begin specification with negative SELEX. In any case, the theory Levine and Seo (submitted), tells us that the result of limiting ultimate SELEX will lead to a pool consisting of exactly one NA if the elements of the set $\{\overrightarrow{A}^j\cdot \widehat{\Omega }_\nu | \ j\in \mathcal {L}_\lambda (m_0)\}$ are all distinct (i.e., the cardinality of this set is the same as that of $\mathcal {L}_\lambda (m_0)\subset \mathcal {L}_\lambda (1)$).

Appendix B: An Algorithm for Determining the Final Values in Positive Selection

A given affinity matrix $A$ defines two functions, one of which, $\varphi :\mathcal {S}\rightarrow (0,\infty )$, is convex and continuous on $\mathcal {S}$ and the second, $\mathcal {L}:\mathcal {S}\rightarrow P(\mathcal {N})$ where $P(\mathcal {N})$ is the set of all subsets of $\mathcal {N}=\{1,2,\dots , N\}.$ The function $\varphi$ is called the maximal target affinity function and is defined by $\varphi (\widehat{\omega })= \max \{\overrightarrow{A}^j\cdot \widehat{\omega } \ \ | j\in \mathcal {N}\}$ where $\overrightarrow{A}^j$ is the jth column of $A$. The set function is defined by $\mathcal {L}(\widehat{\omega })=\{l\in \mathcal {N}\ | \ \overrightarrow{A}^l\cdot \widehat{\omega }=\varphi (\widehat{\omega }) \}$.

Let $\mathcal {L}$ be an increasingly ordered subset of $\mathcal {N}$ with $L$ elements. The following subset of the graph of $\varphi$ given by $\Phi (\mathcal {L})=\overline{\{(\widehat{\omega },\varphi (\widehat{\omega })) \ \ | \overrightarrow{A}^l\cdot \widehat{\omega }=\varphi (\widehat{\omega }),\ \widehat{\omega }\in \mathcal {S}_M, \ l\in \mathcal {L} \}}$, where the over line denotes the set of limit points of the set under it together with the set itself and where the set of vectors $\{\overrightarrow{A}^l | \ \ l\in \mathcal {L}\}$ is linearly independent, is called an $L$ face of the graph of $\varphi$. It is a subset of $R^M$ and can be thought of as a polygon in $R^{M-L}$. An $L$ face is called proper if the set of indices that describes it is unique. That is, $\Phi (\mathcal {L})=\Phi (\mathcal {L}')$ implies $\mathcal {L}=\mathcal {L}'$. If this fails, we say that the face is improper.

If for all $L=1,2,\dots M$, $\varphi$ has the property that every $L$ face is proper, we say that the maximal target affinity function is proper. Otherwise, we say the maximal target affinity function is improper.

For a given free target vector $\widehat{\omega }_s$, the function $\mathcal {W}_s=\mathcal {W}_s([NA],\widehat{\omega }_s)=1/(1+[NA]\varphi (\widehat{\omega }_s))$ and unit vectors in $\mathcal {S}$, namely $\widehat{V}^l=\mathcal {W}_s\widehat{\omega }_s+(1-\mathcal {W}_s)\widehat{A^l\omega _s}$ for $l\in \mathcal {L}$ are defined. The convex hull of this set of vectors is denoted by $\mathcal {H}([NA],\widehat{\omega }_s)$ and is the set from which the starting target fractions must come if and only if the final free target fraction is $\widehat{\omega }_s$. It can be defined regardless of whether or not the set of vectors $\{\overrightarrow{A}^{l} \ | \ l\in \mathcal {L}\}$ is linearly independent. To find the final NA fraction vector $\widehat{F_s}$ when $\widehat{\Omega }_{s}\in \mathcal {H}([NA],\widehat{\omega }_{s})$, we set $F_{s,j}=0$ if $j\notin \mathcal {L}$ and use

$$\begin{aligned} \widehat{F_s}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \widehat{V}^{l_1}\cdot \widehat{V}^{l_1}&{} \widehat{V}^{l_1}\cdot \widehat{V}^{l_2} &{} \dots &{} \widehat{V}^{l_1}\cdot \widehat{V}^{l_{L}} \\ \widehat{V}^{l_2}\cdot \widehat{V}^{l_1}&{} \widehat{V}^{l_2}\ \cdot \widehat{V}^{l_2} &{} \dots &{} \widehat{V}^{l_2}\cdot \widehat{V}^{l_{L}} \\ \vdots &{} \vdots &{} \dots &{}\vdots \\ \widehat{V}^{l_{L}}\cdot \widehat{V}^{l_1}&{} \widehat{V}^{l_{L}}\cdot \widehat{V}^{l_2} &{} \dots &{} \widehat{V}^{l_{L}}\cdot \widehat{V}^{l_{L}}\\ \end{array}\right) ^{-1}\left( \begin{array}{c} \widehat{\Omega }_{s}\cdot \widehat{V}^{l_1}\\ \widehat{\Omega }_{s}\cdot \widehat{V}^{l_2} \\ \vdots \\ \widehat{\Omega }_{s}\cdot \widehat{V}^{l_{L}}\\ \end{array} \right) \end{aligned}$$

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to evaluate the $F_{s,j}$ when $j\in \mathcal {L}$. This clearly requires that the vectors in $\{\overrightarrow{A}^l \ | \ l\in \mathcal {L}\}$ be linearly independent.

The function $\varphi$ is said to be proper if the set $\{\widehat{V}^l | \ l\in \mathcal {L}\}$ is a linearly independent set, i.e., the rank of the Grammian is precisely $L$ and the final NA fractions are uniquely determined by $\widehat{\Omega }_{s}, \ \widehat{\omega }_s.$ However, if $\varphi$ is improper and the set $\mathcal {L}$ does not uniquely determine the $L$ face to which $(\widehat{\omega }_s,\varphi (\widehat{\omega }_s))$ belongs, then the Grammian will not be invertible and two different starting nucleic acid fraction vectors $\widehat{F_s}$ will determine two different sets of final NA fractions.

If we know the distribution of NA fractions in the initial pool, we can use the SELEX iteration scheme to determine both vector quantities simultaneously regardless of whether or not $\varphi$ is proper.

Suppose we are given $\varphi (\widehat{\omega }_s)$ and we wish to determine the final fractions for the NA pool when the initial pool of NAs is unknown. As in Seo et al. (2010), we argue as follows:

• 1. Calculate $\varphi$ and determine its faces. In principle, this could be a tedious task. However, if $M$ is not too large, it is quite doable. The simplex interior $\mathcal {S}_0$ can be projected onto the interior of the unit cube in $R^{M-1}$ via the transformation $\omega _1=1-s_1, \omega _2= s_2(1-s_1), \dots , \omega _{M-1}=s_1\cdots s_{M-2}s_{M-1},\omega _M=s_1\cdots s_{M-2}s_{M-1}$. A rectangular grid can then be imposed on this cube. The pointwise evaluation of $\varphi$ is then carried out over this grid.

• 2. Use equation $\widehat{\Omega }_s=\mathcal {W}_s\biggl ( \sum _{l\in \mathcal {L}} F_{s,l}\{\widehat{\omega }_s+[NA]\overrightarrow{A^{l} \omega _s}\}\biggr )$ to compute the convex hull of the set of vectors $\{\widehat{V}^l(\widehat{\omega }_s) | (\widehat{\omega }_s,\varphi (\widehat{\omega }_s))\in \Phi (\mathcal {L})\}.$ Call this hull $\mathcal {H}([NA], \Phi (\mathcal {L}))$. Then, the simplex $\mathcal {S}$ of target fractions $\widehat{\Omega }_s$ can be written as $\mathcal {S}=\cup \{\mathcal {H}([NA], \Phi (\mathcal {L})) | \ \ \Phi (\mathcal {L})\ \mathrm{is a face of the graph of}\ \varphi (\cdot )\}.$

• 3. Suppose that $\widehat{\Omega }_s \in \mathcal {H}([NA], \Phi (\mathcal {L}))$. Then, $\varphi (\widehat{\omega }_s)=K_{a}$ for some free target vector $\widehat{\omega }_s$ and $\Omega _{s,i}=(1+[NA]\overrightarrow{A}_{i}\cdot \widehat{F_s})\omega _{s,i}\mathcal {W}_s$ where $\mathcal {W}_s=1/(1+[NA]K_{a})$ and where $F_{s,j}=0$ if $j\notin \mathcal {L}.$

When the face in question is proper, the system of $L$ equations defined by (96) has one and only one solution, and the components of the free target are found from $\omega _{s,i}=\Omega _{s,i}/(1+[NA]\overrightarrow{A}_{i}\cdot \widehat{F_s})\mathcal {W}_s$. When the face is improper, the system (96) fails to have a unique solution, i.e., there will be a several parameter family of free targets corresponding to the family of solutions of (96). In Seo et al. (2010), we proved

Theorem 3

The maximal target affinity function $\varphi$ is proper in the sense that defined above if and only if the chemical potential at infinite target dilution defined by each $L$ face has a unique minimum point. (The chemical potential is defined by $\mathcal {R}(\widehat{F_s})=-RT\sum _{i=1}^M\Omega _{s,i}\ln (1+[NA]\overrightarrow{A}_i\cdot \widehat{F_s}),$ where all of the components of $\widehat{F_s}$ vanish if the corresponding index is not in $\mathcal {L}$. Here, $R$ is the gas constant and $T$ is the Kelvin temperature.)

Appendix C: Summary of the Mathematical Results in Terms of Dynamical Systems

We say that any positive (or negative) SELEX process has a single-point asymptotically stable global attractor for a fixed pool concentration $[NA]$, in the following sense: Given any initial target fraction $\widehat{\Omega }_s$ (or $\widehat{\Omega }_\nu$), the final (limiting) free target vector and the distribution of the limiting NA fractions do not depend on the distribution of the NAs present in the initial pool. (The assumption here is that the initial NA fraction vector is in the interior of the simplex $\mathcal {S}_{N}$.) The limit will depend on the initial target fraction vector.

For alternate target SELEX, the definition of the operator sequence must be changed. A single grand round of alternate SELEX begins with a NA fraction vector, $\widehat{F}$, to which are applied successively $R_s=m\lambda R_0$ positive SELEX operators, and $R_\nu =m(1-\lambda )R_0$ negative SELEX operators. Thus, the alternate SELEX operator can be viewed as a composition of $R_s$ positive SELEX operators followed by the composition of $R_\nu$ negative SELEX operators (as defined with the same target component removed for all $R_\nu$ of them) and the image of this operator when applied to the input vector $\widehat{F}$ will be defined as $\widehat{F}_{\nu }^{mR_{\nu ,0}}$, the last of the $R$ recursively defined output vectors $\widehat{F}_s^{1},\dots , \widehat{F}_s^{mR_{s,0}}, \widehat{F}_{\nu }^{1},\dots , \widehat{F}_{\nu }^{mR_{\nu ,0}}$. Let $\mathcal {B}_{k,m}$ be the alternate SELEX operator for the kth grand round and $[T_s](k)$ denote the target concentration after the kth grand round.

For each triple $(m,\lambda , \widehat{\Omega }_s)$, using the approximation we made earlier, $[T_{\nu }f]\thickapprox [T_{\nu }]$, recursively define

$$\begin{aligned} \mathcal {A}_{k}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])=\mathcal {B}_{k,m}\mathcal {A}_{k-1}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }]) \end{aligned}$$

with $\mathcal {A}_{0}=I$. The input NA vector $\widehat{F}$ in the first positive SELEX step in the application of $\mathcal {A}_{k}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])$ will be the last output vector from the application of negative SELEX in $\mathcal {A}_{k-1}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])$. Denote this output vector by $\widehat{F}_{\nu }^{mR_{\nu ,0},k}.$

Next, we define limiting operators $\mathcal {A}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])$ by

$$\begin{aligned} \lim _{k\rightarrow +\infty }\mathcal {A}_{k}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])(\widehat{F})&= \lim _{k\rightarrow +\infty }\widehat{F}_{\nu }^{mR_{\nu ,0},k}= \widehat{F}_{\nu }^{mR_{\nu ,0}}\nonumber \\&= \mathcal {A}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},0, [T_{\nu }]).\nonumber \\ \end{aligned}$$

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(From the results of Levine and Seo (submitted), these limits exist.)

Theorem 2 is equivalent to the following theorem.

Theorem 4

The following limits exist and are continuous functions of $\widehat{F}$:

$$\begin{aligned} \mathcal {A}(\lambda ,\widehat{\Omega }_s)(\widehat{F})&= \lim _{m\rightarrow \infty }\mathcal {A}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},0, [T_{\nu }])(\widehat{F})\nonumber \\&= \lim _{m\rightarrow +\infty } \lim _{k\rightarrow +\infty }\mathcal {A}_k({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }])(\widehat{F}).\qquad \quad \end{aligned}$$

(104)

Now, we define “ultimate single point global asymptotic stability.” We say that the double sequence

$$\begin{aligned} \{\mathcal {A}_{k}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},[T_s](k), [T_{\nu }]), 1\le k\le m, m=1,\dots ,\infty \} \end{aligned}$$

is ultimately single-point globally asymptotically stable for fixed $\lambda , \widehat{\Omega }_s$ if the sequence of limiting operators $\{\mathcal {A}({mR_s,mR_\nu ,\lambda ,\widehat{\Omega }_s},0, [T_{\nu }])\}_{m=1}^\infty$ has a single-point asymptotically stable global attractor, i.e., if $\mathcal {A}(\lambda ,\widehat{\Omega }_s)(\widehat{F})$ exists for every $\widehat{F}\in \mathcal {S}_N^{(0)}$ and does not depend on $\widehat{F}\in \mathcal {S}_N^{(0)}.$

Remark 12

Ultimate single-point global asymptotic stability does not imply limiting ultimate specificity as the discussion in Remark 13 makes clear. However, if we have limiting ultimate specificity to the same nucleic acid on the interior of the simplex $\mathcal {S}_N$, then limiting ultimate specificity and limiting ultimate single-point asymptotic stability are one and the same.

In Appendix B (Sect. 16) and in Seo et al. (2010), we saw that the necessary and sufficient condition for single-point global asymptotic stability for positive multiple-target SELEX was that the maximal target affinity function be proper. Whether or not this is the case, the graph of $\varphi$ is convex and made up of finitely many closed faces. If an $L$ face is proper, then the set $\mathcal {L}$ is maximal with respect to the defining property. However, a set can be maximal with respect to this property without uniquely determining the face if the face is not proper.

The property of “properness” is an algebraic property. It is easy to give simple examples using values for target affinities that are physically meaningful for which the property fails. However, if $M<N$ as is usually the case, then, on average, given a set of $M$ or fewer $N$ vectors with positive entries, the set of vectors is more likely than not to be linearly independent. However, unless $M=1$, there is no simple ordering of vectors to force this to hold that has physical meaning.

Referring to Appendix B again, the projections of the $L$ faces on $\mathcal {S}_M$ partition $\mathcal {S}_M$ into polygonal subsets of varying dimension that in turn determine a decomposition of the initial target space $\mathcal {S}_M$ into a number of convex polygons, namely the convex hulls of the vectors $\{ \widehat{V}^l\ |\ l\in \mathcal {L} \}$, namely $\mathcal {H}([NA], \Phi (\mathcal {L}))$. We say that such a convex hull is proper if and only if it was induced by a proper $L$ face through this construction.

The operator $\mathcal {A}(m, \lambda , \widehat{\Omega }_{s})$ maps $\mathcal {S}_{N}$ onto a subset $\mathcal {S}_{\mathcal {L}(\widehat{\Omega }_{s})}$. Now, fix a given set of indices $\mathcal {L}$, i.e., any subset of $\mathcal {N}$. We define the subsets of $S_M$ (the simplex of target fractions for positive SELEX) by

$$\begin{aligned} \mathcal {H}(m,\lambda ,\mathcal {L}) \equiv \{\widehat{\Omega }_{s} \ | \ \mathcal {L}= \mathcal {L}(\widehat{\Omega }_{s}) \}. \end{aligned}$$

(105)

(We have suppressed the dependence on $[NA]$ here.) The sets defined by (105) will be empty if none of the elements in $\mathcal {L}$ are selected under alternate selection for any $\widehat{\Omega }_{s}$. However, when $\lambda =1$, and $\mathcal {L}$ is a maximal set of indices defining an $L$ face, $\mathcal {H}(m,1,\mathcal {L})$ = convex hull of the set of vectors $\{\widehat{V}_l \ | \ l\in \mathcal {L}\}$.

We say that the set $\mathcal {H}(m,\lambda ,\mathcal {L})$ is proper if and only if $\mathcal {H}([NA], \Phi (\mathcal {L}))$ is proper. Otherwise, $\mathcal {H}(m,\lambda ,\mathcal {L})$ is improper.

Suppose we have a given $L$ face with index set $\mathcal {L}$ and that $\mathcal {J}$ is the set of indices for the nucleic acids that bind best to the removed target. There are four possibilities:

• 1. $\mathcal {H}(m,\lambda ,\mathcal {L})$ is proper, and the intersection $\mathcal {J}\cap \mathcal {L}$ is empty.

• 2. $\mathcal {H}(m,\lambda ,\mathcal {L})$ is improper, and the intersection $\mathcal {J}\cap \mathcal {L}$ is empty.

• 3. $\mathcal {H}(m,\lambda ,\mathcal {L})$ is proper, and the intersection $\mathcal {J}\cap \mathcal {L}$ is not empty.

• 4. $\mathcal {H}(m,\lambda ,\mathcal {L})$ is improper, and the intersection $\mathcal {J}\cap \mathcal {L}$ is not empty.

(Notice that we are using the convex hulls as defined for positive SELEX.)

We see from Figs. 19 and 21 that the following statements should hold:

Conjecture 1

Suppose that $1/2 < \lambda < 1$.

• 1. The hulls, $\mathcal {H}(m,\lambda , \mathcal {L})$, defined by $L$ faces satisfying (1) or(2) above, satisfy $\mathcal {H}(m,\lambda , \mathcal {L})=\mathcal {H}(1,\lambda , \mathcal {L})$. If the $L$ face is proper, for all $m=1, 2,\dots ,$ and $\widehat{\Omega }_s\in \mathcal {H}(m,\lambda , \mathcal {L})$, alternate SELEX is single-point globally asymptotically stable on $\mathcal {H}(m,\lambda ,\mathcal {L})$. If the $L$ face is improper, for all $m=1, 2,\dots ,$ and $\widehat{\Omega }_s\in \mathcal {H}(m,\lambda , \mathcal {L})$, alternate SELEX is not single-point globally asymptotically stable.

• 2. Suppose the relative complement $\mathcal {L}\backslash \mathcal {L}\cap \mathcal {J}\ne \emptyset$. Suppose either of the last two possibilities hold. Then,

  $$\begin{aligned} \displaystyle \bigcap _{m=1}^\infty \mathcal {H}(m,\lambda , \mathcal {L}\backslash \mathcal {L}\cap \mathcal {J})=\emptyset . \end{aligned}$$

  Moreover, for all $m=1,2,3,\dots ,$

  $$\begin{aligned} \bigcup _{\{ \mathcal {L}\ | \ \mathcal {L}\cap \mathcal {J}\ne \emptyset \}}\mathcal {H}(m,\lambda , \mathcal {L})= \bigcup _{\{ \mathcal {L}\ | \ \mathcal {L}\cap \mathcal {J}\ne \emptyset \}}\mathcal {H}(1,\lambda , \mathcal {L}) \equiv \mathcal {H}(\lambda , \mathcal {L}) . \end{aligned}$$

  • a. If case 3 holds, i.e., the hull $\mathcal {H}(m,\lambda , \mathcal {L})$ is proper for some $m$ and $\mathcal {J}\cap \mathcal {L}$ is not empty, then for $\widehat{\Omega }_{s} \in \mathcal {H}(\lambda , \mathcal {L})$, $\mathcal {A}(\lambda , \widehat{\Omega }_{s})\widehat{F}$ does not depend on $\widehat{F}\in \mathcal {S}_N$ when all of the components of $\widehat{F}\in \mathcal {S}_N$ are present in the initial pool. In this case, we have both limiting ultimate specificity and ultimate single-point global asymptotic stability.

  • b. Suppose, in addition, each target component $T_i$ binds best to a single NA $NA_{j(i)}$, i.e., $A_{ij(i)}> A_{ij}$ for all $j\in \mathcal {N}$ such that $j\ne j(i)$. If case 4 holds for all $m$, i.e., the hull $\mathcal {H}(m,\lambda , \mathcal {L})$ is improper, then, the set $\mathcal {L}\cap \mathcal {J}=\{j(i)\}$ (a singleton) and again we have both limiting ultimate specificity and ultimate single-point global asymptotic stability.

We see from Fig. 19 that the proper regions that do not contain the NA with index $3$ but which contain one or more of the other NAs obtained by positive SELEX alone do not shrink to sets of measure zero. For example, in panels (c, f, i, l), the two-dimensional measures of the regions labeled with NA indices $\{1\},\{2\},\{4\},\{1,4\},\{2,4\}$ do not shrink to zero as $m$ becomes large. This is to be expected, because the NAs with indices $1, 2, 4$ are not the best binders to the removed target. That is, the final pool is not ultimately specified for all $m=1, 2, 3, \cdots$.

Remark 13

Suppose the extra hypothesis in item (2b) fails. This could happen in one of two ways. First, the removed sub-target binds equally well to two different NAs. Alternatively, one of two distinct sub-targets that best binds equally well to a single nucleic acid is removed.

In the first case, illustrated in Fig. 22, consider the “limiting” panel (d). There are regions where each of the two NAs ($NA_4, NA_6$) is individually ultimately specified in the limit. However, we do not achieve limiting ultimate specificity in that we obtain both NAs as $m$ becomes large, except on a very small subset of the target set although we do have ultimate single-point global asymptotic stability. Thus, ultimate single-point global asymptotic stability does not imply limiting ultimate specificity.

In the second case, Fig. 23, again consider the “limiting” panel (d). The set of targets for which we select only NA $NA_5$ is a proper subset of the set of targets for which we select $NA_5$ among others. (That is, in the yellow and green regions, we select not only $NA_5$ but also $NA_1$ or $NA_4$.) This again shows that ultimate single-point global asymptotic stability does not imply limiting ultimate specificity.

Appendix D: Affinity Matrices Used to Generate the Figures

For Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 (five-target case), we used the affinity matrix $A=A(1:5,1:20)$ given in (106) and (107) below:

$$\begin{aligned}&A(1:5,1:10)\nonumber \\&=\begin{bmatrix} 822.37&618.81&521.92&984.25&759.88&1938.00&3164.60&1623.40&4629.60&2403.80\\ 2403.80&1091.70&1396.60&659.63&521.92&1225.50&706.21&8620.70&4629.60&1623.40 \\ 4629.60&759.88&521.92&706.21&582.75&3164.60&984.25&550.66&1938.00&1396.60\\ 2403.80&1225.50&3164.60&984.25&1091.70&521.92&582.75&1396.60&4629.60&8620.70\\ 759.88&896.06&659.63&1623.40&1225.50&1091.70&618.81&8620.70&984.25&582.75\\ \end{bmatrix},\nonumber \\ \end{aligned}$$

(106)

$$\begin{aligned}&A(1:5,11:20)\nonumber \\&=\begin{bmatrix} 896.06&550.66&1225.50&1396.60&496.03&8620.70&659.63&706.21&582.75&1091.70 \\ 759.88&896.06&496.03&1938.00&984.25&822.37&618.81&3164.60&582.75&550.66 \\ 1091.70&8620.70&496.03&618.81&896.06&1623.40&822.37&1225.50&2403.80&659.63\\ 896.06&706.21&659.63&822.37&1623.40&1938.00&759.88&496.03&618.81&550.66 \\ 706.21&822.37&550.66&496.03&2403.80&4629.60&3164.60&1396.60&521.92&1938.00 \\ \end{bmatrix}.\nonumber \\ \end{aligned}$$

(107)

The values for the first row were chosen randomly over the range that corresponds to the range of values used in Irvine et al. (1991). Each of the remaining rows was obtained from the first by doing a random reordering of the values of the first row.

For Figs. 14, 15, 16 and 17 (three-target case), we used

$$\begin{aligned} A =\begin{bmatrix} 5538.00&2289.00&7513.00&6702.00&2435.00&2518.00&4791.00 \\ 7265.00&3991.00&5896.00&8119.00&2925.00&8698.00&6062.00 \\ 8641.00&3603.00&7822.00&6315.00&3663.00&3795.00&1420.00 \\ \end{bmatrix}. \end{aligned}$$

(108)

For panels (a–l) in Figs. 18 and 19, and panels (a–d) in Fig. 20 (three-target case), we used

$$\begin{aligned} A =\begin{bmatrix} 9355.00&5529.00&1987.00&7468.00&846.20&1000.00&1500.00&600.00 \\ 916.90&8132.00&3038.00&6451.00&5252.00&1400.00&900.00&1800.00 \\ 4993.40&990.00&9722.00&931.80&2026.00&3963.00&3795.00&2518.00 \\ \end{bmatrix}.\nonumber \\ \end{aligned}$$

(109)

For panels (a–d) in Fig. 21 (three-target case), we used

$$\begin{aligned} A =\begin{bmatrix} 4629.0&5492.0&1623.4&3386.5&4420.2&1000.0&1500.0&600.0 \\ 3164.6&792.0&1938.0&4800.0&1925.5&1400.0&900.0&1800.0 \\ 1091.7&793.0&4630.0&2445.0&2103.8&3963.0&3795.0&2518.0 \\ \end{bmatrix}. \end{aligned}$$

(110)

For panels (a–c) in Fig. 22 (three-target case), we used

$$\begin{aligned} A =\begin{bmatrix} 4629.0&4420.2&1000.0&5492.0&3386.5&1623.4&1500.0&600.0 \\ 3164.6&1925.5&1400.0&792.0&2245.0&2938.0&900.0&1800.0 \\ 3091.7&2103.8&3963.0&4630.0&3793.0&4630.0&3795.0&2518.0 \\ \end{bmatrix}. \end{aligned}$$

(111)

For panels (a–c) in Fig. 23 (three-target case), we used

$$\begin{aligned} A =\begin{bmatrix} 4629.0&5492.0&3386.5&4420.2&1623.4&1000.0&1500.0&600.0 \\ 3164.6&792.0&1938.0&1925.5&4630.0&1400.0&900.0&1800.0 \\ 1091.7&793.0&2445.0&2103.8&4630.0&3963.0&3795.0&2518.0 \\ \end{bmatrix}. \end{aligned}$$

(112)