Aptamer-enabled uptake of small molecule ligands.

The relative ease of isolating aptamers with high specificity for target molecules suggests that molecular recognition may be common in the folds of natural RNAs. We show here that, when expressed in cells, aptamers can increase the intracellular concentrations of their small molecule ligands. We have named these aptamers as DRAGINs (Drug Binding Aptamers for Growing Intracellular Numbers). The DRAGIN property, assessed here by the ability to enhance the toxicity of their ligands, was found for some, but not all, aminoglycoside aptamers. One aptamer protected cells against killing by its ligand. Another aptamer promoted killing as a singlemer and protected against killing as a tandemer. Based on a mathematical model, cell protection vs. killing is proposed as governed by aptamer affinity and access to the inner surface of the cell membrane, with the latter being a critical determinant. With RNA molecules proposed as the earliest functional polymers to drive the evolution of life, we suggest that RNA aptamer-like structures present in primitive cells might have selectively concentrated precursors for polymer synthesis. Riboswitches may be the evolved forms of these ancient aptamer-like "nutrient procurers". Aptamers with DRAGIN capability in the modern world could be applied for imaging cells, in synthetic cell constructs, or to draw drugs into cells to make "undruggable" targets accessible to small molecule inhibitors.

A computational study of alternate SELEX.

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.

A mathematical model for selective differentiation of neural progenitor cells on micropatterned polymer substrates.

The biological hypothesis that the astrocyte-secreted cytokine, interleukin-6 (IL6), stimulates differentiation of adult rat hippocampal progenitor cells (AHPCs) is considered from a mathematical perspective. The proposed mathematical model includes two different mechanisms for stimulation and is based on mass-action kinetics. Both biological mechanisms involve sequential binding, with one pathway solely utilizing surface receptors while the other pathway also involves soluble receptors. Choosing biologically-reasonable values for parameters, simulations of the mathematical model show good agreement with experimental results. A global sensitivity analysis is also conducted to determine both the most influential and non-influential parameters on cellular differentiation, providing additional insights into the biological mechanisms.

A mathematical analysis of multiple-target SELEX.

SELEX (Systematic Evolution of Ligands by Exponential Enrichment) is a procedure by which a mixture of nucleic acids can be fractionated with the goal of identifying those with specific biochemical activities. One combines the mixture with a specific target molecule and then separates the target-NA complex from the resulting reactions. The target-NA complex is separated from the unbound NA by mechanical means (such as by filtration), the NA is eluted from the complex, amplified by PCR (polymerase chain reaction), and the process repeated. After several rounds, one should be left with the nucleic acids that best bind to the target. The problem was first formulated mathematically in Irvine et al. (J. Mol. Biol. 222:739-761, 1991). In Levine and Nilsen-Hamilton (Comput. Biol. Chem. 31:11-25, 2007), a mathematical analysis of the process was given. In Vant-Hull et al. (J. Mol. Biol. 278:579-597, 1998), multiple target SELEX was considered. It was assumed that each target has a single nucleic acid binding site that permits occupation by no more than one nucleic acid. Here, we revisit Vant-Hull et al. (J. Mol. Biol. 278:579-597, 1998) using the same assumptions. The iteration scheme is shown to be convergent and a simplified algorithm is given. Our interest here is in the behavior of the multiple target SELEX process as a discrete "time" dynamical system. Our goal is to characterize the limiting states and their dependence on the initial distribution of nucleic acid and target fraction components. (In multiple target SELEX, we vary the target component fractions, but not their concentrations, as fixed and the initial pool of nucleic acids as a variable starting condition). Given N nucleic acids and a target consisting of M subtarget component species, there is an M × N matrix of affinities, the (i,j) entry corresponding to the affinity of the jth nucleic acid for the ith subtarget. We give a structure condition on this matrix that is equivalent to the following statement: For any initial pool of nucleic acids such that all N species are represented, the dynamical system defined by the multiple target SELEX process will converge to a unique subset of nucleic acids, each of whose concentrations depend only upon the total nucleic acid concentration, the initial fractional target distribution (both of which are assumed to be the same from round to round), and the overall limiting association constant. (The overall association constant is the equilibrium constant for the system of MN reactions when viewed as a composite single reaction). This condition is equivalent to the statement that every member of a certain family of chemical potentials at infinite target dilution can have at most one critical point. (The condition replaces the statement for single target SELEX that the dynamical system generated via the process always converges to a pool that contains only the nucleic acid that binds best to the target). This suggests that the effectiveness of multiple target SELEX as a separation procedure may not be as useful as single target SELEX unless the thermodynamic properties of these chemical potentials are well understood.

Diffusion and reaction in the cell glycocalyx and the extracellular matrix.

Many biologically important macromolecular reactions are assembled and catalyzed at the cell lipid-surface and thus, the extracellular matrix and the glycocalyx layer mediate transfer and exchange of reactants and products between the flowing blood and the catalytic lipid-surface. This paper presents a mathematical model of reaction-diffusion equations that simply describes the transfer process and explores its influence on surface reactivity for a prototypical pathway, the tissue factor (Tf) pathway of blood coagulation. The progressively increasing friction offered by the matrix and glycocalyx to reactants and to the product (coagulation factors X, VIIa and Xa) approaching the reactive surface is simulated and tested by solving the equations numerically with both, monotonically decreasing and constant diffusion profiles. Numerical results show that compared to isotropic transfer media, the anisotropic structure of the matrix and glycocalyx sharply decreases overall reaction rates and significantly increases the mean transit time of reactants; this implies that the anisotropy modifies the distribution of reactants. Results also show that the diffusional transfer, whether isotropic or anisotropic, influences reaction rates according to the order at which the reactants arrive at the boundary. Faster rates are observed when at least one of the reactants is homogeneously distributed before the other arrives at the boundary than when both reactants transfer simultaneously from the boundary.

A mathematical analysis of SELEX.

Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure by which a mixture of nucleic acids that vary in sequence can be separated into pure components with the goal of isolating those with specific biochemical activities. The basic idea is to combine the mixture with a specific target molecule and then separate the target-NA complex from the resulting reaction. The target-NA complex is then separated by mechanical means (for example by filtration), the NA is then eluted from the complex, amplified by polymerase chain reaction (PCR) and the process repeated. After several rounds, one should be left with a pool of [NA] that consists mostly of the species in the original pool that best binds to the target. In Irvine et al. [Irvine, D., Tuerk, C., Gold, L., 1991. SELEXION, systematic evolution of nucleic acids by exponential enrichment with integrated optimization by non-linear analysis. J. Mol. Biol. 222, 739-761] a mathematical analysis of this process was given. In this paper we revisit Irvine et al. [Ibid]. By rewriting the equations for the SELEX process, we considerably reduce the labor of computing the round to round distribution of nucleic acid fractions. We also establish necessary and sufficient conditions for the SELEX process to converge to a pool consisting solely of the best binding nucleic acid to a fixed target in a manner that maximizes the percentage of bound target. The assumption is that there is a single nucleic acid binding site on the target that permits occupation by not more than one nucleic acid. We analyze the case for which there is no background loss (no support losses and no free [NA] left on the support). We then examine the case in which such there are such losses. The significance of the analysis is that it suggests an experimental approach for the SELEX process as defined in Irvine et al. [Ibid] to converge to a pool consisting of a single best binding nucleic acid without recourse to any a priori information about the nature of the binding constants or the distribution of the individual nucleic acid fragments.

A mathematical model for the onset of avascular tumor growth in response to the loss of p53 function.

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

A mathematical model for the regulation of tumor dormancy based on enzyme kinetics.

In this paper we present a two-compartment model for tumor dormancy based on an idea of Zetter [1998, Ann. Rev. Med. 49, 407-422] to wit: The vascularization of a secondary (daughter) tumor can be suppressed by an inhibitor originating from a larger primary (mother) tumor. We apply this idea at the avascular level to develop a model for the remote suppression of secondary avascular tumors via the secretion of primary avascular tumor inhibitors. The model gives good agreement with the observations of [De Giorgi et al., 2003, Derm. Surgery 29, 664-667]. These authors reported on the emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter. The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites 5-7 cm from the primary site within a period of 1 month after the excision of the primary site. We attempt to provide a reasonable biochemical/cell biological model for this phenomenon. We show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumor while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumor, even after it is removed. It should be possible to use the model as the basis for a testable hypothesis.

A mathematical model for the role of cell signal transduction in the initiation and inhibition of angiogenesis.

Neovascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel. In two previous papers, [Levine, H.A. and Sleeman, B.D. (1997) "A system of reaction diffusion equations arising in the theory of reinforced random walks" SIAM J. AppL Math. 683-730; Levine, H.A., Sleeman, B.D. and Nilsen-Hamilton, M. (2001b) "Mathematical modelling of the onset of capillary formation initiating angiogenesis." J. Math. Biol. 195-238] the authors introduced a new approach to angiogenesis, based on the theory o f reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial vascular endothelial cell growth factor (VEGF) receptors as the catalyst for transforming into a proteolytic enzyme in order to model the first stage. It is the purpose of this paper to present a more descriptive yet not overly complicated mathematical model of the biochemical events that are initiated when VEGF interacts with endothelial cells and which result in the cell synthesis of proteolytic enzyme. We also delineate via chemical kinetics, three mechanisms by which one may inhibit angiogenesis (inhibition of growth factor, growth factor receptor and protease function).