Conceptualizing a tool to optimize therapy based on dynamic heterogeneity - PubMed (original) (raw)

Conceptualizing a tool to optimize therapy based on dynamic heterogeneity

David Liao et al. Phys Biol. 2012 Dec.

Abstract

Complex biological systems often display a randomness paralleled in processes studied in fundamental physics. This simple stochasticity emerges owing to the complexity of the system and underlies a fundamental aspect of biology called phenotypic stochasticity. Ongoing stochastic fluctuations in phenotype at the single-unit level can contribute to two emergent population phenotypes. Phenotypic stochasticity not only generates heterogeneity within a cell population, but also allows reversible transitions back and forth between multiple states. This phenotypic interconversion tends to restore a population to a previous composition after that population has been depleted of specific members. We call this tendency homeostatic heterogeneity. These concepts of dynamic heterogeneity can be applied to populations composed of molecules, cells, individuals, etc. Here we discuss the concept that phenotypic stochasticity both underlies the generation of heterogeneity within a cell population and can be used to control population composition, contributing, in particular, to both the ongoing emergence of drug resistance and an opportunity for depleting drug-resistant cells. Using notions of both 'large' and 'small' numbers of biomolecular components, we rationalize our use of Markov processes to model the generation and eradication of drug-resistant cells. Using these insights, we have developed a graphical tool, called a metronomogram, that we propose will allow us to optimize dosing frequencies and total course durations for clinical benefit.

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Figures

Figure 1

Figure 1

Fundamental origins of stochasticity and the contribution of “small” numbers to the magnitude of noise. (a) A system of components oscillating individually and deterministically between conditions periodically can sample a variety of waiting times between events. (b) and (c) Small changes in the initial position of the ribosome explosively accumulate in its ensuing trajectory as molecular constituents within a cell ricochet off each other, quickly leading to qualitatively distinct outcomes, i.e. protein production in (b) and the absence of protein production in (c). (d), (e), and (f) The effect of small numbers. The proportion of outcomes that deviate from average behavior decreases as the number of independently fluctuating parts increases. (g), (h), (i), and (j) Central limit theorem and “ N” rule.

Figure 2

Figure 2

Stochastic fluctuations in mRNA and protein level manifest as phenotypic interconversion at the single-cell level, which generates heterogeneity and a tendency to restore homeostatic heterogeneity at the population level. (a) and (b) Local interconversion exploring a graded phenotypic distribution. (c) Rate coefficients for a minimal Markov model with simplified drug-sensitive and drug-resistant compartments. Drug-sensitive cells replicate with rate coefficient rS, are cleared with rate coefficient mS, and convert to the drug-resistant phenotypic state with rate coefficient cS. Analogous rate coefficients describe the dynamics of the drug-resistant cells. (d) Microscopic description of the minimal model in (c) and equations (1) and (2): Individual cells adopt future phenotypes according to the outcomes of spins of wheels of fortune. The configurations of the roulettes depend on the cells’ immediate phenotypes, rather than their historical states. (e) Gradual approach toward steady-state mixture of drug-sensitive and drug-resistant cells from an initial population of 20 drug-sensitive cells (rS = rR = mS = mR = 0 and cS = cR = 1). (f) Gradual approach toward steady-state heterogeneous population composition from an initial population of 15 drug-resistant cells. (g) Approach toward both steady-state population composition and net expansion rate starting from a purely drug-resistant population (rS − mS = 1, rR = mR = 0, and cS = cR = 1).

Figure 3

Figure 3

Experimental examples of the generation of heterogeneity from a purified phenotype and the restoration of homeostatic heterogeneity. Levels of DHFR were measured in single cells from a mammary carcinoma cell line (MDA-MB-231) using flow cytometry. (a)–(g) Generation of heterogeneity from a purified population. (a) Parental cell population with full-width-half-maximum (FWHM) indicated by dots on the logarithmic histogram. (b) The FWHM is narrower after sorting to enrich for the marked fluorescence values (dashed lines). (c), (d), (e), and (f) Histograms with increasing FWHMs at 30h, 51h, 77h, and 95h following sort. (g) Quantification of mean fluorescence (m) and FWHM (w). Broadening FWHM is consistent with generation of phenotypes absent in purified population. (h)–(n) Histograms obtained from the same parental cell population, purified for the most intensely stained members, display a return toward parental mean fluorescence values within a day after sort. The purified population tends to restore the shape and position of the original distribution even though it is initially skewed toward the right. (n) Mean fluorescence values.

Figure 4

Figure 4

Using a metronomogram to graph the dynamics of phenotypic interconversion and expansion in a heterogeneous population of cells. (a) Dynamics of a mixture of drug-sensitive and drug-resistant subpopulations with rest periods of duration Δ_t_ between pulses of cell kill via drug exposure. Therapy continues until all targeted cells have been eradicated. (b) Each curve on a metronomogram expresses the kill fractions fS and population expansion fractions fP explored by varying the interdose rest period Δ_t_ for a particular population of cells. The dashed curve labeled cR = 1.0 corresponds to the solution of the model in Figure 2(c) with parameters rR − mR = rS − mS = 1, cR = 1, and cS = 1. The dotted curve labeled cR = 0.5 and the solid curve labeled cR = 1.5 correspond to these alternative values of cR, with the remaining coefficients unchanged.

Figure 5

Figure 5

Clinical uses of metronomograms. During therapy, metronomograms could facilitate (a) dosing frequency selection and (b) recommending changes in dosing strategy in response to changes in efficacy of therapy over time.

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