Mathematical Biosciences and Engineering
2019, Volume 16, Issue 5: 4477-4490.	doi: 10.3934/mbe.2019223
Research article Special Issues
Nonspecific probe binding and automatic gating in flow cytometry and fluorescence activated cell sorting (FACS)
Bhaven A. Mistry 1 , , ,
Tom Chou 1,2 , ,
Department of Biomathematics, UCLA, Los Angeles, CA, 90095-1766, USA
Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA
Received: 31 December 2018 Accepted: 09 May 2019 Published: 21 May 2019
Flow cytometry is extensively used in cell biology to differentiate cells of interest (mutants) from control cells (wild-types). For mutant cells characterized by expression of a distinct membrane surface structure, fluorescent marker probes can be designed to bind specifically to these structures while the cells are in suspension, resulting in a sufficiently high fluorescence intensity measurement by the cytometer to identify a mutant cell. However, cell membranes may have relatively weak, nonspecific binding affinity to the probes, resulting in false positive results. Furthermore, the same effect would be present on mutant cells, allowing both specific and nonspecific binding to a single cell. We derive and analyze a kinetic model of fluorescent probe binding dynamics by tracking populations of mutant and wild-type cells with differing numbers of probes bound specifically and nonspecifically. By assuming the suspension is in chemical equilibrium prior to cytometry, we use a two-species Langmuir adsorption model to analyze the confounding effects of non-specific binding on the assay. Furthermore, we analytically derive an expectation maximization method to infer an appropriate estimate of the total number of mutant cells as an alternative to existing, heuristic methods. Lastly, using our model, we propose a new method to infer physical and experimental parameters from existing protocols. Our results provide improved ways to quantitatively analyze flow cytometry data.
FACS,
flow cytometry,
automatic gating,
fluorescing antibodies,
Langmuir adsorption,
mixture model,
serial dilution
Citation: Bhaven A. Mistry, Tom Chou. Nonspecific probe binding and automatic gating in flow cytometry and fluorescence activated cell sorting (FACS)[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4477-4490. doi: 10.3934/mbe.2019223
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Bhaven A. Mistry
Tom Chou
Bhaven A. Mistry, Tom Chou. Nonspecific probe binding and automatic gating in flow cytometry and fluorescence activated cell sorting (FACS)[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4477-4490. doi: 10.3934/mbe.2019223
Figure 1. Cartoon of probe molecules binding to wild-type $ W $ and mutant $ M $ cells used in a typical flow cytometry assay. Wild-type cells are assumed to bind probes only nonspecifically while each mutant cell expresses $ I $ receptors to which probes specifically bind. The variables defining all quantities in the kinetic mass-action model analyzed in this paper are given in the table
Figure 2. Fractional occupancy $ \theta_{\mathrm{T}} $ of available binding sites as a function of the free probe concentration $ [N] $. (a) The isotherms for large $ [N] $ with $ [C_{\mathrm{T}}] = 100 $, $ [M] = 10 $, $ K_{\mathrm{s}} = 10^{-1} $, $ J = 10^3 $, $ \left < I\right > = 10 $, and $ K_{\mathrm{n}} = 10^1 $, $ 10^2 $, and $ 10^3 $. The cell membrane reaches saturation of bound probes at a rate dictated by $ K_{\mathrm{n}} $. (b) The isotherms for small $ [N] $ values with $ [C_{\mathrm{T}}] = 100 $, $ [M] = 10 $, $ K_{\mathrm{n}} = 10^{3} $, $ J = 10^3 $, $ \left < I\right > = 10 $, and $ K_{\mathrm{s}} = 10^{-1} $, $ 10^{-2} $, and $ 10^{-3} $. Due to small $ K_{\mathrm{s}} $, the occupancy reaches saturation for all available receptors quickly, then resumes the slower saturation of non-specific binding to membrane
Figure 3. Expected population densities of cells $ C_r $ with exactly $ r $ probes bound. (a) Low concentrations of free probe $ [N] = 1.2 $ with $ C_{\mathrm{T}} = 100 $, $ \left < I\right > = 10 $, $ J = 10^3 $, $ K_{\mathrm{s}} = 10^{-1} $, and $ K_{\mathrm{n}} = 10^3 $ for $ M = 10 $ and $ 90 $ cells. The density will cause clustering of wild-type cells close to $ r = 0 $ and mutants close to $ r = \left < I\right > $, though the non-specific binding allows some of the density associated with the mutants to contribute to the lower $ r $ values of $ C_r $. A clear boundary exists between the two densities and heuristic gating can partition the populations sufficiently. (b) Large concentrations of free probe $ [N] = 60 $ with $ C_{\mathrm{T}} = 100 $, $ \left < I\right > = 10 $, $ J = 10^3 $, $ K_{\mathrm{s}} = 10^{-1} $, and $ K_{\mathrm{n}} = 10^3 $. The population densities of wild-types and mutants are now found in similar values of $ r $ and overlap extensively, causing difficulty in differentiating the two clusters as probes saturate the membrane
Figure 4. (a) Simulated fluorescence data using parameters $ C_{\mathrm{T}} = 100 $, $ [N] = 1.2 $, $ K_{\mathrm{s}} = 10^{-1} $, $ K_{\mathrm{n}} = 10^{3} $, $ J = 10^3 $, $ \left < I\right > = 5 $, $ F_0 = 1 $, $ F_{\mathrm{s}} = F_{\mathrm{n}} = 10^3 $, and $ \sigma_0 = 0.5 $. We set $ M = 0.5C_{\mathrm{T}} $ and assign each cell $ k $ with a number of bound probes $ r_k $ using Eq 2.13 and subsequent fluorescence intensity $ x_k $ using Eq 2.14. At this particular parameter regime, $ 27 $ of the $ 50 $ wild-type cells managed to bind with at least one probe, increasing their relative fluorescence and clustering them with the mutants. A typical gating threshold, shown above, would separate the two clusters and count all $ 87 $ cells on the right hand side as mutants; far larger than the true count of $ 50 $. The iterative estimate using Eq 3.2 returns $ M = 51 $, relatively close to the actual count. (b) Probability that a given cell has one or more probes bound as a function of the dilution number $ d $ as we vary the receptor distribution mean $ \left < I\right > $. Here $ [N] = 1 $, $ K_{\mathrm{n}} = 10^3 $, $ K_{\mathrm{s}} = 10^{-3} $, $ J = 10^3 $, and dilution factor $ D = 10 $