Fractional response analysis

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Fractional Response Analysis (FRA) is an approach in biological studies for interpreting multivariate, high-throughput single-cell signaling responses . It deviates from traditional methods that rely on mean or median dose-response curves, which often fail to capture the complexity of cellular behaviors. Instead, FRA focuses on quantifying changes in fractions of cells exhibiting certain levels of response, accommodating the heterogeneity and dynamics inherent in single-cell data.

Overview

Cellular signaling systems are notoriously complex, with responses varying significantly between seemingly identical cells and involving multiple signaling effectors. Traditional analysis methods using mean or median dose-response curves oversimplify this reality, often leading to incomplete interpretations.

FRA addresses these challenges by incorporating probabilistic modeling and information theory. It quantifies dose-responses based on observable changes in the fractions of cells with a particular response level rather than averaging responses across the entire cell population.

Fractional Response Curves

Central to FRA is the concept of the Fractional Response Curve (FRC), which quantifies how cellular response fractions vary with changes in experimental conditions, such as stimulant dosage. The FRC reflects the diversity of cellular responses within the population, providing insights into how these responses shift when conditions change.

To construct an FRC, researchers first determine the fraction of cells exhibiting each possible response under a single condition. Each subsequent condition (e.g., an increased dose) is then analyzed to see what fraction of the cell population exhibits a different response compared to the previous condition. By stacking these differential response fractions cumulatively, researchers can build a composite picture of how the cellular community's behavior evolves with changing conditions.

Formal definition of the FRC

Consider a series of doses $(x_1,\dots ,x_i,\dots ,x_m)$ and denote a single cell response as $y$. Depending on the context, $(y)$ may be a number or a vector, e.g. the level of one or more measured signaling effectors. Suppose that responses to a given dose, $x_i$, are represented as the probability distribution:

$$P(Y|x_i)$$

The FRC is then formally defined as:

$$r(x_i)=\underset{Y}{\int }\underset{x_k\le x_i}{\max }P(y|x_k)dy$$

where integration takes place over $Y$, the set of all possible responses, $y$. The integral quantifies the area under the curve (or under surface for multivariate data), with respect to $y$, defined as $\underset{x_k\le x_i}{\max }P(y|x_k)$.

The concept is closely related to Rényi min-information.

Applications and Insights

FRA's ability to decipher the behaviors of heterogeneous cellular populations has made significant revelations in biological studies. For example, research on cellular responses to type I interferon in human peripheral blood mononuclear cells showed that fractional responses were surprisingly consistent across different cell types, despite substantial variations in mean or median responses.

Another insight from FRA is the linear scaling of fractional responses with the logarithm of cytokine dose, suggesting that heterogeneous cellular populations are more attuned to fold-changes in dosage levels rather than absolute changes. This finding has profound implications for understanding cellular sensitivity and response mechanisms.

References

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