INVESTIGATION Positional Information, Positional Error, and Readout Precision in Morphogenesis: A Mathematical Framework Gasperˇ Tkacik,*ˇ ,1 Julien O. Dubuis,†,‡ Mariela D. Petkova,† and Thomas Gregor†,‡ *Institute of Science and Technology Austria, A-3400 Klosterneuburg, Austria, and †Joseph Henry Laboratories of Physics and ‡Lewis Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544 ABSTRACT The concept of positional information is central to our understanding of how cells determine their location in a multi- cellular structure and thereby their developmental fates. Nevertheless, positional information has neither been defined mathematically nor quantified in a principled way. Here we provide an information-theoretic definition in the context of developmental gene ex- pression patterns and examine the features of expression patterns that affect positional information quantitatively. We connect positional information with the concept of positional error and develop tools to directly measure information and error from experimental data. We illustrate our framework for the case of gap gene expression patterns in the early Drosophila embryo and show how information that is distributed among only four genes is sufficient to determine developmental fates with nearly single-cell resolution. Our approach can be generalized to a variety of different model systems; procedures and examples are discussed in detail. ENTRAL to the formation of multicellular organisms is mately giving rise to cell fate assignments that are very re- Cthe ability of cells with identical genetic material to producible across the embryos of the same species (Wolpert acquire distinct cell fates according to their position in a de- 2011). veloping tissue (Lawrence 1992; Kirschner and Gerhart The concept of positional information has been widely 1997). While many mechanistic details remain unsolved, used as a qualitative descriptor and has had an enormous there is a wide consensus that cells acquire knowledge about success in shaping our current understanding of spatial pat- their location by measuring local concentrations of various terning in developing organisms (Wolpert 1969; Tickle et al. form-generating molecules, called “morphogens” (Turing 1975; French et al. 1976; Driever and Nüsslein-Volhard 1952; Wolpert 1969). In most cases, these morphogens di- 1988a,b; Meinhardt 1988; Struhl et al. 1989; Reinitz et al. rectly or indirectly control the activity of other genes, often 1995; Schier and Talbot 2005; Ashe and Briscoe 2006; Jaeger coding for transcription factors, resulting in a regulatory and Reinitz 2006; Bökel and Brand 2013; Witchley et al. network whose successive layers produce refined spatial pat- 2013). Mathematically, however, positional information has terns of gene expression (von Dassow et al. 2000; Tomancak not been rigorously defined. Specific morphological features et al. 2007; Fakhouri et al. 2010; Jaeger 2011). The system- during early development have been studied in great detail atic variation in the concentrations of these morphogens with and shown to occur reproducibly across wild-type embryos position defines a chemical coordinate system, used by cells (Gierer 1991; Gregor et al. 2007a; Gregor et al. 2007b; Okabe- to determine their location (Nüsslein-Volhard 1991; St. Oho et al. 2009; Dubuis et al. 2013a; Liu et al. 2013), while Johnston and Nüsslein-Volhard 1992; Grossniklaus et al. perturbations to the morphogen system resulted in systematic 1994). Morphogens are thus said to contain “positional in- shifts of these same features (Driever and Nüsslein-Volhard formation,” which is processed by the genetic network, ulti- 1988a,b; Struhl et al. 1989; Liu et al. 2013). This established acausal—but not quantitative—link between the positional in- formation encoded in the morphogens and the resulting body Copyright © 2015 by the Genetics Society of America doi: 10.1534/genetics.114.171850 plan. Manuscript received April 22, 2014; accepted for publication October 27, 2014; In Drosophila, the body plan along the major axis of the published Early Online October 31, 2014. fl 1Corresponding author: Institute of Science and Technology Austria, Am Campus 1, future adult y is established by a hierarchical network of inter- A-3400 Klosterneuburg, Austria. E-mail: [email protected] acting genes during the first 3 hr of embryonic development Genetics, Vol. 199, 39–59 January 2015 39 (Nüsslein-Volhard and Wieschaus 1980; Akam 1987; Ingham variability across embryos impedes the ability of the pattern- 1988; Spradling 1993; Papatsenko 2009). The hierarchy is ing system to transmit positional information. composed of three layers: long-range protein gradients that This study builds on two previously published articles. In span the entire long axis of the egg (Driever and Nüsslein- Dubuis et al. (2013a), the general data acquisition and error Volhard, 1988a), gap genes expressed in broad bands (Jaeger analysis frameworks were presented with a subset of the 2011), and pair-rule genes that are expressed in a regular data analyzed here (referred to as data set A below); here seven-striped pattern (Lawrence and Johnston 1989). Posi- we analyze roughly eight times as many samples, to assess tional information is provided to the system solely via the first the experimental reproducibility of the results and perform layer, which is established from maternally supplied and tests that would be impossible with the original small sam- highly localized messenger RNAs (mRNAs) that act as protein ple. In Dubuis et al. (2013b), data set A was used to estimate sources for the maternal gradients (Nüsslein-Volhard 1991; positional information and positional error, as outlined in St. Johnston and Nüsslein-Volhard 1992; Ferrandon et al. greater detail here. The study aimed at understanding posi- 1994; Anderson 1998; Little et al. 2011). The network uses tional information in the Drosophila gap gene network, but these inputs to specify a blueprint for the segments of the the treatment of the conceptual and data analysis frame- adult fly in the form of gene expression patterns that cir- works was very cursory. In the present methodological arti- cumferentially span the embryo in the transverse direc- cle, we provide a full account of both frameworks and tion to the anterior–posterior axis. These patterns define include a number of previously unreported results, relating distinct single-nucleus wide segments with nuclei expressing to (i) the mathematical connection between positional error the downstream genes in unique and distinguishable combi- and information, (ii) different estimators of information nations (Gergen et al. 1986; Gregor et al. 2007a; Dubuis et al. from data, (iii) various data normalization techniques, (iv) com- 2013a,b). bining data from multiple experiments, and (v) the validity of It is remarkable that such precision can be achieved in various approximation schemes. We report on these techniques such a short amount of time, using only a few handfuls of in detail and benchmark them on real data to prepare our genes. Gene expression is subject to intrinsic fluctuations, approach for a straightforward generalization to other devel- which trace back to the randomness associated with regula- opmental systems. tory interactions between molecules present at low absolute copy numbers (van Kampen 2011; Tsimring 2014). Moreover, Results there is random variability not only within, but also between, Theoretical foundations embryos, for instance in the strength of the morphogen sources (Bollenbach et al. 2008). These biophysical limitations—e.g., In this section we establish the information-theoretic frame- in the number of signaling molecules, the time available for work for positional information carried by spatial patterns of morphogen readout, and the reproducibility of initial and gene expression. To develop an intuition, we start with environmental conditions—place severe constraints on the a one-dimensional toy example of a single gene, which will ability of the developmental system to generate reproduc- be generalized later to a many-gene system. We present ible gene expression patterns (Gregor et al. 2007a; Tkaˇcik scenarios where positional information is stored in different et al. 2008). qualitative features of gene expression patterns. To capture Given these constraints, how precisely can gene expression that intuition mathematically, we give a precise definition of levels encode information about position in the embryo? To positional information for one and for multiple genes. address this question rigorously, we need to be able to make Finally, we show how a quantitative formulation of posi- quantitative statements about the positional information of tional information is related to “decoding,” i.e., the ability of spatial gene expression profiles without presupposing which the nuclei to infer their position in the embryo. features of the profile (e.g., sharpness of the boundary, size of the domains, position-dependent variability, etc.) encode the Positional information in spatial gene expression profiles: information. Here we make the case that the relevant mea- Let us consider the simplest possible example where the sure for positional information is the mutual information expression of a single gene G(x) varies with position x along I—a central information-theoretic quantity (Shannon 1948)— the axis of a one-dimensional embryo. We choose units of between expression profiles of the gap genes and position length such