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Positional information, in bits

Julien O. Dubuis, Gasper Tkacik, Eric F. Wieschaus, Thomas Gregor, William Bialek

arXiv:1201.0198v1q-bio.MN

TL;DR

Embryonic cells lack a direct positional measurement, so the paper quantifies how gene-expression levels encode position in the Drosophila embryo. Using gap-gene expression and information-theoretic analysis, it finds nearly two bits per gene and approximately 1% positional precision from four genes. This precision matches later pattern reproducibility and is nearly enough to distinguish every cell uniquely.

  • Problem

    The central problem is determining how much positional information noisy gene-expression levels represent and whether it is sufficient for cell-by-cell pattern specification.

  • Method

    The paper measures mutual information between gap-gene expression levels and anterior–posterior position using immunofluorescence profiles across embryos.

  • Results

    2.26 ± 0.04 bits for Hunchback and nearly two bits for each measured single gap gene show that intermediate expression levels carry substantial positional information.

  • Takeaways & Limitations

    Four gap genes provide approximately 1% positional precision, matching the reproducibility of later pattern elements and approaching unique cell identities.

  • Takeaways & Limitations

    The measured information is not quite sufficient to identify every cell uniquely because distribution tails create small overlaps, although correlations among errors could contain missing information.

Abstract

from arXiv · show

Cells in a developing embryo have no direct way of "measuring" their physical position. Through a variety of processes, however, the expression levels of multiple genes come to be correlated with position, and these expression levels thus form a code for "positional information." We show how to measure this information, in bits, using the gap genes in the Drosophila embryo as an example. Individual genes carry nearly two bits of information, twice as much as expected if the expression patterns consisted only of on/off domains separated by sharp boundaries. Taken together, four gap genes carry enough information to define a cell's location with an error bar of ~1% along the anterior-posterior axis of the embryo. This precision is nearly enough for each cell to have a unique identity, which is the maximum information the system can use, and is nearly constant along the length of the embryo. We argue that this constancy is a signature of optimality in the transmission of information from primary morphogen inputs to the output of the gap gene network.

I. INTRODUCTION

Embryonic cells must adopt fates appropriate to position, while gene-expression levels provide a readable positional blueprint along the Drosophila anterior–posterior axis. The paper asks quantitatively how much positional information gap genes represent and how noise limits its transmission.

  • Gene-expression levels vary systematically along the Drosophila body axis, forming an approximate blueprint for the segmented larval body.
  • The key quantitative question is whether broad, smooth gap-gene profiles can specify development cell by cell along the anterior–posterior axis.
  • Low concentrations and copy numbers of regulatory molecules introduce expression noise that can limit information transmission.

II. QUANTIFYING INFORMATION

The paper formalizes positional information as the reduction in uncertainty about cell position after observing gene expression. Mutual information measures this reduction in bits while accounting for expression range and noise.

  • Observing expression level g narrows the probability distribution of position x from the prior P_x(x) to the conditional P(x|g).
  • The information supplied by g is the entropy reduction ΔS = S_i − S_f between prior positional uncertainty and conditional uncertainty.
  • Mutual information is symmetric: information expression provides about position equals information position provides about expression.
  • The measurable information is constrained by overall expression-level dynamic range and variability at fixed position.
  • Immunofluorescence measurements across embryos estimate P(g|x), enabling direct information measurement and decoding-based position estimates.
  • The information measure is independent of whether spatial patterns arise from primary morphogens or communication between neighboring cells.

III. INFORMATION CARRIED BY SINGLE GAP GENES

Single gap genes carry nearly two bits of positional information, substantially exceeding the one-bit limit of idealized on/off domains. Intermediate expression levels therefore provide reproducible positional information beyond sharp boundaries.

  • 2.26 ± 0.04 bits is the positional information carried by Hunchback across the middle 80% of the anterior–posterior axis.
  • 1.95 ± 0.07, 1.84 ± 0.05, and 1.75 ± 0.05 bits are carried by Krüppel, Giant, and Knirps, respectively.
  • The measurements have very small statistical errors after potential systematic errors from finite sampling are controlled.
  • Each single gap gene carries nearly two bits, whereas a perfect on/off pattern with an infinitely sharp boundary could provide at most one bit.
  • Reproducible intermediate expression levels carry substantial positional information that a domains-and-boundaries description misses.

IV. HOW MUCH INFORMATION DOES THE EMBRYO USE?

The paper tests whether embryonic pattern elements are positioned with approximately single-cell precision along the anterior–posterior axis. Pair-rule elements provide a direct reproducibility test against the information required to distinguish nuclei.

  • Four idealized on/off genes could encode at most four bits, or 16 reliably distinguishable states, under favorable domain alignment.
  • 5.9 ± 0.1 bits corresponds to the information required to distinguish nuclei individually across the middle 80% of the embryo.
  • The cephalic furrow is positioned reproducibly with approximately 1% accuracy along the anterior–posterior axis.
  • Seven pair-rule peaks and six troughs are used to ask whether approximately 1% positioning precision applies beyond the cephalic furrow.
  • Pattern-element positions are reproducible within 1% of embryo length, strongly suggesting that cells know their anterior–posterior positions with approximately 1% precision.

V. DECODING THE POSITIONAL INFORMATION CARRIED BY MULTIPLE GENES

The four gap genes jointly encode embryonic position by combining spatial expression profiles with their variability and correlations. Their combined positional error is nearly constant at about 1% of egg length, and the information estimates agree.

  • Encoding position: The joint expression distribution of four genes incorporates both individual variances and gene-to-gene covariances through the matrix Cij(x).Simultaneous or pairwise staining experiments provide the experimentally accessible covariance elements.
  • Encoding position: Positional error σx decreases with lower expression variability, steeper mean spatial profiles, and contributions from more genes.Equation (10) defines the precision of position estimates from the observed expression levels.
  • Measurements: 100 points along the axis are used to summarize Hunchback-based positional estimates, while four-gene estimates are shown with bootstrap error bars.The four-gene estimate is compared with lighter individual-gene estimates across position.
  • Measurement considerations: Modest spectral crosstalk does not change the estimate of σx, although avoiding crosstalk is the major difficulty in quadruple staining.The imaging protocol compares sample and control embryos to estimate channel crosstalk.
  • Measurements: ∼1% is the nearly constant positional error obtained from the four gap genes together across the embryo.The result is consistent with the observed reproducibility of pattern-element positioning.
  • Information estimate: 4.57 ± 0.02 bits from the positional-error approximation agrees with 4.97 ± 0.23 bits from direct expression distributions.Agreement between the two estimates supports the approximations used to characterize positional encoding with σx.

VI. A SIGNATURE OF OPTIMIZATION?

The paper considers whether the gap-gene network efficiently transmits positional information from a primary morphogen despite molecular noise. An optimization argument predicts constant positional uncertainty, matching the observed nearly uniform accuracy along the embryo.

  • Motivation: Molecular copy-number limitations constrain information capacity, but cells may use that capacity more efficiently through input-output matching.The relevant noise sources include low transcription-factor concentrations and small output-protein copy numbers.
  • Input-output model: The four gap-gene levels can be treated as encoding a primary morphogen concentration c, with decoding accuracy σeff_c(c).This reframes the network as a communication channel from morphogen input to gene-expression output.
  • Optimization principle: Information-transmission optimization predicts that input symbols should be distributed in inverse proportion to their variability.The prediction applies when noise levels are small.
  • Optimization principle: For a morphogen with c = c(x) and uniformly distributed cells, the input concentration distribution is constrained by its mapping to position.Uniform cell density gives P(x) = 1/L along the embryo.
  • Prediction: Constant positional uncertainty σx(x) follows when the morphogen distribution required by optimal transmission is combined with its positional encoding.The prediction agrees with the nearly constant positional accuracy observed in Fig. 4.

VII. DISCUSSION

Four gap genes provide about 1% positional precision along the embryo’s anterior–posterior axis, matching the reproducibility of later pattern elements. This precision is nearly uniform and may reflect optimal information transmission despite molecular noise.

  • VII. DISCUSSION: ∼1% positional precision from four gap genes matches the observed reproducibility of anterior–posterior pattern elements.The authors argue this is strong evidence that the gap genes carry enough information to specify the full pattern.
  • VII. DISCUSSION: The same ∼1% precision is available for later pair-rule stripes and the cephalic furrow through local gap-gene readout.
  • VII. DISCUSSION: The observed information locates nuclei with error bars smaller than neighbor spacing, but not quite enough to assign every cell a unique identity.Tail overlap may explain the gap, although correlations among neighboring-cell errors could contain missing information.
  • VII. DISCUSSION: Nearly uniform positional precision is consistent with an optimal network because cells are almost uniformly distributed along the embryo.Noise limits information, but matching input-signal distributions to network reliability can improve transmission.

APPENDIX

The appendix derives information transmission from an input morphogen concentration to multiple noisy gap-gene outputs. Under small, approximately Gaussian noise, optimal transmission uses input symbols in proportion to their reliability.

  • APPENDIX: The framework considers an input concentration c, multiple output genes g1 through gK, and an input distribution Pin(c).Different embryo cells experience different input concentrations according to position.
  • APPENDIX: Information flowing from input to output is expressed as a difference of entropies involving the posterior input distribution.
  • APPENDIX: The transmitted information depends on both the gene network’s conditional output distribution and the distribution of input signals.Finite molecule numbers contribute irreducible noise through P({gi}|c).
  • APPENDIX: With approximately Gaussian noise, the effective input uncertainty σeff depends on the actual input value.
  • APPENDIX: Optimal transmission uses input symbols in proportion to their reliability, matching the input distribution to network noise characteristics.
  • APPENDIX: The noise magnitude can be summarized by σx, which is experimentally nearly constant and smaller than the spatial scale of single-gene expression changes.
  • APPENDIX: The derivation assumes noise is small relative to expression changes driven by varying inputs, allowing outputs to be approximated by their mean responses.The authors state that the measured effective noise is small enough to justify this approximation.

METHODS

The study fixed, stained, imaged, and computationally profiled embryos to quantify gap-gene expression along embryonic axes. Embryo age was calibrated from cellularization, and mutual information was computed from gene-expression and position distributions.

  • METHODS: Embryos were collected at 25 C, bleach-dechorionated, heat-fixed, methanol-treated, and fluorescently labeled with antibodies against four gap-gene products.
  • METHODS: Confocal imaging acquired high-resolution fluorescence images along the anteroposterior axis, followed by Matlab-based image analysis.
  • METHODS: Expression profiles were extracted by sliding a nucleus-sized disk along the embryo edge and averaging pixel intensity, with coordinates projected onto embryonic axes.
  • METHODS: Embryo age was inferred from dorsal cellularization-membrane length using a wildtype calibration, with an age-estimation error of ±3 min.
  • METHODS: Mutual information between expression level g and position x was estimated from multiple embryos as samples of the joint distribution P(g, x), after discretizing both continuous axes.
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