### Abstract

The spatial relationships between molecules can be quantified in terms of information. In the case of membranes, the spatial organization of molecules in a bilayer is closely related to biophysically and biologically important properties. Here, we present an approach to computing spatial information based on Fourier coefficient distributions. The Fourier transform (FT) of an image contains a complete description of the image, and the values of the FT coefficients are uniquely associated with that image. For an image where the distribution of pixels is uncorrelated, the FT coefficients are normally distributed and uncorrelated. Further, the probability distribution for the FT coefficients of such an image can readily be obtained by Parseval's theorem. We take advantage of these properties to compute the spatial information in an image by determining the probability of each coefficient (both real and imaginary parts) in the FT, then using the Shannon formalism to calculate information. By using the probability distribution obtained from Parseval's theorem, an effective distance from the uncorrelated or most uncertain case is obtained. The resulting quantity is an information computed in k-space (kSI). This approach provides a robust, facile and highly flexible framework for quantifying spatial information in images and other types of data (of arbitrary dimensions). The kSI metric is tested on a 2D Ising model, frequently used as a model for lipid bilayer; and the temperature-dependent phase transition is accurately determined from the spatial information in configurations of the system.

Original language | English (US) |
---|---|

Pages (from-to) | 59-68 |

Number of pages | 10 |

Journal | Journal of Membrane Biology |

Volume | 241 |

Issue number | 2 |

DOIs | |

State | Published - May 2011 |

### Fingerprint

### Keywords

- Information theory
- Ising model
- Membrane
- Spatial organization

### ASJC Scopus subject areas

- Biophysics
- Physiology
- Cell Biology

### Cite this

*Journal of Membrane Biology*,

*241*(2), 59-68. https://doi.org/10.1007/s00232-011-9362-x

**Computing spatial information from Fourier coefficient distributions.** / Heinz, William F.; Werbin, Jeffrey L.; Lattman, Eaton; Hoh, Jan H.

Research output: Contribution to journal › Article

*Journal of Membrane Biology*, vol. 241, no. 2, pp. 59-68. https://doi.org/10.1007/s00232-011-9362-x

}

TY - JOUR

T1 - Computing spatial information from Fourier coefficient distributions

AU - Heinz, William F.

AU - Werbin, Jeffrey L.

AU - Lattman, Eaton

AU - Hoh, Jan H.

PY - 2011/5

Y1 - 2011/5

N2 - The spatial relationships between molecules can be quantified in terms of information. In the case of membranes, the spatial organization of molecules in a bilayer is closely related to biophysically and biologically important properties. Here, we present an approach to computing spatial information based on Fourier coefficient distributions. The Fourier transform (FT) of an image contains a complete description of the image, and the values of the FT coefficients are uniquely associated with that image. For an image where the distribution of pixels is uncorrelated, the FT coefficients are normally distributed and uncorrelated. Further, the probability distribution for the FT coefficients of such an image can readily be obtained by Parseval's theorem. We take advantage of these properties to compute the spatial information in an image by determining the probability of each coefficient (both real and imaginary parts) in the FT, then using the Shannon formalism to calculate information. By using the probability distribution obtained from Parseval's theorem, an effective distance from the uncorrelated or most uncertain case is obtained. The resulting quantity is an information computed in k-space (kSI). This approach provides a robust, facile and highly flexible framework for quantifying spatial information in images and other types of data (of arbitrary dimensions). The kSI metric is tested on a 2D Ising model, frequently used as a model for lipid bilayer; and the temperature-dependent phase transition is accurately determined from the spatial information in configurations of the system.

AB - The spatial relationships between molecules can be quantified in terms of information. In the case of membranes, the spatial organization of molecules in a bilayer is closely related to biophysically and biologically important properties. Here, we present an approach to computing spatial information based on Fourier coefficient distributions. The Fourier transform (FT) of an image contains a complete description of the image, and the values of the FT coefficients are uniquely associated with that image. For an image where the distribution of pixels is uncorrelated, the FT coefficients are normally distributed and uncorrelated. Further, the probability distribution for the FT coefficients of such an image can readily be obtained by Parseval's theorem. We take advantage of these properties to compute the spatial information in an image by determining the probability of each coefficient (both real and imaginary parts) in the FT, then using the Shannon formalism to calculate information. By using the probability distribution obtained from Parseval's theorem, an effective distance from the uncorrelated or most uncertain case is obtained. The resulting quantity is an information computed in k-space (kSI). This approach provides a robust, facile and highly flexible framework for quantifying spatial information in images and other types of data (of arbitrary dimensions). The kSI metric is tested on a 2D Ising model, frequently used as a model for lipid bilayer; and the temperature-dependent phase transition is accurately determined from the spatial information in configurations of the system.

KW - Information theory

KW - Ising model

KW - Membrane

KW - Spatial organization

UR - http://www.scopus.com/inward/record.url?scp=79959701278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959701278&partnerID=8YFLogxK

U2 - 10.1007/s00232-011-9362-x

DO - 10.1007/s00232-011-9362-x

M3 - Article

C2 - 21544539

AN - SCOPUS:79959701278

VL - 241

SP - 59

EP - 68

JO - Journal of Membrane Biology

JF - Journal of Membrane Biology

SN - 0022-2631

IS - 2

ER -