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Micropatterns of an Extracellular Matrix Protein with Defined Information Content Jeffrey L. Werbin,† William F. Heinz,† Lewis H. Romer,‡ and Jan H. Hoh†,* Departments of Physiology and Anesthesiology, Johns Hopkins School of Medicine, Baltimore, Maryland ReceiVed May 31, 2007. In Final Form: August 20, 2007 One powerful approach to understanding how cells process spatially variant signals is based on using micropatterned substrates to control the distribution of signaling molecules. However, quantifying spatially complex signals requires an appropriate metric. Here we propose that the Shannon information theory formalism provides a robust and useful way to quantify the organization of proteins in micropatterned systems. To demonstrate the use of informational entropy as a metric, we produced patterns of lines of fibronectin with varying information content. Fibroblasts grown on these patterns were sensitive to very small changes in informational entropy (6.6 bits), and the responses depended on the scale of the pattern.
In multicellular organisms, cells have highly specialized functions that depend on signals from their local microenvironment. Understanding the relationship between these signals in the cellular responses requires the ability to quantify the incoming signals. In many simple cases, there are direct ways of doing so, such as measuring the concentration of a ligand or the steepness of a gradient. However, important signals are often compositionally and spatially complex. For example, in a developing nervous system axonal growth is directed by the composition and distribution of molecules in the extracellular environment.1 Likewise, the migration of cancer cells involves responses to the organization of specific molecules in the surroundings.2 In Vitro, the importance of the spatial organization of extracellular matrix (ECM) proteins has been demonstrated using micropatterning approaches. For example, the spatial distribution of fibronectin (FN) directs lamellipodial extension3 and also controls the position of cellular organelles.4 Studies of how cells interpret and respond to specific protein distributions would benefit from a broadly applicable metric for quantifying spatially complex signals. Here we develop an information theory approach for quantifying spatial information in 2D patterns of an ECM protein and demonstrate that this approach can be used to measure information-dependent responses of cells interacting with spatially variant patterns. Shannon’s information theory provides a well-tested and robust formalism for measuring the amount of information in a “message” or a signal.5,6 There is longstanding interest in using information theory in the biological sciences.7,8 In a few areas such as sequence analysis, information theory has found notable applications;9 however, more broadly its use has been limited. Information can be classified into observer-independent informa* Corresponding author. E-mail:
[email protected]. Tel: 410-614-3795. † Department of Physiology. ‡ Department of Anesthesiology. (1) Plachez, C.; Richards, L. J. Curr. Top. DeV. Biol. 2005, 69, 267-346. (2) Yamaguchi, H.; Wyckoff, J.; Condeelis, J. Curr. Opin. Cell. Biol. 2005, 17, 559-564. (3) Brock, A.; Chang, E.; Ho, C. C.; LeDuc, P.; Jiang, X.; Whitesides, G. M.; Ingber, D. E. Langmuir 2003, 19, 1611-1617. (4) Thery, M.; Racine, V.; Piel, M.; Pepin, A.; Dimitrov, A.; Chen, Y.; Sibarita, J. B.; Bornens. M. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19771-19776. (5) Shannon, C. E. Bell Syst. Tech. J. 1948, 27, 379-423. (6) Applebaum, D. Probability and Information : An Integrated Approach; Cambridge University Press: New York, 1996. (7) Essays on the Use of Information Theory in Biology; Quastler, H., Ed; University of Illinois Press: Urbana, IL, 1953. (8) Gatenby, R. A.; Frieden, B. R. Bull. Math. Biol. 2007, 69, 635-657. (9) Schneider, T. D. Appl. Bioinf. 2002, 1, 111-119.
tion, which is the amount of information in the absence of an observer, and observer-dependent information. The latter is the subset of observer-independent information that is “meaningful” to the observer, such a predetermined code (e.g., language for humans or cell-type-specific signaling molecules). Shannon’s theory establishes a metric for quantifying the total information inherent in a signal. This observer-independent information is quantified by the probability of the specific signal occurring out of all possible signals. Shannon showed that the information in a signal is
Isignal ) -K logb(Psignal)
(1)
where I is the information, K and b are arbitrary constants that set the units of the information (for K ) 1 and b ) 2, the units are bits), and P is the probability of the signal (out of all possible signals). The weighted average of information is referred to as informational entropy (H), where
H)
∑j PjIj ) -K ∑j Pj logb Pj
(2)
The entropy of a set of signals gives a quantitative measure of structure within the ensemble of signals. The entropy is maximized when all outcomes are equally likely and minimized (H ) 0) when a single outcome is certain. In this work, we adapt Shannon’s information theory to quantify the information in 2D patterns of the ECM protein fibronectin. Because cells are known to respond to geometric features of ECM proteins by assuming the shape of the features,3,4 our approach was to produce substrates with controlled distributions of fibronectin with specified informational entropy. We made substrates with patterns that vary from well-defined lines to randomly distributed spots by adjusting the amount of informational entropy in one axis. A microfluidic patterning tool that can place spots of fibronectin of ∼1-10 µm diameter on a substrate in user-defined patterns was employed.10 Cells were then grown on these substrates, and the response of cells to specific patterns was evaluated. The patterns were created using a simple Monte Carlo-based approach. Spots (N ) 999) were placed onto an 800 point × 800 (10) Xu, J.; Lynch, M.; Huff, J. L.; Mosher, C.; Vengasandra, S.; Ding, G.; Henderson. E. Biomed. MicrodeV. 2004, 6, 117-123.
10.1021/la701605s CCC: $37.00 © 2007 American Chemical Society Published on Web 09/22/2007
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Figure 1. Computed line patterns with varying amounts of informational entropy. (A) A pattern with a strong biasing potential produces a distribution of spots that form well-defined lines. (B) A somewhat weaker biasing potential produces perceptibly less-well-defined lines. (C) The absence of a biasing potential produces a completely random “gas” of spots. (D) Graph of informational entropy as a function of the bias potential weighting factor, σ. The theoretical relationship is calculated from the normalized biasing potential (b). The entropy in the computed patterns (9) is obtained by summing over at least 80 000 patterns for each point, thus creating an effective potential. The effective potential takes into account the area exclusion step in generating the patterns. The computed curve reaches a minimum because at the highest biasing (lowest σ) crowding prevents additional spots in the lines and they are forced into the space between the lines.
point grid with a 1D biasing potential such that the distributions of spots varied from high entropy to low entropy (Figure 1). Because of technical patterning limitations, each point that was placed had to be treated as a hard circle with an eight-point radius. The resulting exclusion area placed a limit on the range of informational entropy that could be examined; the lower limit was 9.3878 kbits, and the upper limit was 9.6276 kbits. Patterns of fibronectin type III repeats 7-10, which contain the cellbinding domain,11 were produced using a microfluidic patterning tool (Nano eNabler, BioForce Nanosciences).10 Swiss 3T3 cells were plated on the patterns and were allowed to spread for 2 h before they were fixed and stained for the fibronectin fragment (monoclonal antibody), focal adhesions (vinculin antibody), and cell membrane (DiI). Cell shape relative to the pattern was established by immunofluorescence microscopy. Cellular re(11) Leahy, D. J.; Aukhil, I.; Erickson, H. P. Cell 1996, 84, 155-164.
sponses were quantified by determining the extent to which the cell elongated along the axis of the lines. The aspect ratio of the smallest possible rectangle, with one side parallel to the lines in the pattern, that contained the entire cell was used as a simple, robust metric. For a spatial signal of the type used here, the length scale of the signal is critical to the cell. For example, if the distance between spots is much larger than the cell dimensions, then the cell cannot see the information in the pattern. Conversely, if the distance between spots is too small, then the information will become homogeneous relative to the dimensions of the cell. Thus for any given system, there is a length scale at which cells can detect and respond to information in a pattern. Swiss 3T3 cells grown on flat substrates have lateral dimensions of 65 ( 10 µm, and one would expect the relevant length scale to be smaller than this. We examined line patterns with two average interspot distances, 25 and 15 µm (minimum distances of 16 and
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Figure 2. Fluorescence micrographs of Swiss 3T3 cells on FN patterns with informational entropies of (A) 9.3878 and (B) 9.6275 kbits. The rectangles around the cells are used to evaluate the cellular shape response. (C and D) Formation of focal adhesions on the 25 and 15 µm FN patterns, respectively, using vinculin as a marker. Focal adhesions form on FN spots as indicated by the arrows. FN appears in green, and vinculin appears in red. The scale bars represents 35 um.
9.6 µm, respectively). These variations in spacing were achieved by simply scaling the line patterns. By scaling a pattern with defined informational entropy to two different length scales, two patterns with different average entropy densities were produced. The results show that the cells clearly respond to the fibronectin spots on the substrate, typically forming well-defined focal adhesions with individual spots and adopting shapes that are influenced by the underlying pattern (Figure 2). An analysis of the cell shape as a function of informational entropy in the pattern for the 25 µm spacing shows no tendency to align with the pattern. However, at 15 µm spacing there is a clear response of cell shape to changing information. The cells changed from an aspect ratio of ∼1 to ∼1.3 across a change of 240 bits of informational entropy or an entropy density change of 0.9 mbits/ µm2 (Figure 3). The smallest increment over which there is a statistically significant change in cell shape is 6.6 bits (from H ) 9.6210 to 9.6276 kbits). This corresponds to an information density change of 0.03 mbits/µm2. These results yield two important findings: that cells can recognize changes in informational entropy in 2D patterns and that the cellular response is highly sensitive to informational entropy. Despite an appreciable amount of noise in the measurements, a change of 6.6 bits of informational entropy is enough to induce a significant response in the cells. This change is especially small, considering that it is approximately 1/1000 of the maximum entropy of 9.6276 kbits of the system. It should be noted that we have not explored the full range of informational entropy of the system, and thus we do not know where the most information-sensitive region lies. Furthermore, here we have not attempted to determine the molecular systems responsible for the differential response to the patterns. The spatial information in a pattern can be changed either by varying the local density of isotropic “pattern elements”, thereby making the pattern anisotropic, or by making the pattern elements themselves anisotropic. Here we have used circular spots that are roughly isotropic as pattern elements, and the density of spots therefore varies with the information. Thus, one explanation for the change in shape is that the cells are trying to maximize the number of focal contacts. However, the microfluidic patterning approach used does not allow anisotropic features of sufficiently small dimensions to be produced, thus the role of focal adhesion density will have to be examined using other methods.
Figure 3. (A) Cell aspect ratios versus informational entropy, in kilobits, for 15 µm (9) and 25 µm patterns (b). There is no statistically significant geometric response in the 25 µm pattern. The cells responded geometrically to the informational entropy in the 15 µm patterns. The change in aspect ratio becomes statistically different from disorder (H ) 9.6276 kbits) at H ) 9.6210 kbits. Note that at the highest informational entropy the average aspect ratio is less than 1, which indicates that the cells spread to a greater extent perpendicular to the lines than along the lines. However, the data is noisy, and these values are not statistically different from 1. (B) Plot of aspect ratio versus informational entropy density. The extremes in informational entropy densities are 41.8 and 40.7 mbits/µm2 for the 15 µm patterns and 15.0 and 14.7 mbits/µm2 for the 25 µm patterns. A significant response to changes in information density is seen above 40 mbits/µm2 but not below 15 mbits/µm2, which suggests that the information entropy density threshold for the 3T3 cells lies in this range.
The other finding is, as expected, that the cell’s response depends on the scale at which the signal is delivered. In the case of a 2D pattern, the information is distributed over an area, and hence variations in dimensions (scale) appear as differences in the density of informational entropy (Figure 3b). Here, the lower density shows no significant response, but the higher density does, suggesting that these entropy densities lie close to the limit at which these cells can detect a difference in entropy. The consideration of scale, in our case the area or density, is analogous to Shannon’s consideration of time in communication
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systems5 and is akin to the idea of the Nyquist frequency. Scale is ultimately related to the observer dependence of the information because the scale can determine if the signal can be detected. The approach here opens the door to quantifying almost any 2D pattern of biomolecules. This in turn will allow functional responses, ranging from simple functions such as the secretion of a protein to complex outputs such as changes in microarray profiles or directional migration, to be examined as a function of the amount of information in the pattern. The approach is also sufficiently general that it holds in many other settings, including multicomponent 3D structures. On a broader level, reducing a biological system to a mathematical representation, such as a transfer function, offers a powerful approach to simplifying descriptions of biological interactions.12 Treating an entire cell as a transfer function, or a state function, offers the possibility of examining higher-level biological functions that would (12) Sakamoto, N. Biosystems 1987, 20, 317-327.
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otherwise be intractable. By using information theory to quantify the input and measuring any one of hundreds or thousands different functional outputs, one can attempt to derive the cellular transfer function. This in turn will provide a new type of understanding of how cells work and may suggest novel approaches to reverse engineer cellular processes. Acknowledgment. We thank Dr. David Haviland for interesting and helpful discussions, Dr. Harold Erikson for the gift of the 7-10 fibronectin construct and the associated antibody and Ryan Bloom for technical assistance. Supporting Information Available: The supplemental document contains detailed descriptions of how the patterns were computationally generated, made, and analyzed, along with the results of the pairwise Student’s t tests that were used to draw some of the conclusions of this work. This material is available free of charge via the Internet at http://pubs.acs.org. LA701605S
