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Analysis of biological time-lapse microscopic experiment from the point of view of the information theory
Micron 42 (2011) 360–365
Contents lists available at ScienceDirect
Micron journal homepage: www.elsevier.com/locate/micron
Analysis of biological time-lapse microscopic experiment from the point of view of the information theory Dalibor Sˇtys *, Jan Urban, Jan Vaneˇk, Petr Cı´sarˇ Institute of Physical Biology, University of South Bohemia, Academic and University Center, Za´mek 136, 373 33 Nove´ Hrady, Czech Republic
A R T I C L E I N F O
Article history: Received 11 December 2009 Accepted 29 January 2010 Keywords: Information entropy Cell monolayer Intracellular compartments Image analysis
Abstract: We report objective analysis of information in the microscopic image of the cell monolayer. The process of transfer of information about the cell by the microscope is analyzed in terms of the classical Shannon information transfer scheme. The information source is the biological object, the information transfer channel is the whole microscope including the camera chip. The destination is the model of biological system. The information contribution is analyzed as information carried by a point to overall information in the image. Subsequently we obtain information reflection of the biological object. This is transformed in the biological model which, in information terminology, is the destination. This, we propose, should be constructed as state transitions in individual cells modulated by information bonds between the cells. We show examples of detected cell states in multidimensional state space. This space is reflected as colour channel intensity phenomenological state space. We have also observed information bonds and show examples of them. ß 2010 Elsevier Ltd. All rights reserved.
1. Introduction Biological imaging can be, from the information point of view, split into two parts: the physical acquisition of the information and retrieval of biological function. These two distinct views correspond vaguely to distinction between engineers who construct the device and the users – medical practitioners, researchers in applied and pure research, but also for example operators of water cleaning stations using biological detection systems. Systems biology and its close companion biological engineering utilizes engineering approaches to put the biological view on a more firm ground. In this article we wish to outline theoretical grounds of these analyses. The practical part, based on engineering stochastic system theory, will be presented in subsequent article since in our opinion it requires separate discussion. This work is motivated by a highly practical consideration: a cell culture is usually available for a limited period of time. In this time we may acquire data of given quality. If we focus on obtaining maximum information about the actual cell state, we need to acquire data containing maximum information in a given time, and extract this information. This we propose to do using the methods of information theory. Theory of information transmission dates back to Nyquist (1928) and Hartley (1928) who analyzed the information transmission by telegraph lines. The most celebrated article is that of Shannon (1948) which is also among the earliest which discusses jointly the physical
* Corresponding author. Tel. +420 777729581; fax: +420 386361219. 0968-4328/$ – see front matter ß 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.micron.2010.01.012
information transmission and the mathematical theory. Shannon introduced the information measure S and due to its analogy (in fact identity) with Boltzmann entropy of thermodynamics (Boltzmann, 1866) called it information entropy. Re´nyi (1961) introduced a more general information measure but in his time the mathematical theory became too complex for the user to see its practical advantage. The information measure, Shannon entropy, was utilized blindly if at all and theoretical discussion remained the domain of statistical physics. In our opinion first this is a shame, since over the years a lot of information has been misinterpreted or not interpreted at all. Secondly now, with the availability of high performance computers, it is less a technical problem to utilize advanced mathematical concepts. Classical Shannon scheme of information transmission is given at Fig.. The general communication scheme consists of Information source (Fig. 1A), Transmitter (Fig. 1B), Noise source (Fig. 1C), Receiver (Fig. 1D) and Destination (Fig.E). The connection between (B) and (D) is the communication channel. We propose that the Information source is the observed biological object. Transmitter plus the communication channel is our microscope. Receiver is the camera chip and destination is the representation of the biological object from captured images. For the moment, we shall not discuss the noise source separately because it requires additional extensive concept building. In general, noise arises in all components of the communication scheme. Shannon considers the information source to have a certain statistical structure described by probabilities p(i, j) that a given information, for example a letter, j is followed by information i. The information channel has transmission probability Pi for informa-
[()TD$FIG]
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361
Fig. 1. Modification of the original Shannon scheme for signal transmission. We propose that the Information source is the observed biological object, transmitter plus the communication channel is our microscope, the receiver is the camera chip. These are elements which we utilize for theoretical analysis and experimental view in the article.
if we then re-write (4) for non-limit case as
tion i and we consider that Pj ¼
X
P i pði; jÞ
(1)
1
From this approach Shannon derived properties of the information measure for independent probabilities pi of events i and came to definition of information entropy. H¼K
n X
pi log pi
(2)
i¼1
which we use in the form n X
S¼
pi log 2 pi
1 log 2 pqi q1
n X
pqi
i¼1
log 2 r
which may be re-written if we understand average of piq1
i¼1
pqi ¼
n X i¼1
i¼1
¼
log 2 r
1 p1i
log 2 D¼ log 2 r
log 2
pi piq1 ¼
D
pq1 i
D E1 q1 log 2 pq1 i log 2 r
¼
n X
(7)
pi p1i
i¼1
(8)
log 2 r
1 1 log 2 p2i 2 ¼ 2 log 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X log 2 pi p2i i¼1
log 2 r
E
(5) Pn
i¼1
pqi as weighted
(6)
(9)
which is root mean square weighting of probabilities which may, in a naive approach, be viewed as average distance in a (probabilistic) plane. Equally interesting for our case we find the case q = 4 which leads to
(4)
at q ! 1. Re´nyi entropy – or information has been mathematically proven to be the most general information measure (Jizba and Arimitsu, 2004). But practical use of generalization made by Re´nyi was not recognized. For the context of this article we point out the connection between information entropy and space dimensionality summarized by Theiler (1990). Based on older derivations of Hentschel and Procaccia (1983) and Grassberg (1983) Theiler suggests that the connection between generalized or information dimension and the information entropy lies in the appropriate averaging of measure in the particular space. The derivation is as follows: We start with Re´nyi information measure (4) which he may use for definition of dimensionality as ‘‘appropriate averaging’’
n X
(3)
The original derivation has long been time forgotten and overshadowed by the finding of Re´nyi who showed that the Shannon measure (3) is only a limit case of more general information measure
1 D¼ lim q 1 r!0
pqi
and for example for q = 2 and fixed finite scaling factor r we get
D¼
log 2
n X
which is arithmetic average of probabilities. For q = 3 we get
i¼1
Iq ¼
1 D¼ q1
log 2
D¼
1 1 log 2 p3i 3 ¼ 3 log 2 r
log 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X 3 pi p2i i¼1
log 2 r
(10)
which might be called cubic average and may be imagined as average distance in the three dimensional case. For the purpose of this article we propose the following hypothesis. Let us assume that the information source is a section of 3D space. The distance between points in the 3D space is the spatial diagonal in a cube whose outer edges are Cartesian coordinates of points. Each point is transformed by the optical path to a set of points. The transformation function is called the point spread function. It is a complicated function imaging the point in the original object to a set of points at the camera chip plane. The point spread function is different for each wavelength and for each point in the observed region – focal volume. The dimension of the point spread function may be, to some extent, assessed from theoretical analysis given by Nijboer and Zernike (1949) whose contemporary form is presented by Braat et al. (2008). For the case of phase contrast microscopy demonstrated in this article the construction of point spread function is far more complicated (i.e. Masato et al., 2005). In confocal microscopy the point spread function may be calculated rather precisely for the object in focus, below and above the focus and used productively for the object reconstruction (Braat et al., 2008).
362
[()TD$FIG]
D. Sˇtys et al. / Micron 42 (2011) 360–365
All super-resolution microscopy techniques focus on transformation of the point spread function for a single point emitter/ reflector placed in the focal point of the lens (Lipson, 2003). For this purpose, the information theory based analysis was developed. The problem of ‘‘general’’ optical microscope, which transfers information from any point in the focal volume at multiple wavelengths is much more complicated, perhaps insoluble in classical analytical terms. But it is of high practical importance. We expect that for a single image a set of multifractals – cells and their interior – are transmitted by a transmission channel which also has a multifractal character, to a receiver. Receiver – camera chip – provides discrete resolution. From the view the multifractal statistic (Jizba and Arimitsu, 2004) we may assume to observe spectrum of fractal dimensions of the resulting image. In practical terms, we may observe ‘‘best resolution’’ of different types of information observed in the image when different information representation is given. Using standard samples such as nanoparticles, some degree of calibration of the system may be achieved. However, any clear-cut analysis may be done only in case of the existence of isolated points, e.g. dilute appearance of individual fluorescence proteins in the cell. The relation to the real case of living cell interior, where there is a mixture of transparent regions with significantly different refractivity indexes which change at a scale of 10–20 nm is by this analytical approach insoluble. In this article we discuss the possibilities of the calibration-free approach, global observation of detectable intracellular compartments and its dynamics. The problem in such a complicated environment is how to define the best resolution at the destination. In another words, we do not have a clear definition of the information source model – the destination. For the development of the method, we replace objective information resolution (in fact in full accordance with Shannon) by the operators opinion. He or she would decide which is the observable object in the cell interior, how long its lasts, whether it expands or shrinks. In this article, we describe the method and some elementary result using the information content calculation using Shannon entropy. Use of different Re´nyi entropy measures and the attempt to create a dynamic model of biological cell development will be discussed in a separate paper. 2. Method 2.1. Biological experiment 2.1.1. Growth conditions HeLa cells (Human Negroid cervix epitheloid carcinoma)were obtained from ECACC – European collection of cell cultures. The cells were grown at low optical density overnight in a 37 8 C temperature in a synthetic dropout media with 30% raffinose as the sole carbon source. The nutrient solution for HeLa cells consists of: 86% EMEM, 10% newborn-calf serum, 1% antibiotics and antimycotics, 1% L-glutamine, 1% non essential amino acids, 1% NaHCO3 (all components from the PAA company). 2.1.2. Microscopy 5000 original phase-contrast images of growing cells were taken at intervals of 1 minute. The image collection was performed at the temperature of 37 8 C using the Olympus IX 51 with an automated stage, integrated incubator and photographic camera Camedia C7070. The objective with 20 times magnification was used in image capture. 2.2. Information entropy contribution of an image point The information contribution of a point was performed as described earlier (Urban et al., 2009). In brief, we represent the
Fig. 2. Expansion of a region in histograms of intensity occurrences. Contribution of a point was calculated from comparison of two histograms with and without the examined point. The figure shows expansion of the histogram in the region containing the point for which the information contribution was calculated. The information contribution was calculated as Shannon entropy difference for the normalized histogram containing the point in question (occurrence of intensity 72 with gray bar) and without it (occurrence of intensity 72 with gray bar).
image at the camera screen – receiver data – by a cross whose shanks intersect in the point whose information contribution we calculate. We create a histogram of frequencies of occurrence fi of given intensity in the given colour channel (red, green, or blue) with and without the respective data point (Fig. ). Intensities were normalized to give frequency probabilities pi using the formula pi ¼
fi n X
(11) fj
j¼0
where n is number of intensity levels. Shannon entropy was obtained from probabilities using the formula (3), we denote the information with the central point S+ and that without the central point S. The contribution of one point to information in the image is
DS ¼ Sþ S
(12)
3. Results 3.1. Single point information contribution At Fig. 2 it we demonstrate the sensitivity of image entropy to intensity levels which occur in the image infrequently. Fig. 2 is expansion of a segment of combination of the two images showing the small intensity contribution. For the observed point was DS = 0.00287735 bit/pixel when DSmin = 0, 0011595 and was DSmax = 0, 0053682. Thus information value of this pixel was in the middle of the scale. 3.2. Information-based image representation Information carried by different colour channels is, as expected, significantly different. In the case of phase contrast microscopy the difference reflects deep differences in the construction of the image. The phase plate and other parts of the light path are optimized for certain wavelength in the green region. There the
[()TD$FIG]
D. Sˇtys et al. / Micron 42 (2011) 360–365
363
colour channels are clearly visible. In this one still image the differences must be assigned to the different optical path, or in another words, to different properties of the information transfer channel. As derived in the introduction, the properties of the channel have direct relation to the statistics by which the probability of information transfer is distributed. In the calculation presented in this paper, we assume the Shannon–Gibbs statistics. The image is a result of overlap of intensities resulting from a set of point spread function as the statistics are far more likely not to follow Gibbsian conditions than to follow them. Due to multifractality of the problem we may assume that there will be a spectrum of different statistics which expands the computational intensiveness of the analysis. From the point of view of generalized dimension spectrum, we may expect to have set non-integer q value by which the information space will be adequately represented. In case that calculation using only one q is feasible, the Shannon entropy is equally appropriate as any other.
Fig. 3. Composite representation of information carried in the phase contrast microscopic image of the cell. The pseudocolours represent composition of intensity information carried by the particular point in the image. Black points mean that the part of the image is information equivalent to the background.
intensity may be clearly assigned to degree of productive or destructive interference. Fig. 3 shows sections of a microscopic image transformed by image entropy calculations presented in composite form in which the pseudocolours. At Fig., section of an image depicting cell nucleus, are shown details of the image at which differences in the information content carried by different
[()TD$FIG]
3.3. Biological model As discussed in the introduction, the goal of the analysis is to obtain a model of the biological object. In another words the model of a biological system is the destination. It is a general agreement that living organisms are among the prominent examples of nonlinear dynamics (Kaneko, 2006). We should then seek major features predicted by non-linear dynamics, for example limit cycles (Cvitanovic et al., 2008). How shall these features appear in an image? What are their parameters? At Fig. 5 we show sketch of a selected dynamic feature of a cell at 0, 4, 9 and 14 minute of the experiment. Inspection of the sequence indicates that in the time of
Fig. 4. Comparison of a section of transformed image depicting cell nucleus. It is clearly seen that certain features in the highlighted part of the nucleus (probably chromosome) are significantly differently represented in the blue than in the red an green channel. This difference is a consequence of significantly different point spread function for wavelengths represented in the blue channel than those represented by other two channels. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
[()TD$FIG]
D. Sˇtys et al. / Micron 42 (2011) 360–365
364
Fig. 5. Image of a selected region from image 3 showing high dynamics. Series of image sections at 0, 4, 9 and 14 minute of the experiment depicting a highly dynamic region of the cell. The identity of the image features is not known since it represents a mixture of objects at different distances from the focal plane and at which the shape of point spread function representation is different from each other. However, its colour composition and shape dynamics may be utilized as phenomenological system variable. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
the experiment the region does not significantly expand in space in either of the information channels which we observe. That is exactly what we may expect from a limit cycle. We should thus examine the state space in which the limit cycle evolves. In case of infinite resolution, we should be able to determine complete time dynamics of a dynamic region. In most real cases this is not feasible and, in fact, also not necessary. It is sufficient to be able to identify a region, distinguish it from another and quantify its role in the symbolic dynamic of the cell. Such symbolic dynamic information should be a parameter of the biological model. The biological model we propose in analogy to equilibrium thermodynamics we may assume to have constituents and components (i.e. Atkins and De Paula, 2006). Constituents, in textbook sense, are all independent chemical individui in the system. Components are sets of constituents which change independently, for example a set of chemical compounds bound together by a chemical reaction. Already in one component system the detection of its state by a simple device, i.e. single wavelength spectrometer, it is impossible to detect any ordinary chemical equilibrium. We most need to detect at least two reactants. With more components and more phases we need more parameters. Yet, there is one parameter, the Gibbs (or Helmholtz) energy which characterizes the system unanimously. From another point of view, taken to the extreme, the idea of a component is only statistical. The activity ai, the correct representation of concentration ci, of each constituent i is connected to the chemical potential
mi ¼ mi;0 þ RTln ai
(13)
while ai = gici and gi = f(c1, c2, . . ., cn) where n is number of constituents of the system. Finally, Gibbs energy G of a phase is G¼
n X
mj
(14)
j¼1
The components is the sub-set of constituents k for which it holds that
g k f ðcr ; . . . ; cm Þ
(15)
where concentrations (cr, . . ., cm) are sub-set of set of all concentrations (c1, . . ., cn). While in very dilute solutions some approximate solutions may be found, in real life concentrated solutions the component is very difficult to define. And the cell interior is highly concentrated. From the point of view of equilibrium thermodynamics, attempt to characterize the cell interior by one constituent is hopeless. We must seek the Gibbs energy which determines the phase equilibrium. For systems obeying the rules of nonequilibrium dynamics we do not expect one Gibbs energy value for each point in the state space. The state space is segregated into regions of attraction of individual dynamic features, attractors. The attractors may be of different kind, attractors of limit cycles, chaotic attractors, strange attractors etc. In each case, there is a sub-set of the state space which is much more populated than the rest and this we observe. The internal structure of living cells invokes the idea that individual structures are those regions of attraction and their dynamics reflects the dynamics of these asymptotically stable objects (i.e. Kaneko, 2006). Such objects no longer should have the same Gibbs energy and the different chemical composition may mean different region of the state space. The problem of identity of the coordinates in the state space remains but must not necessarily be crucial. As described for example in Cvitanovic et al. (2008) but introduced in principle much earlier (i.e Poincare´, 1892–1989; Sinai, 1968), the system behavior may be analyzed as symbolic trajectory. That precisely saying means that state space is separated into regions which are assigned a symbol. The symbols in our approach would come from identification of detectable and distinguishable regions. Not all transitions between symbols are available, there is a set of trajectories which are possible, the state space is pruned. In case that the partition is performed in the way that the transitions are independent of previous parts of the trajectory, we call the partition Markovian and may build statistical function characterizing it (i.e. Sinai, 1972; Ruelle, 1978). In order to identify the symbolic trajectory, all we need is to be able to identify individual states, give them a name, and to determine the rules of transition. By counting of trajectories we may obtain their probability and even construct a characteristic
[()TD$FIG]
D. Sˇtys et al. / Micron 42 (2011) 360–365
Fig. 6. Building of intracellular bridge – cell-to-cell intersubsystem information bond.
function which is the dynamic equivalent of Gibbs energy (Cvitanovic et al., 2008). Yet, the non equilibrium dynamics is perhaps not readily available either. Individual cells would be representing the individual systems only in isolated form. In the monolayer or a culture, cells represents sub-systems of and the monolayer is a system. In our opinion it is more appropriate to use the general stochastic systems theory derived for behavior of cybernetic systems (Zˇampa and Arnosˇt, 2004). The general principle is the same, we have to identify the state. In practical terms it means to determine all measurable – phenomenological – variables in sufficient time preceding the change of state to assure that the state change is independent from any preceding history. By counting number of transitions and state lifetimes we obtain probability distribution functions for transitions between states. In the example given at Fig. 5 it means that we follow the oscillations of the observed object form their start up to the time they cease or significantly change. General stochastic systems theory is superior to non-linear dynamics in consideration of the structure of the system. It considers the system to be composed of sub-systems which in isolation would behave as independent systems, as the cells. The bond between them is then called information bond. We believe that this theory may be directly applied to analysis of cell monolayer dynamics. For cells in contact one may obviously consider information exchange. At Fig. we show example of information exchange between two cells mediated by matter transfer. 4. Conclusion Results presented in this article show the framework of the objective analysis of cell monolayer dynamics. We analyze methods of extraction of information from the microscopic image from the theoretical point of view and give example of its practical use. The use of Shannon entropy for the information content analysis is based on consideration of normal Gibbs–Boltzmann statistics of the information channel behavior. This is clearly not adequate since the information transfer by microscope has complicated, multidimensional, character. We outline methods by which the analysis might be developed to give better representation of certain features. In another words, it is possible to calculate the spectrum of information features from a
365
combination of observed object (information source), transmitter and information channel (the microscope). Next step is the analysis of information receipt at destination by which we understand the building and parameterization of a model of the observed organism, in this case of the culture of somatic cells. We compared obtained results with quantitative theories which may be utilized for description of the system dynamics. The general stochastic systems theory might be able to comprehend both the relatively independent behavior of individual cells and the cell-tocell communication. We present examples of detection of several image features which might be representations of this behavior, intracellular oscillating object (Fig.) and information bond (Fig. 6). Such detailed analysis is extremely laborious and needs to be automated. We believe that in the process of automation we shall begin to understand the system itself in much better way. The final goal is to diagnose the state of tissue culture in more objective manner. However, it is not quite clear whether the necessary theoretical apparatus is available.
Acknowledgments This work was partly supported by the Ministry of Education, Youth and Sports of the Czech Republic under the grant MSM 6007665808 and grant HCTFOOD A/CZ0046/1/0008 of EEA funds. References Atkins, P., De Paula, J., 2006. Atkins’ Physical Chemistry. Oxford University Press, March. ¨ ber die Mechanische Bedeutung des Zweiten Hauptsatzes der Boltzmann, L., 1866. U Wa¨rmetheorie. Wiener Berichte 53, 195–220. Braat, J.M., van Haver, S., Janssen, A.J.E.M., Dirksen, P., 2008. Assessment of optical systems by means of point-spread functions. In: Wolf, E. (Ed.), Progress in Optics, vol. 51. Amsterdam/The Netherlands, Elsevier. Cvitanovic, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G., 2008. Chaos: Classical and Quantum, Niels Bohr Institute, Copenhagen (ChaosBook. org/version12) . Grassberg, P., 1983. Generalized dimension of strange attractors. Phys. Lett. A97, 227–230. Hartley, J.V., 1928. Bell Syst. Tech. J. 7 . Hentschel, H.G.E., Procaccia, I., 1983. Fractal nature of turbulence as manifested in turbulent diffusion. Phys. Rev. A 27, 1266–1269. Jizba, P., Arimitsu, T., 2004. The world according to Re´nyi: thermodynamics of multifractal systems. Ann. Phys. 312, 17–595. Kaneko, K., September 2006. Life: An Introduction to Complex Systems Biology (Understanding Complex Systems). Springer. Lipson, S.G., 2003. Why is super-resolution so inefficient? Micron 34, 309–312. Masato, S., Hiroshi, O., Kimihiro, S., Suezou, N., 2005. Generalizing effective point spread function and its application to the phase-contrast microscope. Opt. Rev. 12 (2), 105–108. Nijboer, B.R.A., Zernike, F., 1949. Contribution in La The´orie des Images Optiques. E´ditions de la Revue d’Optique, Paris. Nyquist, H., 1928. Certain topics in telegraph transmission theory. Trans. AIEE 47, 617–644. Poincare´, H., 1892–1899. Les me´thodes nouvelles de la me´chanique ce´leste, Guthier-Villars, Paris (1968. English translation: Funct. Anal. Appl. 2, 245). Re´nyi, A., 1961. On measures of information and entropy. In: Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability. pp. 547–561. Ruelle, D., 1978. Statistical Mechanics, Thermodynamic Formalism. Addison-Wesley, Reading MA. Shannon, C.E., 1948. A mathematical theory of communication. Bell Syst. Tech. J. 27 379–423, 623–656, July and October. Sinai, Ya.G., 1968. ‘‘Construction of Markov partitions’’, Funkts. Analiz i Ego. Sinai, Ya.G., 1972. Gibbs measures in ergodic theory. Russ. Math. Surveys 166, 21. Theiler, J., 1990. Estimating fractal dimension. J. Opt. Soc. Am. A 7, 1055–1073. Urban, J., Vaneˇk, J., Sˇtys, D., 2009. Preprocessing of microscopy images via Shannon’s entropy. In: Proceedings of Pattern Recognition and Information Processing, Minsk, Belarus, pp. 183–187. ˇZampa, P., Arnosˇt, R., 2004. Alternative approach to continuous-time stochastic systems definition. ISBN:111-6789-99-3 In: Proceedings of the 4th WSEAS conference, Wisconsin, USA.
Contents lists available at ScienceDirect
Micron journal homepage: www.elsevier.com/locate/micron
Analysis of biological time-lapse microscopic experiment from the point of view of the information theory Dalibor Sˇtys *, Jan Urban, Jan Vaneˇk, Petr Cı´sarˇ Institute of Physical Biology, University of South Bohemia, Academic and University Center, Za´mek 136, 373 33 Nove´ Hrady, Czech Republic
A R T I C L E I N F O
Article history: Received 11 December 2009 Accepted 29 January 2010 Keywords: Information entropy Cell monolayer Intracellular compartments Image analysis
Abstract: We report objective analysis of information in the microscopic image of the cell monolayer. The process of transfer of information about the cell by the microscope is analyzed in terms of the classical Shannon information transfer scheme. The information source is the biological object, the information transfer channel is the whole microscope including the camera chip. The destination is the model of biological system. The information contribution is analyzed as information carried by a point to overall information in the image. Subsequently we obtain information reflection of the biological object. This is transformed in the biological model which, in information terminology, is the destination. This, we propose, should be constructed as state transitions in individual cells modulated by information bonds between the cells. We show examples of detected cell states in multidimensional state space. This space is reflected as colour channel intensity phenomenological state space. We have also observed information bonds and show examples of them. ß 2010 Elsevier Ltd. All rights reserved.
1. Introduction Biological imaging can be, from the information point of view, split into two parts: the physical acquisition of the information and retrieval of biological function. These two distinct views correspond vaguely to distinction between engineers who construct the device and the users – medical practitioners, researchers in applied and pure research, but also for example operators of water cleaning stations using biological detection systems. Systems biology and its close companion biological engineering utilizes engineering approaches to put the biological view on a more firm ground. In this article we wish to outline theoretical grounds of these analyses. The practical part, based on engineering stochastic system theory, will be presented in subsequent article since in our opinion it requires separate discussion. This work is motivated by a highly practical consideration: a cell culture is usually available for a limited period of time. In this time we may acquire data of given quality. If we focus on obtaining maximum information about the actual cell state, we need to acquire data containing maximum information in a given time, and extract this information. This we propose to do using the methods of information theory. Theory of information transmission dates back to Nyquist (1928) and Hartley (1928) who analyzed the information transmission by telegraph lines. The most celebrated article is that of Shannon (1948) which is also among the earliest which discusses jointly the physical
* Corresponding author. Tel. +420 777729581; fax: +420 386361219. 0968-4328/$ – see front matter ß 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.micron.2010.01.012
information transmission and the mathematical theory. Shannon introduced the information measure S and due to its analogy (in fact identity) with Boltzmann entropy of thermodynamics (Boltzmann, 1866) called it information entropy. Re´nyi (1961) introduced a more general information measure but in his time the mathematical theory became too complex for the user to see its practical advantage. The information measure, Shannon entropy, was utilized blindly if at all and theoretical discussion remained the domain of statistical physics. In our opinion first this is a shame, since over the years a lot of information has been misinterpreted or not interpreted at all. Secondly now, with the availability of high performance computers, it is less a technical problem to utilize advanced mathematical concepts. Classical Shannon scheme of information transmission is given at Fig.. The general communication scheme consists of Information source (Fig. 1A), Transmitter (Fig. 1B), Noise source (Fig. 1C), Receiver (Fig. 1D) and Destination (Fig.E). The connection between (B) and (D) is the communication channel. We propose that the Information source is the observed biological object. Transmitter plus the communication channel is our microscope. Receiver is the camera chip and destination is the representation of the biological object from captured images. For the moment, we shall not discuss the noise source separately because it requires additional extensive concept building. In general, noise arises in all components of the communication scheme. Shannon considers the information source to have a certain statistical structure described by probabilities p(i, j) that a given information, for example a letter, j is followed by information i. The information channel has transmission probability Pi for informa-
[()TD$FIG]
D. Sˇtys et al. / Micron 42 (2011) 360–365
361
Fig. 1. Modification of the original Shannon scheme for signal transmission. We propose that the Information source is the observed biological object, transmitter plus the communication channel is our microscope, the receiver is the camera chip. These are elements which we utilize for theoretical analysis and experimental view in the article.
if we then re-write (4) for non-limit case as
tion i and we consider that Pj ¼
X
P i pði; jÞ
(1)
1
From this approach Shannon derived properties of the information measure for independent probabilities pi of events i and came to definition of information entropy. H¼K
n X
pi log pi
(2)
i¼1
which we use in the form n X
S¼
pi log 2 pi
1 log 2 pqi q1
n X
pqi
i¼1
log 2 r
which may be re-written if we understand average of piq1
i¼1
pqi ¼
n X i¼1
i¼1
¼
log 2 r
1 p1i
log 2 D¼ log 2 r
log 2
pi piq1 ¼
D
pq1 i
D E1 q1 log 2 pq1 i log 2 r
¼
n X
(7)
pi p1i
i¼1
(8)
log 2 r
1 1 log 2 p2i 2 ¼ 2 log 2 r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X log 2 pi p2i i¼1
log 2 r
E
(5) Pn
i¼1
pqi as weighted
(6)
(9)
which is root mean square weighting of probabilities which may, in a naive approach, be viewed as average distance in a (probabilistic) plane. Equally interesting for our case we find the case q = 4 which leads to
(4)
at q ! 1. Re´nyi entropy – or information has been mathematically proven to be the most general information measure (Jizba and Arimitsu, 2004). But practical use of generalization made by Re´nyi was not recognized. For the context of this article we point out the connection between information entropy and space dimensionality summarized by Theiler (1990). Based on older derivations of Hentschel and Procaccia (1983) and Grassberg (1983) Theiler suggests that the connection between generalized or information dimension and the information entropy lies in the appropriate averaging of measure in the particular space. The derivation is as follows: We start with Re´nyi information measure (4) which he may use for definition of dimensionality as ‘‘appropriate averaging’’
n X
(3)
The original derivation has long been time forgotten and overshadowed by the finding of Re´nyi who showed that the Shannon measure (3) is only a limit case of more general information measure
1 D¼ lim q 1 r!0
pqi
and for example for q = 2 and fixed finite scaling factor r we get
D¼
log 2
n X
which is arithmetic average of probabilities. For q = 3 we get
i¼1
Iq ¼
1 D¼ q1
log 2
D¼
1 1 log 2 p3i 3 ¼ 3 log 2 r
log 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X 3 pi p2i i¼1
log 2 r
(10)
which might be called cubic average and may be imagined as average distance in the three dimensional case. For the purpose of this article we propose the following hypothesis. Let us assume that the information source is a section of 3D space. The distance between points in the 3D space is the spatial diagonal in a cube whose outer edges are Cartesian coordinates of points. Each point is transformed by the optical path to a set of points. The transformation function is called the point spread function. It is a complicated function imaging the point in the original object to a set of points at the camera chip plane. The point spread function is different for each wavelength and for each point in the observed region – focal volume. The dimension of the point spread function may be, to some extent, assessed from theoretical analysis given by Nijboer and Zernike (1949) whose contemporary form is presented by Braat et al. (2008). For the case of phase contrast microscopy demonstrated in this article the construction of point spread function is far more complicated (i.e. Masato et al., 2005). In confocal microscopy the point spread function may be calculated rather precisely for the object in focus, below and above the focus and used productively for the object reconstruction (Braat et al., 2008).
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All super-resolution microscopy techniques focus on transformation of the point spread function for a single point emitter/ reflector placed in the focal point of the lens (Lipson, 2003). For this purpose, the information theory based analysis was developed. The problem of ‘‘general’’ optical microscope, which transfers information from any point in the focal volume at multiple wavelengths is much more complicated, perhaps insoluble in classical analytical terms. But it is of high practical importance. We expect that for a single image a set of multifractals – cells and their interior – are transmitted by a transmission channel which also has a multifractal character, to a receiver. Receiver – camera chip – provides discrete resolution. From the view the multifractal statistic (Jizba and Arimitsu, 2004) we may assume to observe spectrum of fractal dimensions of the resulting image. In practical terms, we may observe ‘‘best resolution’’ of different types of information observed in the image when different information representation is given. Using standard samples such as nanoparticles, some degree of calibration of the system may be achieved. However, any clear-cut analysis may be done only in case of the existence of isolated points, e.g. dilute appearance of individual fluorescence proteins in the cell. The relation to the real case of living cell interior, where there is a mixture of transparent regions with significantly different refractivity indexes which change at a scale of 10–20 nm is by this analytical approach insoluble. In this article we discuss the possibilities of the calibration-free approach, global observation of detectable intracellular compartments and its dynamics. The problem in such a complicated environment is how to define the best resolution at the destination. In another words, we do not have a clear definition of the information source model – the destination. For the development of the method, we replace objective information resolution (in fact in full accordance with Shannon) by the operators opinion. He or she would decide which is the observable object in the cell interior, how long its lasts, whether it expands or shrinks. In this article, we describe the method and some elementary result using the information content calculation using Shannon entropy. Use of different Re´nyi entropy measures and the attempt to create a dynamic model of biological cell development will be discussed in a separate paper. 2. Method 2.1. Biological experiment 2.1.1. Growth conditions HeLa cells (Human Negroid cervix epitheloid carcinoma)were obtained from ECACC – European collection of cell cultures. The cells were grown at low optical density overnight in a 37 8 C temperature in a synthetic dropout media with 30% raffinose as the sole carbon source. The nutrient solution for HeLa cells consists of: 86% EMEM, 10% newborn-calf serum, 1% antibiotics and antimycotics, 1% L-glutamine, 1% non essential amino acids, 1% NaHCO3 (all components from the PAA company). 2.1.2. Microscopy 5000 original phase-contrast images of growing cells were taken at intervals of 1 minute. The image collection was performed at the temperature of 37 8 C using the Olympus IX 51 with an automated stage, integrated incubator and photographic camera Camedia C7070. The objective with 20 times magnification was used in image capture. 2.2. Information entropy contribution of an image point The information contribution of a point was performed as described earlier (Urban et al., 2009). In brief, we represent the
Fig. 2. Expansion of a region in histograms of intensity occurrences. Contribution of a point was calculated from comparison of two histograms with and without the examined point. The figure shows expansion of the histogram in the region containing the point for which the information contribution was calculated. The information contribution was calculated as Shannon entropy difference for the normalized histogram containing the point in question (occurrence of intensity 72 with gray bar) and without it (occurrence of intensity 72 with gray bar).
image at the camera screen – receiver data – by a cross whose shanks intersect in the point whose information contribution we calculate. We create a histogram of frequencies of occurrence fi of given intensity in the given colour channel (red, green, or blue) with and without the respective data point (Fig. ). Intensities were normalized to give frequency probabilities pi using the formula pi ¼
fi n X
(11) fj
j¼0
where n is number of intensity levels. Shannon entropy was obtained from probabilities using the formula (3), we denote the information with the central point S+ and that without the central point S. The contribution of one point to information in the image is
DS ¼ Sþ S
(12)
3. Results 3.1. Single point information contribution At Fig. 2 it we demonstrate the sensitivity of image entropy to intensity levels which occur in the image infrequently. Fig. 2 is expansion of a segment of combination of the two images showing the small intensity contribution. For the observed point was DS = 0.00287735 bit/pixel when DSmin = 0, 0011595 and was DSmax = 0, 0053682. Thus information value of this pixel was in the middle of the scale. 3.2. Information-based image representation Information carried by different colour channels is, as expected, significantly different. In the case of phase contrast microscopy the difference reflects deep differences in the construction of the image. The phase plate and other parts of the light path are optimized for certain wavelength in the green region. There the
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colour channels are clearly visible. In this one still image the differences must be assigned to the different optical path, or in another words, to different properties of the information transfer channel. As derived in the introduction, the properties of the channel have direct relation to the statistics by which the probability of information transfer is distributed. In the calculation presented in this paper, we assume the Shannon–Gibbs statistics. The image is a result of overlap of intensities resulting from a set of point spread function as the statistics are far more likely not to follow Gibbsian conditions than to follow them. Due to multifractality of the problem we may assume that there will be a spectrum of different statistics which expands the computational intensiveness of the analysis. From the point of view of generalized dimension spectrum, we may expect to have set non-integer q value by which the information space will be adequately represented. In case that calculation using only one q is feasible, the Shannon entropy is equally appropriate as any other.
Fig. 3. Composite representation of information carried in the phase contrast microscopic image of the cell. The pseudocolours represent composition of intensity information carried by the particular point in the image. Black points mean that the part of the image is information equivalent to the background.
intensity may be clearly assigned to degree of productive or destructive interference. Fig. 3 shows sections of a microscopic image transformed by image entropy calculations presented in composite form in which the pseudocolours. At Fig., section of an image depicting cell nucleus, are shown details of the image at which differences in the information content carried by different
[()TD$FIG]
3.3. Biological model As discussed in the introduction, the goal of the analysis is to obtain a model of the biological object. In another words the model of a biological system is the destination. It is a general agreement that living organisms are among the prominent examples of nonlinear dynamics (Kaneko, 2006). We should then seek major features predicted by non-linear dynamics, for example limit cycles (Cvitanovic et al., 2008). How shall these features appear in an image? What are their parameters? At Fig. 5 we show sketch of a selected dynamic feature of a cell at 0, 4, 9 and 14 minute of the experiment. Inspection of the sequence indicates that in the time of
Fig. 4. Comparison of a section of transformed image depicting cell nucleus. It is clearly seen that certain features in the highlighted part of the nucleus (probably chromosome) are significantly differently represented in the blue than in the red an green channel. This difference is a consequence of significantly different point spread function for wavelengths represented in the blue channel than those represented by other two channels. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article.)
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Fig. 5. Image of a selected region from image 3 showing high dynamics. Series of image sections at 0, 4, 9 and 14 minute of the experiment depicting a highly dynamic region of the cell. The identity of the image features is not known since it represents a mixture of objects at different distances from the focal plane and at which the shape of point spread function representation is different from each other. However, its colour composition and shape dynamics may be utilized as phenomenological system variable. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
the experiment the region does not significantly expand in space in either of the information channels which we observe. That is exactly what we may expect from a limit cycle. We should thus examine the state space in which the limit cycle evolves. In case of infinite resolution, we should be able to determine complete time dynamics of a dynamic region. In most real cases this is not feasible and, in fact, also not necessary. It is sufficient to be able to identify a region, distinguish it from another and quantify its role in the symbolic dynamic of the cell. Such symbolic dynamic information should be a parameter of the biological model. The biological model we propose in analogy to equilibrium thermodynamics we may assume to have constituents and components (i.e. Atkins and De Paula, 2006). Constituents, in textbook sense, are all independent chemical individui in the system. Components are sets of constituents which change independently, for example a set of chemical compounds bound together by a chemical reaction. Already in one component system the detection of its state by a simple device, i.e. single wavelength spectrometer, it is impossible to detect any ordinary chemical equilibrium. We most need to detect at least two reactants. With more components and more phases we need more parameters. Yet, there is one parameter, the Gibbs (or Helmholtz) energy which characterizes the system unanimously. From another point of view, taken to the extreme, the idea of a component is only statistical. The activity ai, the correct representation of concentration ci, of each constituent i is connected to the chemical potential
mi ¼ mi;0 þ RTln ai
(13)
while ai = gici and gi = f(c1, c2, . . ., cn) where n is number of constituents of the system. Finally, Gibbs energy G of a phase is G¼
n X
mj
(14)
j¼1
The components is the sub-set of constituents k for which it holds that
g k f ðcr ; . . . ; cm Þ
(15)
where concentrations (cr, . . ., cm) are sub-set of set of all concentrations (c1, . . ., cn). While in very dilute solutions some approximate solutions may be found, in real life concentrated solutions the component is very difficult to define. And the cell interior is highly concentrated. From the point of view of equilibrium thermodynamics, attempt to characterize the cell interior by one constituent is hopeless. We must seek the Gibbs energy which determines the phase equilibrium. For systems obeying the rules of nonequilibrium dynamics we do not expect one Gibbs energy value for each point in the state space. The state space is segregated into regions of attraction of individual dynamic features, attractors. The attractors may be of different kind, attractors of limit cycles, chaotic attractors, strange attractors etc. In each case, there is a sub-set of the state space which is much more populated than the rest and this we observe. The internal structure of living cells invokes the idea that individual structures are those regions of attraction and their dynamics reflects the dynamics of these asymptotically stable objects (i.e. Kaneko, 2006). Such objects no longer should have the same Gibbs energy and the different chemical composition may mean different region of the state space. The problem of identity of the coordinates in the state space remains but must not necessarily be crucial. As described for example in Cvitanovic et al. (2008) but introduced in principle much earlier (i.e Poincare´, 1892–1989; Sinai, 1968), the system behavior may be analyzed as symbolic trajectory. That precisely saying means that state space is separated into regions which are assigned a symbol. The symbols in our approach would come from identification of detectable and distinguishable regions. Not all transitions between symbols are available, there is a set of trajectories which are possible, the state space is pruned. In case that the partition is performed in the way that the transitions are independent of previous parts of the trajectory, we call the partition Markovian and may build statistical function characterizing it (i.e. Sinai, 1972; Ruelle, 1978). In order to identify the symbolic trajectory, all we need is to be able to identify individual states, give them a name, and to determine the rules of transition. By counting of trajectories we may obtain their probability and even construct a characteristic
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Fig. 6. Building of intracellular bridge – cell-to-cell intersubsystem information bond.
function which is the dynamic equivalent of Gibbs energy (Cvitanovic et al., 2008). Yet, the non equilibrium dynamics is perhaps not readily available either. Individual cells would be representing the individual systems only in isolated form. In the monolayer or a culture, cells represents sub-systems of and the monolayer is a system. In our opinion it is more appropriate to use the general stochastic systems theory derived for behavior of cybernetic systems (Zˇampa and Arnosˇt, 2004). The general principle is the same, we have to identify the state. In practical terms it means to determine all measurable – phenomenological – variables in sufficient time preceding the change of state to assure that the state change is independent from any preceding history. By counting number of transitions and state lifetimes we obtain probability distribution functions for transitions between states. In the example given at Fig. 5 it means that we follow the oscillations of the observed object form their start up to the time they cease or significantly change. General stochastic systems theory is superior to non-linear dynamics in consideration of the structure of the system. It considers the system to be composed of sub-systems which in isolation would behave as independent systems, as the cells. The bond between them is then called information bond. We believe that this theory may be directly applied to analysis of cell monolayer dynamics. For cells in contact one may obviously consider information exchange. At Fig. we show example of information exchange between two cells mediated by matter transfer. 4. Conclusion Results presented in this article show the framework of the objective analysis of cell monolayer dynamics. We analyze methods of extraction of information from the microscopic image from the theoretical point of view and give example of its practical use. The use of Shannon entropy for the information content analysis is based on consideration of normal Gibbs–Boltzmann statistics of the information channel behavior. This is clearly not adequate since the information transfer by microscope has complicated, multidimensional, character. We outline methods by which the analysis might be developed to give better representation of certain features. In another words, it is possible to calculate the spectrum of information features from a
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combination of observed object (information source), transmitter and information channel (the microscope). Next step is the analysis of information receipt at destination by which we understand the building and parameterization of a model of the observed organism, in this case of the culture of somatic cells. We compared obtained results with quantitative theories which may be utilized for description of the system dynamics. The general stochastic systems theory might be able to comprehend both the relatively independent behavior of individual cells and the cell-tocell communication. We present examples of detection of several image features which might be representations of this behavior, intracellular oscillating object (Fig.) and information bond (Fig. 6). Such detailed analysis is extremely laborious and needs to be automated. We believe that in the process of automation we shall begin to understand the system itself in much better way. The final goal is to diagnose the state of tissue culture in more objective manner. However, it is not quite clear whether the necessary theoretical apparatus is available.
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