Identification and Modeling of Co-Rhythmic Genes from Micro-array Time Series Data

Transcription

1 Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 8 Identification and Modeling of Co-Rhythmic Genes from Micro-array Time Series Data Wenxue Wang Bijoy K. Ghosh Texas Tech University, Department of Mathematics and Statistics, Lubbock, TX 7949 USA ( wenxue.wang@ttu.edu. Texas Tech University, Department of Mathematics and Statistics, Lubbock, TX 7949 USA ( bijoy.ghosh@ttu.edu Abstract: Circadian Rhythm is a biological phenomenon observed in a large number of organisms ranging from unicellular bacteria to human beings. In this paper, transcriptome data from Cyanothece, a photosynthetic cyanobacteria, has been analyzed for the purpose of discovering genes whose expressions are rhythmically close (co-rhythmic. Subsequently we study if these rhythms can be modeled, up to, using a cascade of three oscillators. One of the oscillator in the network is derived from the model of a limit cycle oscillator using KaiC protein (the master clock. We conclude that Circadian Rhythms in Cyanothece transcriptome data can be dynamically modeled up to using a single master clock derived from limit cycle oscillator using KaiC protein cascaded with a pair of interconnected oscillators. Biologically substrates of the oscillators are presently unknown. Keywords: Modeling and Identification, Bioinformatics, Dynamics and Control, System Biology, Circadian Oscillation, Network. INTRODUCTION Circadian Rhythm is a biological phenomenon observed in a large number of organisms ranging from unicellular bacteria to human beings. The underlying biochemical mechanism is also understood for many of the organisms to a varying degree of details. It is unclear, however, how a circadian clock controls various different metabolic processes. In particular, it is of importance to understand if only one clock is enough or if multiple clocks are required. In this paper, transcriptome data from Cyanothece, a photosynthetic cyanobacteria, has been analyzed for the purpose of discovering genes whose expressions are rhythmically close (co-rhythmic. Subsequently we study if these rhythms can be modeled, up to, using a cascade of three oscillators. One of the oscillators in the network is derived from the model of a limit cycle oscillator using KaiC protein (the master clock. We conclude that Circadian Rhythms in Cyanothece transcriptome data can be dynamically modeled up to using a single master clock derived from the limit cycle oscillator using KaiC protein cascaded with a pair of interconnected oscillators. Biologically substrates of the oscillators are presently unknown. Cyanothece, a photosynthetic cyanobacteria, is grown in ASP- N medium under / hour light/dark cycle. Under light conditions, cells are kept under 5 µe m s irradiance while it is µe m s for the dark condition. Samples for mrna are extracted every 4 hours, at points (hour, hour 5, through hour 45, to generate the micro array data. The This work is part of a Membrane Biology EMSL Scientific Grand Challenge project at the W. R. Wiley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the U.S. Department of Energy s Office of Biological and Environmental Research (BER program located at Pacific Northwest National Laboratory. PNNL is operated for the Department of Energy by Battelle. This work was also partially supported by funding from the NSF-FIBR program (EF Gene 3 pattern with Linear Trend Removed Linear Rhythmic Fig.. The log ration pattern of micro data (black and its linear trend (red and rhythmic component (blue. data are associated with more than 6 genes. A mixture of mrna extractions from different points has been used as reference channel. The data are normalized using Lowess algorithm and quality assessment is performed using T-test algorithm (see Quackenbush []. Log ratios of the data from control signals and target signals are obtained at points and they show rhythmic patterns. The rhythmicity has been displayed in Fig.. In the figure, the black pattern is one example from more than 6 log ratios of the normalized micro array series data. The rhythmic component (the blue curve can be obtained by removing the linear trend in the pattern. The rhythmic component can be reconstructed with two sinusoidal functions of a pair of optimum frequencies. A gene clustering strategy is proposed based on the optimum frequency pairs, instead of patten similarity. Circadian rhythms are a central feature of biological systems and are environmentally adaptive, and homeostatically regulated oscillators have been found in many organisms (cyanobacteria, fungi, and flies. Oscillation in KaiC phosphorylation /8/$. 8 IFAC /876-5-KR-.495

2 th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8. Gene 3 pattern estimation with Ω unchanged Gene 3 pattern estimation error over frequency pairs Estimation Error Ω Ω..4.6 Fig.. The rhythmic component in the micro array expression data and its estimates with two sinusoidal functions over different pairs of frequencies. The red curve is the estimate with the optimum frequency pair. is a key regulator for the clock in vivo. A suitable circadian oscillation in cyanothece has been built using the phosphorylation/dephosphorylation cycle of KaiC protein. Simulation of this cycle using a dynamic model for the in vitro KaiC circadian clock (see Mehra et al. [6] shows convergence to a limit cycle. Phase activity of the KaiC oscillation, obtained after a suitable linear transformation, is modeled using a simple evolution equation described in Tass [6]. The typical rhythmic pattern of micro array expression data can be modeled up to its. A oscillatory network is proposed with three oscillators. The oscillations in micro array series data are reconstructed up to s by a cascade of three oscillators. One of the oscillator is a model of master clock obtained from phosphorylation/dephophorhylation cycle of KaiC protein. The other two oscillators are the model of the activities of two rhythmic components with the optimum frequency pairs in the micro array expression patterns and Kuramoto model has been utilized (see Kuramoto [984].. ESTIMATION OF MICRO ARRAY EXPRESSION PATTERNS WITH OPTIMUM FREQUENCY PAIR. The log ratio patterns of points collected in micro array experiments show rhythmicity after the process of Lowess normalization and T-test. We conjecture that the rhythmic pattern in the micro array data can be modeled using a pair of fundamental frequencies (two sinusoids with unknown amplitudes and s. In this study we estimate the micro array pattern using linear combination of a linear trend and two sinusoidal functions with a pair of optimum frequencies described as follows: g(t = a+bt + α sin(ω t + θ +α sin(ω t + θ where a+bt is the linear component in the estimation, ω and ω are the frequencies of the rhythmic component, α and α are the amplitudes, θ and θ are the s related to two sinusoidal components. Fig. shows one example log ratio patten (black curve that displays rhythmicity and its linear trend (red and rhythmic component (blue. The part of linear trend can be obtained using linear regression. For the pairs {(t i,g i,i =,,N}, we can fit the curve {(t i,g i } using a linear function: g(t = a+bt where the parameter a and b are chosen to minimize the cost function: Fig. 3. The estimation errors over the pairs of frequencies. The optimum frequency pair is chosen that caused the smallest error and the optimum frequency pair makes best estimate (red curve in Fig.. C(a,b = N i= The least square method gives: (g i a bt i. a = g i b t i N. b = N g it i g i t i N ti ( t i The rhythmic component in the micro array expression pattern then can be obtained by removing the linear trend: ḡ(t i = g(t i a bt i. with obtained a and b (see Fig.. The rhythmic component can be estimated using a linear combination of two sinusoidal functions described as follows: ḡ(t i = α sin(ω t i + θ +α sin(ω t i + θ. Given a pair of frequencies, (ω,ω, the parameters α,α,θ,θ are chosen in such a way that the cost function is minimized: E(ω,ω = min G(α,α,θ,θ α,α,θ,θ where G(α,α,θ,θ is the cost function: N G(α,α,θ,θ = (ḡ(t i α sin(ω t i + θ α sin(ω t i + θ. i= For the given frequency pair, (ω,ω, the optimal parameters α,α,θ,θ are obtained using the gradient method: α α α θ = α θ µ G(α,α,θ,θ θ θ α α θ θ k+ k where µ is the size of change in the vector (α,α,θ,θ and G(α,α,θ,θ is the gradient of the function G(α,α,θ,θ : N i= sin(ω t i + θ e i (α,α,θ θ G = = N i= sin(ω t i + θ e i (α,α,θ θ α N i= cos(ω t i + θ e i (α,α,θ θ α N i= cos(ω t i + θ e i (α,α,θ θ where e i (α,α,θ θ = ḡ(t i α sin(ω t i + θ α sin(ω t i + θ. The frequency pairs, (ω,ω, are varied and the estimation errors over (ω,ω have been computed. The optimum frequency pair is chosen that make the smallest errors. Fig. shows the rhythmic component of one micro array expression pattern (blue and its estimates with two sinusoidal functions over different pairs of frequencies (green smooth curves. The red smooth curve is 9696

3 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 Number of genes over optimal frequency pairs 3.5 KaiC KaiC* kai data.5 Number of genes ω.8.9. ω Fig. 4. The number of genes over optimum frequency pairs. 5 clusters of genes using optimal frequency pairs Fig. 6. The oscillatory activities of unphosphorylated KaiC and phosphorylated KaiC (KaiC in..6 Cluster contains 8 genes Cluster contains 6 genes Cluster 3 contains 443 genes Cluster 4 contains 34 genes Cluster 5 contains 9 genes ω.3 KaiC* ω Fig. 5. Five clusters of genes are chosen based on optimum frequency pair. the best estimate with the optimum frequency pair. The estimation error over frequency pairs is shown in Fig. 3. The frequency pair that cause the smallest estimation error is the optimum frequency pair that makes the best estimate (red curve in Fig.. The genes that have the same optimum frequency pair are counted together. Fig. 4 shows the number of genes over the optimum frequency pair. The optimum frequency pairs provides a method of gene clustering based on the similarity of rhythmicity contained in the genetic expression patterns of more than 6 genes. Fig. 5 shows four clusters of genes based on the optimum frequency pairs. 3. SIMULATION OF CIRCADIAN PHASE PATTERN OF KAIC PROTEIN PHOSPHORYLATION Circadian rhythms are a central feature of biological systems and are environmentally adaptive. Homeopathically regulated oscillators have been found in many organisms (cyanobacteria, fungi, and flies. Oscillation in KaiC phosphorylation is a key regulator for the clock in vivo. KaiC is an enzyme with autokinase and autophosphatase activities. A synthetic, predictive and dynamic model for the in vitro KaiC phosphorylation activity has been built based on the self-amplifying response ( autocatalysis of autophosphorylating kinases (see Mehra et al. [6], described as follows: ẋ = k 5 x 8 k x x 3 k 3 x 6 x x 3 ẋ = k 6 x 7 k 4 x 6 x ẋ 3 = k 7 x 4 k x x 3 k 3 x 6 x x 3 ẋ 4 = k 6 x 7 k 7 x 4 ẋ 5 = k x x 3 k x 5 ẋ 6 = k x 5 + k 3 x 6 x x 3 k 4 x 6 x ẋ 7 = k 5 x 8 k 6 x 7 ẋ 8 = k 4 x 6 x k 5 x 8 where x,...,x 8 represent KaiA, KaiB, KaiC, KaiC, KaiAC, KaiAC*, KaiBC* and KaiABC* respectively. The model was described in details in Mehra et al. [6]. Among these variables, x 3 and x 4, i.e., unphosphorylated KaiC protein and phosphorylated KaiC protein (KaiC are of interest in this study. Activities of KaiC and KaiC are oscillatory and they converge to a limit cycle on the plane with proper initial conditions. Fig. 6 shows the oscillatory activities KaiC Fig. 7. An oscillatory behavior around the origin on the plane is obtained by taking a proper linear transformation. velocity Fig. 8. The artificial pattern (top and its velocity pattern (bottom in. of KaiC and KaiC in. The activities of unphosphorylated KaiC and phosphorylated KaiC (KaiC forms a limit cycle oscillations. Taking a proper linear transformation of the oscillatory activities, we can have the oscillatory behavior around the origin on the plane. Then we can obtain an monotonic artificial activity of KaiC phosphorylation/dephosphrylation cycle. Fig. 7 shows the oscillatory behavior of KaiC and KaiC protein around the origin on the plane obtained by taking a proper linear transformation, and Fig. 8 shows the artificial pattern of the KaiC phosphorylation/dephosphrylation cycle with the linear transformation and its velocity pattern in. In Fig. 9 (top, the velocity pattern is plotted versus the pattern. Since the oscillatory activities of KaiC and KaiC proteins converge to a limit cycle, the pattern contains the clock information of the circadian regulator. One important and interesting problem is if the activity of KaiC phosphorylation/dephosphrylation oscillation can be modeled with some dynamical model. In this study we utilize the simple evolution equation described in Tass [6]: ż = (α + iωz βz z + S(z,z 9697

4 th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 velocity simulation velocity Fig. 9. The velocity is plotted versus the pattern (top, The section of velocity pattern (red part has been used to compute the Fourier coefficients in the evolution equation (. Fig.. The reproduction of the using the evolution equation ( with the obtained Fourier coefficients. The blue curve is the pattern and the red curve is the simulated pattern. The s are plotted in π modulus at the bottom. error fourier simulation.4 velocity number of fourier terms velocity Fig.. The estimation of the part of velocity pattern using the Fourier Transform in with the coefficients obtained. On the top is the estimation error versus the number of Fourier transform coefficients and on the bottom is the estimate (green color of the actual velocity (red color using 3 pairs of Fourier transform coefficients. where z is a complex variable, and z denotes the complex conjugate of z. ω is the eigenfrequency, α is a real parameter and β a nonnegative real parameter. S(z,z, depending on z and z, represents a class of stimulation mechanisms which merely influence the. One special form of choice of S is given by with complex coefficients S(z,z = z (η m z m+ z ηm(z m+ z m= η m = w m + iv m (w m,v m R. Introduction of polar coordinates r and φ by inserting the hypothesis z(t = r(texp(iφ(t gives the evolution equations of the amplitude r and the φ: ṙ = αr βr 3, φ = ω + m= (rm+ w m sin(mφ+r m+ v m cos(mφ. The stimulus S(z,z only acts on the oscillator s whereas the amplitude remains unaffected. For this reason the amplitude r relaxes towards the stable value α/β irrespective of the initial value r( and irrespective of the stimulus action. So the evolution equation of the φ can be simplified into: φ = ω + (a m sin(mφ+b m cos(mφ ( m= where a m and b m are real constant coefficients irrespective of the amplitude r. In Fig. 9 (top, the plot of the velocity pattern versus the pattern shows that the velocity is a periodic function of the pattern. In the evolution equation of the (, the derivative of is represented in the form of Fourier expansion in. These observations tell that the pattern can be reproduced with the evolution equation (. To reproduce the Fig.. Phase velocity pattern (in comparison: velocity (blue, the estimate of velocity using the Fourier expansion (green, and the estimate of velocity using the evolution equation (red. pattern using the evolution equation (, the parameters in the model, ω, a m and b m have to be found. A period of velocity pattern has been used to compute those Fourier coefficients with DFT algorithm. Fig. 9 shows the section of velocity pattern (red that is used to compute the coefficients. Fig. shows the estimation of velocity pattern using the Fourier expansion in with the coefficients obtained. On the top is the estimation error versus the number of Fourier expansion coefficients and on the bottom is the estimate (green color of the actual velocity (red color using 3 pairs of Fourier coefficients. With the obtained Fourier coefficients, we can reproduce the pattern by simulating the evolution equation (. Fig. shows the simulation of pattern using the evolution equation, in which the blue curve is the pattern and the red curve is the simulated pattern. Fig. shows the comparison of velocity and its estimates. 4. OSCILLATORY NETWORK WITH CIRCADIAN CLOCK OSCILLATOR In Section, we studied the micro array expression pattern fitting using the linear combination of a linear trend and two sinusoidal components with an optimum frequency pair. In Section 3, we simulated the circadian activities of KaiC and KaiC proteins using a dynamical model and the artificial activity of KaiC phosphorylation/dephosphorylation oscillation was reproduced using an model of evolution equation. A typical rhythmic pattern of an expression data can be modelled up to its. In this section, we propose a network of 3 oscillators that reproduce the circadian pattern and the patterns of two sinusoidal components in the micro array expression pattern fitting, shown in Fig. 3. Then the rhythmic component of micro array expression pattern with linear trend removed can be estimated with the sinusoids of the patterns of two sinusoidal components. One oscillator of the network is described by the evolution equation that reproduce master circadian clock and control the 9698

5 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 6 the of circadian oscillator network master clock first component second component Fig. 3. The oscillatory network including one master clock (φ and two Kuramoto oscillators (φ and φ. Fig. 4. The circadian pattern φ (t(blue and two patterns, φ (t and φ (t, for the sinusoidal components (green and blue generated by simulating the oscillatory network (.5 s of sinusoidal components. The other two oscillators of the network are described by a modified Kuramoto model of two unit with extra terms which are the functions of master circadian. The oscillatory network is described as follows: φ = ω + M m= (a m sin(mφ +b m cos(mφ φ = ω + ε sin(k k φ φ + ψ +ε sin(φ k φ + ψ ( φ = ω + ε 3 sin(k φ φ + ψ 3 +ε 4 sin(k φ φ + ψ 4, with k = ω ω and k = ω ω, where φ is the of the master clock with ω <, φ and φ are the s of the two sinusoidal components with natural frequencies, ω >, ω >, the optimum frequency pair obtained via pattern fitting of the micro array data. ω < is the average velocity of the master clock, which is the slope of the straight line that interpolates the activity of the master clock (the circadian pattern in Fig. 8. There exists a constraint, ψ 4 = ψ = ψ ψ k. The connection parameters - s, i =,,4, are positive. The above interconnected dynamics ( is a variation of the well known Kuramoto s models (see Kuramoto [984]. The rhythmic component of the micro array data now can be estimated using the sinusoidal functions of the s φ and φ as follows: ḡ(t = α sin(φ (t+α sin(φ (t, Here we picked up one micro array expression pattern, as shown in Fig., for example. Using the optimum frequency pair in the micro array pattern fitting obtained in Section and the Fourier coefficients obtained in Section 3, and choosing values for other parameters in the dynamical model of oscillatory network, the patterns are simulated. Fig. 4 shows the master circadian pattern and two patterns for the sinusoidal components simulated with the model of oscillatory network (. Fig. 5 shows the corresponding sinusoids of the patterns. Fig. 6 shows the estimation of the genetic pattern using sinusoids of the s, φ (t and φ (t, generated by the oscillatory network (green curve. One important problem in the dynamical model ( is the stability of the scaled differences, k k φ φ and k φ φ, that is, k k φ (t φ (t and k φ (t φ (t converge to some equilibrium points. Another difference in the model is φ k φ, which is dependent on the differences k k φ φ and k φ φ. We start this analysis with a simpler model with those Fourier terms removed in the first equation, described as follows: φ = ω φ = ω + ε sin(k k φ φ + ψ +ε sin(φ k φ + ψ φ = ω + ε 3 sin(k φ φ + ψ 3 +ε 4 sin(k φ φ + ψ 4, in which ω = ω. Multiplying k k on the second equation and k on the third equation, we have: ( sinusoid of the of master clock sinusoid of the of the first frequency component sinusoid of the of the second frequency component Fig. 5. The sinusoids of patterns shown in Fig. 4. The blue curve on the top is the sinusoid of the circadian sin(φ (t. The other two curves (green and red are the sinusoids of the patterns, sin(φ (t and sin(φ (t, for the two sinusoidal components in the genetic pattern Optimum Frequencies Phase Network Fig. 6. Estimation of micro array expression pattern. The blue curve is the rhythmic component of the micro array expression pattern, as shown in Fig. ; the red curve is the estimate using two sinusoidal component with the optimum frequency pair; the green curve is the estimate using the sinusoids of the s generated with the model of oscillatory network (. φ = ω k k φ = k k ω + k k ε sin(k k φ φ + ψ +k k ε sin(φ k φ + ψ k φ = k ω + k ε 3 sin(k φ φ + ψ 3 +k ε 4 sin(k φ φ + ψ 4 where k k ω = k ω = ω by the definitions of k and k. Define η = k k ε,η = k k ε,η 3 = k ε 3, and, η 4 = k ε 4, then η i -s, i =,,4, are (4 9699

6 7th IFAC World Congress (IFAC'8 Seoul, Korea, July 6-, 8 negative because k is negative. Define Φ = k k φ φ +ψ and Φ = k φ φ + ψ 3, and by the constraint ψ 4 = ψ = ψ ψ k, we have φ k φ + ψ = Φ Φ k and k φ φ + ψ 4 = Φ Φ k. From (4 and the new variables defined above, we have differential equations on Φ and Φ as follows: with η i -s and k negative. Φ = η sin(φ +η sin( Φ Φ k Φ = η 3 sin(φ +η 4 sin( Φ Φ k Claim: The system (5 has a locally stable equilibrium at (,. Proof: Firstly, we can easily verify that Φ (, = and Φ (, =, which means the origin (, is an equilibrium. Then we need to prove the equilibrium (, is stable locally. Define the Lyapunov function V(Φ,Φ as follows: (5 =. = k k φ φ k φ φ k φ φ k k φ φ k φ φ k φ φ V(Φ,Φ = α Φ + Φ with α = η 4 η = η 4 η >. It is obvious that V(Φ,Φ is positive definite. Then we have V(Φ,Φ = αφ Φ + Φ Φ = αφ (η sin(φ +η sin( Φ Φ k +Φ (η 3 sin(φ +η 4 sin( Φ Φ = αη Φ sin(φ +η 3 Φ sin(φ + η 4 (Φ Φ sin( Φ Φ k with η i -s and k negative, we can easily verify that αη Φ sin(φ < η 3 Φ sin(φ < η 4 (Φ Φ sin( Φ Φ k < k for (Φ,Φ with Φ and Φ small enough. Then V(Φ,Φ is negative definite for (Φ,Φ in the neighborhood of the origin (,. By Lyapunov Theorem, the equilibrium (, is locally stable in the system (5. Hence, the scaled differences, k k φ φ and k φ φ, converge to ψ and ψ 3 in the system (3. In the dynamical model of oscillatory network with the Fourier terms in the first equation (, the scaled differences, k k φ φ + ψ and k φ φ + ψ 3, can not converge to a certain equilibrium, but they can be controlled within certain range by choosing stronger connection parameters, -s, in the model (. For simplicity, we choose ψ i = and = ε for all i in the model (. By increasing the connection coefficients,, the scaled differences, k k φ φ and k φ φ can be reduced within smaller range. Fig. 7 shows the scaled difference patterns k k φ φ (red, k φ φ (green and k φ φ (blue, for three different connection coefficients, = ε =.,.8 and.6 from top to bottom. and maximal differences become smaller as the connection coefficients,, increase, which is shown in Fig CONCLUSION We have analyzed the transcriptome data from Cyanothece, a photosynthetic cyanobacteria, for the purpose of discovering genes whose expressions are rhythmically close (co-rhythmic. The rhythmic component of the micro array expression patterns can be reconstructed using two sinusoidal functions with a pair of optimum frequencies. Subsequently we utilized an evolution equation to model the activity of KaiC phosphorylation/dephosphorylation cycle. At last we proposed an oscillatory network, using a cascade of three oscillators, to model the patterns of rhythmic components in the micro array expression data. One of the oscillators in the network is derived from the model of a limit cycle oscillator using KaiC protein (the master clock. We conclude that Circadian Rhythms in Cyanothece transcriptome data can be dynamically modeled up to using a single master clock derived from limit cycle oscillator using KaiC protein cascaded with a pair of interconnected oscillators. Further work on the oscillatory network will be done. The work will include how the master clock controls the s of the other two oscillator, if one master clock is enough to drive patterns of rhythmic components in multiple micro array expression data, and how those parameters in the model are chosen. = k k φ φ k φ φ k φ φ Fig. 7. The scaled difference patterns k k φ φ (red, k φ φ (green and k φ φ (blue, for three different connection coefficients, = ε =.,.8 and.6 from top to bottom..5 k k φ φ k φ φ Fig. 8. The maximal differences become smaller as the connection coefficients,, increase REFERENCES Y. Kuramoto. Chemical Oscillations, Waves, and Turbulence. Springer Verlag, New York, 984. A. Mehra, C. I. Hong, M. Shi, J. J. Loros, J. C. Dunlap and P. Ruoff. Circadian rhythmicity by autocatalysis. PLos Computational Biology, (7:86 83, 6. J. Quackenbush. Microarray data normalization and transformation. Nature Genetics, 3:496 5, P. A. Tass. Phase Resetting in Medicine and Biology: Stochastic Modelling and Data Analysis. Springer Verlag, New York, 6. 97