The ultimate aim of using the cdna microarray technology to

Transcription

1 Matching gene activity with physiological functions Wei Huang*, Yuh-Pyng Sher, Konan Peck, and Yuan Cheng B. Fung* *Department of Bioengineering, University of California at San Diego, La Jolla, CA ; and Institute of Biomedical Sciences, Academia Sinica, Taipei, Taiwan 11529, Republic of China Contributed by Yuan Cheng B. Fung, December 19, 2001 Matching the activity of the genes with biomechanics and physiology is an effective way to use cdna microarray technology. Required are data on the change of activities of genes associated with specific physiological functions with respect to a continuous variable such as time. For each pair of data (gene and physiological function) as functions of time, we can compute a coefficient of correlation, R. The correlation is perfect if R is 1 or 1; it is nonexistent if R 0. By evaluating R for every gene in a microarray, we can arrange the genes in the order of the number R, thus learning which genes are best correlated with the mechanical or physiological function. We illustrate this procedure by studying the blood vessels in the lung in response to pulmonary hypoxic hypertension, including the remodeling of vascular morphometry, the elastic moduli, and the zero-stress state of the vessel wall. For each physiological function, we identify the top genes that correlate the best. We found that different genes correlate best with a given function in large and small arteries, and that the genes in pulmonary veins which respond to arterial functions are different from those in pulmonary arteries. We found one set of genes matching the remodeling of arterial wall thickness, but another set of genes whose integral of activity over time best fit the wall thickness change. Our method can be used to study other thought-provoking problems. cdna microarray blood vessel morphometric parameters blood vessel elasticity blood vessel opening angle at zero-stress state pulmonary hypertension The ultimate aim of using the cdna microarray technology to obtain gene expression data is to correlate genes with physiological functions. The purpose of this article is to show how this correlation can be pursued. We describe both the chosen physiological functions of a tissue and the activities of the genes of the tissue as continuous functions of time. Then we compare these continuous functions and determine their correlation coefficients. The correlation coefficients allow us to line up the genes in the order of the correlation coefficients for each physiological function. In this article, a survey of the association of gene activity with the functions of blood vessels is presented. For each blood vessel, we consider the following functions of time, x i (t), i 1, 2, 3,... 8: x 1 (t) the blood pressure; x 2 (t) the oxygen concentration in the blood; x 3 (t) the opening angle of a blood vessel defining its zero-stress state; x 4 (t) the thickness of the intima-media layer; x 5 (t) the thickness of the adventitia layer; x 6 (t) the inner circumferential length at zero-stress state; x 7 (t) the outer circumferential length at zero-stress state; x 8 (t) Young s moduli of pulmonary arteries. Each of these variables has a stable static equilibrium (homeostatic) value in vivo under physiological conditions. When a disturbance is introduced at a time t 0, these variables change; and the changes are correlated with gene activity. We took tissue samples at time t, and measured the gene activity with a cdna microarray method to obtain the history of the activity of each gene. Let the results be denoted by y j (t), which are functions of time, and the gene numbers j, j 1, 2, 3,... 9,600 in our case. Now we have two sets of function x i (t) and y j (t). We can compare any pair of functions to see whether they are similar or not. For this purpose, we define a correlation coefficient R(x i, y j )bythe formula R x i, y j 0 0 T x i t y j t dt, [1] T T x 2 i y j t dt 0 2 t dt in which T is the total period covered by the correlation. The correlation is perfect if R 1; it is poor if the absolute value of R is small. For any given x i (t), we can arrange the genes in the order of the cardinal number R(x i, y j ). To understand the function x i (t), we focus our attention on those genes that are closer to R 1orR 1. All of the functions x i (t) are not of the same character. Some morphometric functions such as x 3...x 8 that describe the local tissue remodeling are strictly local. These variables are probably more directly correlated with the gene expressions of the local tissue. The blood pressure x 1 (t), however, varies with the total circuit. It would be interesting to inquire whether the genes sense x i, or its rate of change, dx i (t) dt, and control x i (t)byy j,orbythe integral of y j. Thus, we should also investigate the correlation coefficient of the derivatives and integrals of x i and y j, e.g., R ij (dx i dt, y j ), R ij (x i, dy j dt), R ij (dx i dt, dy j dt), and R ij (x i, 0 T y j (t)dt). Correlation of the derivatives and integrals of x i and y j can be done according to Eq. 1 by appropriate substitutions of x i, y j with their derivatives or integrals. Getting physiological and gene expression data on tissue remodeling in vivo of the type studied here is very expensive in time, labor, and cost; therefore, the data on tissue remodeling are often sparse. Blood pressure data recorded digitally are, however, extremely rich, but stochastic and nonstationary. To use Eq. 1, we handled the sparse data by curve fitting with proposed analytical formulas, whereas the stochastic data were dealt with by using our intrinsic mode function method (1 3), which is explained in Materials and Methods. The principle we put forward is that to discover which genes activity matches a physiological function best, we first look for a continuous variable such as time, then perform experiments to obtain gene activities and physiological functions as functions of time. The correlation allows an assessment of coupling. Identification of genes with physiology and pathology is common. A popular method is the self-organizing map (SOM) method of Tamayo et al. (4). We tried the SOM method on our specimens (5, 6), found it interesting, but not sufficiently specific. The method here is simpler to interpret. Materials and Methods We induced a rapid increase of pulmonary arterial blood pressure in the lungs of rats living in a modified commercial chamber To whom reprint requests should be addressed. ycfung@bioeng.ucsd.edu. The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C solely to indicate this fact. MEDICAL SCIENCES ENGINEERING cgi doi pnas PNAS March 5, 2002 vol. 99 no

2 (Snyder, Denver, CO) by suddenly decreasing the oxygen concentration in the chamber (2, 3, 7, 8). The cardiac output and systemic blood pressure did not change much (9, 10). We studied the remodeling of vascular tissues in response to this sudden change of blood pressure and the associated gene activities as functions of time. We then calculated the correlation coefficients of these functions of time and determined the genes that are most relevant to physiology. The details of the animal protocol and the regulatory approval were reported (1 3, 5, 6, 11). Under anesthetization, a catheter was implanted into the pulmonary arterial trunk of the rat through its jugular vein and sutured to the back so that after awakening the rat could move freely and eat and drink normally. Blood pressure data were recorded at 100 points per sec chronically. Each oxygen level change was accomplished in min. At a scheduled time, the rat was killed with an overdose of an i.p. injection of pentobarbital sodium (50 mg kg of body weight). Then, vascular tissues were collected and prepared for measurements of morphometric and mechanical properties and gene expression. The total RNAs from the left pulmonary arteries and veins ( 50 mg of tissue per sample) were extracted by using the Stratagene Micro RNA isolation kit (no ). We measured the variables x 1 (t), x 2 (t),...x 8 (t) (3, 5, 6, 11) and used them in this article. Perhaps x 3 (t), the opening angle, needs an explanation. When an artery is cut into rings and a ring is cut along a radial line, it opens up into a sector, which represents the vessel wall at zero-stress. By taking the midpoint of the inner wall as an origin and erecting two radius vectors from this origin to the tips of the inner wall in a normal cross section, we obtain a subtended angle, x 3 (t), the opening angle of the vessel. Rounding up the opening sector back into a tube would generate residual stresses. Hence, the opening angle is a measure of the residual stresses and one of the best measures of tissue remodeling (12). Measurement of Gene Activity. Gene activity was measured by a cdna microarray method with enzyme colorimetry detection (13). The mrnas of the pulmonary arteries and veins were amplified by using an in vitro transcription method (14) before they were labeled with biotin during the reverse transcription process. To have better signal to noise ratios, the colorimetric signals were amplified by a modified catalyzed reporter deposition (CARD) method (15, 16). The details of the procedures for in vitro transcription and the CARD signal amplification methods are described in ref. 17. The labeled cdna derived from the tissues were then hybridized to 9,600 probes in a microarray. The array images were digitized by a 3,000 dots per inch flatbed scanner (PowerLook 3000, UMAX, Taiwan), and quantitative information was obtained by an image analysis software GENEPIX PRO 3.0 (Axon Instruments, Foster City, CA). The hybridization condition and data analysis were described in detail (5). The gene activity was expressed in an arbitrary scale of 3,000 35,000. The sum total of the scores of all genes on a chip was considered as an indication of the total amount of a sample. The data from samples of the same tissue were normalized for sample sizes. Fig. 1. A record of the pulmonary arterial blood pressure of a rat (rat code: ) subjected to step lowering of oxygen concentration in breathing gas. y t A 1 A 2 t A 3 t 2 A 4 te t T1 A 5 te t T2, [3] where A 1,...A 5 are constants, T 1 is the time for the first peak, T 2 is the time for the second peak. We arrived at these formulas after trying many other methods. The blood pressure signal, as shown in Fig. 1, is very complex. It is nonstationary and stochastic. To explain the matching of gene activity and the blood pressure as presented in Figs. 3 and 4 and Tables 1 and 2, an explanation of the mathematics whose details are given in refs. 1 3 is given here. Very briefly, denoting the blood pressure as X(t), we compute an upper envelope that connects the successive local maxima of X(t), and a lower envelope that connects all of the minima. The mean of the envelopes is designated as m(t). We then compute X(t) m(t) h(t) and treat h(t) as new data, compute the new envelopes and a new mean, and iterate until it converges. The convergent result is called the first intrinsic mode C 1 (t), which has a zero local mean. Next, we compute X(t) C 1 (t) and treat it as new data, for which the second intrinsic mode C 2 (t) is determined. The process ends after n steps, when C n (t) is nonoscillatory, X t C 1 t C 2 t C n t. [4] Every term has a zero local mean. The successive modes have successively fewer zero crossings. C n (t) is a trend. C n C n-1 is also a trend with some oscillations whose local mean is zero. We define the mean trend of order k by the formula M k t C k t C k 1 t C n t [5] and the oscillations about the mean M k (t) as X k t C 1 t C k 1 t. [6] Mathematical Representation of Experimental Results. To use Eq. 1, we first fit the physiological and gene activity data with mathematical expressions by the method of least-square errors. The expression for the mean pulmonary blood pressure, M k (t), after a step decrease of oxygen tension in breathing gas, is as given in ref. 2, for t 0, M k t A Bte t T1 C 1 e t T2, [2] in which A, B, C, T 1, and T 2 are constants. The expression for the remodeling change of the opening angle, as well as that of the thickness of the media and adventitia, is given by the following equation as in ref. 5: Fig. 2. The distribution of the genes as a function of the correlation coefficient R. The ordinate N is the number of genes in the interval (R, 1)for positive R or in ( 1, R) for negative R. BP, blood pressure cgi doi pnas Huang et al.

3 Fig. 3. Activity of the gene PIX no (phenylalanyl-trna synthetase -subunit gene) vs. the 16th order mean of the blood pressure, M 16 BP(t) above the steady in vivo value at t 0. E, normalized M 16 BP, defined as [(1 N) N 1 ( M 16 BP) 2 ] 1 2, fitted by a solid line, one SEM flag up. }, normalized gene activity, defined as (Gene Activity) [(1 N) N 1 ( (Gene Activity) 2 ] 1 2, fitted by dash line, one SEM flag down. Then we computed the Hilbert transform of X k (t), whose amplitude is a function of frequency and time, H k (, t). The integral of the square of H k (, t) over all frequencies is defined as the oscillatory energy about the kth order mean: E k t H k 2, t d. [7] E k (t) is a stochastic variable that can be handled in the same way as X(t), and orders of its mean trend are defined in turn. Results (i) Positive and Negative Correlation Coefficients. As defined in the introduction, the correlation coefficients R(x, y) can be positive or negative. A physiological function may be caused by an increase of a gene activity or by a decrease. It is interesting to know the sign. Hence, in listing the genes according to the values of the correlation coefficients, we tabulate the genes for positive R separately from those having negative R. This is done in Tables 1 4. (ii) The Distribution of the Genes Lined Up in a Column According to the Correlation Coefficients with Respect to a Physiological Function. The gene having the largest correlation coefficient is called the top gene, the others are called no. 2, no. 3, etc. The names of these genes and their identification numbers (PIX no.) are listed in the software GENEPIX PRO 3.0 (Axon Instruments). The distribution of the number of genes as function of the correlation coefficient is illustrated in Fig. 2 for two cases: (i) the opening angle and (ii) the mean pulmonary blood pressure, both following a step decrease of oxygen concentration in the breathing gas. The abscissa is the correlation coefficient R. The ordinate is N, the number of genes in the interval (R, 1) for positive R or in ( 1, R) for negative R. Table 1. Top 5 pulmonary artery genes of R[x i (t)y j (t)] Physiol. entity Range of correlation coefficient Identification PIX no. of the 5 top genes x 1 (t)-kth mean of the blood pressure M k (0 24 h) M 10 BP(t) (0.9997; ) M 14 BP(t) (0.9997; ) M 16 BP(t) (0.9999; ) x 1 (t)-kth mean of energy of the oscillations E 6 (0 24 h) M 9 of E 6 (t) (0.9999; ) M 13 of E 6 (t) (0.9986; ) x 3 (t)-opening angle of pulmonary arteries (0 30 d) OA-r1(t) (0.9341; ) OA-r2(t) (0.9677; ) OA-r3(t) (0.9096; ) OA-r4(t) (0.9149; ) OA-r5(t) (0.8404; ) OA-r6(t) (0.8903; ) OA-r7(t) (0.9403; ) OA-r8(t) (0.9350; ) x 4 (t)-media thickness of pulmonary arteries (0 30 d) MT-r1(t) (0.9700; ) MT-r2(t) (0.9425; ) MT-r3(t) (0.9632; ) MT-r4(t) (0.9548; ) MT-r5(t) (0.9732; ) MT-r6(t) (0.9640; ) MT-r7(t) (0.9595; ) MT-r8(t) (0.9550; ) x 5 (t)-adventitia thickness of pulmonary arteries (0 30 d) AT-r1(t) (0.9691; ) AT-r2(t) (0.9669; ) AT-r3(t) (0.9524; ) AT-r4(t) (0.9652; ) AT-r5(t) (0.9704; ) AT-r6(t) (0.9703; ) x 6 (t)-inner circumferential length of pulmonary arteries (0 24 h) Li-r1(t) (0.9995; ) Li-r2(t) (0.9999; ) Li-r3(t) (0.9999; ) Li-r4(t) (0.9998; ) Li-r5(t) (0.9998; ) Li-r6(t) ( ; ) Li-r7(t) ( ; ) Li-r8(t) ( ; ) x 7 (t)-outer circumferential length of pulmonary arteries (0 24 h) Lo-r1(t) (0.9999; ) Lo-r2(t) (0.9996; ) Lo-r3(t) (0.9992; ) Lo-r4(t) (0.9989; ) Lo-r5(t) ( ; ) Lo-r6(t) (0.9997; ) Lo-r7(t) ( ; ) Lo-r8(t) ( ; ) x 8 (t)-young s moduli of pulmonary arteries (0 24 h) Y (t) (0.9998; ) Y zz (t) (0.9998; ) Y z (t) (0.9999; ) MEDICAL SCIENCES ENGINEERING Fig. 4. Activity of the gene PIX no (wee1 homolog gene) vs. the blood pressure oscillation energy, the 13th order mean of the 6th order of energy of oscillation, E 6,defined by Eqs. 4 7 and ref. 3. E, [(1 N) N 1 ( E 6 ) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. (iii) Correlation of Activity of Pulmonary Arterial Genes with the Changes of Pulmonary Arterial Blood Pressure. We induced a rapid rise in pulmonary arterial blood pressure by a step decrease of the oxygen concentration in the breathing gas. This is the well-known high altitude disease (7 10). The details of the control of gene expression by this process, which are of great Huang et al. PNAS March 5, 2002 vol. 99 no

4 Table 2. Top 5 pulmonary artery genes of R[x i (t)y j (t)] Physiol. entity Range of correlation coefficient Identification PIX no. of the 5 top genes x 1 (t)-kth mean of the blood pressure M k (0 24 h) M 10 BP(t) ( ; ) M 14 BP(t) ( ; ) M 16 BP(t) ( ; ) x 1 (t)-kth mean of energy of the oscillations E 6 (0 24 h) M 9 of E 6 (t) ( ; ) M 13 of E 6 (t) ( ; ) x 3 (t)-opening angle of pulmonary arteries (0 30 d) OA-r1(t) ( ; ) OA-r2(t) ( ; ) OA-r3(t) ( ; ) OA-r4(t) ( ; ) OA-r5(t) ( ; ) OA-r6(t) ( ; ) OA-r7(t) ( ; ) OA-r8(t) ( ; ) x 4 (t)-media thickness of pulmonary arteries (0 30 d) MT-r1(t) ( ; ) MT-r2(t) ( ; ) MT-r3(t) ( ; ) MT-r4(t) ( ; ) MT-r5(t) ( ; ) MT-r6(t) ( ; ) MT-r7(t) ( ; ) MT-r8(t) ( ; ) x 5 (t)-adventitia thickness of pulmonary arteries (0 30 d) AT-r1(t) ( ; ) AT-r2(t) ( ; ) AT-r3(t) ( ; ) AT-r4(t) ( ; ) AT-r5(t) ( ; ) AT-r6(t) ( ; ) x 6 (t)-inner circumferential length of pulmonary arteries (0 24 h) Li-r1(t) ( ; ) Li-r2(t) ( ; ) Li-r3(t) ( ; ) Li-r4(t) ( ; ) Li-r5(t) ( ; ) Li-r6(t) ( ; ) Li-r7(t) ( ; ) Li-r8(t) ( ; ) x 7 (t)-outer circumferential length of pulmonary arteries (0 24 h) Lo-r1(t) ( ; ) Lo-r2(t) ( ; ) Lo-r3(t) ( ; ) Lo-r4(t) ( ; ) Lo-r5(t) ( ; ) Lo-r6(t) ( ; ) Lo-r7(t) ( ; ) Lo-r8(t) ( ; ) x 8 (t)-young s moduli of pulmonary arteries (0 24 h) Y (t) ( ; ) Y zz (t) ( ; ) Y z (t) ( ; ) interest, are unknown. Fig. 1 shows a typical pulmonary blood pressure trace. We analyzed the features of the blood pressure by the intrinsic mode functions outlined in Eqs. 4 7 (ref. 1 3). We fitted these features with Eq. 3 and fitted the gene activities with the same Eq. 3. The plots of the fitted curves of the second genes are shown in Figs. 3 and 4. The corresponding numerical Table 3. Best correlation of pulmonary artery function x i (t) with the cumulative integrated gene activity 0 T (pulmonary artery gene activity(t))dt x 3 (t)-opening angle (0 30 d), (corr. coef.), gene PIX no. Artery region 1 (0.5352), 7589 ( ), 6420 Artery region 4 (0.7102), 4988 ( ), 5330 Artery region 8 (0.7070), 8919 ( ), 4988 x 4 (t)-media thickness (0 30 d), (corr. coef.), gene PIX no. Artery region 1 (0.9886), 1535 ( ), 730 Artery region 4 (0.9837), 735 ( ), 730 Artery region 8 (0.9809), 735 ( ), 730 x 5 (t)-adventitia thickness (0 30 d), (corr. coef.), gene PIX no. Artery region 1 (0.9967), 9365 ( ), 4988 Artery region 4 (0.9937), 410 ( ), 4988 Artery region 6 (0.9841), 735 ( ), 4988 corr. coef., correlation coefficient. listing of the top five genes that correlate best with the blood pressure data is presented in Table 1 for positive correlation and Table 2 for negative correlation. The M 10 BP(t) represents the order 10 of the mean blood pressure as a function of time. The M 9 of E 6 (t) represents the order 9 of the energy of oscillation E of order 6. See Materials and Methods for definitions of these terms. The parentheses indicate the range of the correlation coefficients for the top five genes whose gene identification numbers (PIX no.) are listed. From this study we learn that the activities of many genes correlated extremely well with the time courses of the change of blood pressure, and that the genes which match the mean pressure best are not the same genes that best match the energy of oscillations about the mean. Table 4. Top pulmonary venous genes correlating with pulmonary arterial function x i (t) x 1 (t)-kth mean of blood pressure, (corr. coef.), gene PIX no. M 10 BP(t) (0.9999), 4745 ( ), 2249 M 14 BP(t) (0.9999), 3301 ( ), 8113 x 1 (t)-kth mean of energy of oscillation, (corr. coef.), gene PIX no. M 9 of E 6 (t) (0.9999), 7724 ( ), 802 M 13 of E 6 (t) (0.9998), 3165 ( ), 9488 x 3 (t)-opening angle of pulmonary artery, (corr. coef.), gene PIX no. Artery region 1 (0.9525), 4754 ( ), 9158 Artery region 4 (0.9263), 870 ( ), 9156 Artery region 8 (0.9496), 369 ( ), 2021 x 4 (t)-media thickness of pulmonary artery, (corr. coef.), gene PIX no. Artery region 1 (0.9792), 7360 ( ), 27 Artery region 4 (0.9676), 4082 ( ), 27 Artery region 8 (0.9735), 8715 ( ), 27 x 5 (t)-adventitia thickness of pulmonary artery, (corr. coef.), gene PIX no. Artery region 1 (0.9488), 8339 ( ), 870 Artery region 4 (0.9440), 8715 ( ), 1648 Artery region 6 (0.9502), 3073 ( ), 1648 x 6 (t)-inner circ. length of pulmonary artery, (corr. coef.), gene PIX no. Artery region 1 (0.9999), 4052 ( ), 6612 Artery region 4 (0.9999), 977 ( ), 7594 Artery region 8 (0.9998), 7203 ( ), 6146 x 7 (t)-outer circ. length of pulmonary artery, (corr. coef.), gene PIX no. Artery region 1 (0.9999), 3655 ( ), 7282 Artery region 4 (0.9992), 6567 ( ), 1968 Artery region 8 (0.9999), 9317 ( ), 7496 corr. coef., correlation coefficient cgi doi pnas Huang et al.

5 Fig. 5. Activity of the gene PIX no (pleckstrin 2 homolog gene) vs. the opening angle of pulmonary arterial trunk. E, (open angle) [(1 N) N 1 ( open angle) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. Fig. 7. Activity of the gene PIX no (osteoblast-specific factor 2, fasciclin I-like gene) vs. the thickness of adventitia in pulmonary arterial trunk. E, (H adv ) [(1 N) N 1 ( H adv ) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. (iv) Correlation of the Activity of Pulmonary Arterial Genes with the Oxygen Concentration in Breathing Gas. When po 2 level is a step function, the top five genes whose activity matches the hypoxic levels are listed in Tables 1 and 2. (v) Correlation of the Activity of Pulmonary Arterial Genes with Pulmonary Arterial Opening Angle. The opening angle defines the zero-stress state. Its change indicates tissue remodeling. The distribution of the number of genes as a function of the correlation coefficient R is shown in Fig. 2. The top pulmonary arterial genes that correlate with the opening angle changes are listed in Tables 1 and 2; the correlation is illustrated in Fig. 5. The pulmonary arteries are labeled by Region numbers. Region 1 is the pulmonary arterial trunk. Region 1 bifurcates into two smaller Region 2 vessels, and so on. Eight consecutive regions of the main pulmonary artery were studied. The relationship between region numbers and generation numbers is given in W.H. et al. (5). (vi) Correlation of the Activity of Pulmonary Arterial Genes with Arterial Intima-Media Layer Thickness. The endothelium and intima of the pulmonary artery have a thickness of only 2 5 m, hence measurements were made on the combined thickness of intimamedia layer. The media layer is composed of smooth muscle cells and elastin. The thickness change is a measure of tissue remodeling. Data are given in Tables 1 and 2 and illustrated in Fig. 6. (vii) Correlation of the Activity of the Pulmonary Arterial Genes with Arterial Adventitia Layer Thickness. The adventitia is the outer layer of the artery. It is composed of collagen, fibroblasts, and ground substances. The remodeling of this layer after a step hypoxia is shown in Fig. 7 and Tables 1 and 2. (viii) Correlation of the Activity of Pulmonary Arterial Genes with the Circumferential Lengths of the Inner and Outer Walls of the Pulmonary Arteries at the Zero-Stress State. We measured the remodeling of the circumferential length of the inner and outer arterial walls at the zero-stress state. The results are summarized in Fig. 8 and Tables 1 and 2. These, and the opening angle, are fundamental characteristics of tissue remodeling. (ix) Correlation of the Activity of Pulmonary Arterial Genes with the Young s Modulus of Elasticity of the Arteries. Elasticity measurements were done on the Region 2 pulmonary artery. The results are presented in ref. 11. The matching with gene activities is shown in Fig. 9 and Tables 1 and 2. The vessel wall is treated as a circular cylindrical shell with anisotropic elasticity. Under a transmural pressure, the radial stress is much smaller than the circumferential stress and can be neglected. The circumferential and longitudinal stresses and strains are related by a biaxial relationship, which can be linearized in the neighborhood of the in vivo state. In Fung and Liu (18) it is shown that the constitutive equation requires three elastic moduli. These are the Young s modulus in the circumferential direction, Young s modulus in the longitudinal direction, and a Cross modulus between the longitudinal and circumferential directions. These three moduli were measured on specimens with samples collected at specific instants of time after the initiation of step hypoxic breathing and pulmonary arterial hypertension. Data on all three moduli are given in ref. 11. Fig. 9 illustrates the matching of the time course of the Young s modulus in the circumferential direction with gene activity. (x) Correlation of Various Rates and Integrals of Pulmonary Arterial Gene Activities and Physiological Parameters. It is reasonable to expect that the remodeling of the thickness of the media and adventitia layers of the blood vessel and the opening angle at zero-stress state, which involves the addition or subtraction of materials, be the result of cumulative integrated effect of gene activities. Hence we examined the correlation of the integral of the gene activity with physiological parameters. The experimental data were fitted with Eq. 3, integrated from 0 to t, then substituting y j in Eq. 1 by 0 T y j (t)dt and computing the correlation over a total period T. The results are illustrated in Table 3. The correlation between the integral of gene activities and the opening angle lies in the range of at best. On the other hand, the integrated activity of some arterial genes shows MEDICAL SCIENCES ENGINEERING Fig. 6. Activity of the gene PIX no (inorganic pyrophosphatase gene) vs. the media thickness of pulmonary arterial trunk. E, (H med ) [(1 N) N 1 ( H med ) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. Fig. 8. Activity of the gene PIX no (GTP-binding protein 2 gene) vs. the inner circumference of the pulmonary arterial trunk at zero-stress state. E, (Inner Circumference) [(1 N) N 1 ( Inner Circumference) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. Huang et al. PNAS March 5, 2002 vol. 99 no

6 Fig. 9. Activity of the gene PIX no. 44 (dynein, cytoplasmic light polypeptide gene) vs. the change of Young s modulus of elasticity of the pulmonary artery in region 2, Y, relating circumferential stress and strain. E, (Y ) [(1 N) N 1 ( Y ) 2 ] 1 2, one SEM flag up; }, gene activity, one SEM flag down. excellent correlation with the thicknesses of the arterial media and adventitia layers; with positive correlation coefficients well above 0.98, but the genes with the best correlation for the integral test were not the same as those for the straight contest listed in Tables 1 and 2. There was no good negative correlation between the integrated gene activity and wall thickness. We tested also the correlation of the first derivatives of the gene activity with the first derivatives of the mean blood pressure. We found remarkably high values of positive and negative correlation coefficients in the ranges of ( ) and ( ), respectively. But again these genes are not the same as those with the highest coefficient of correlation when tested without differentiation (listed in Tables 1 and 2). (xi) Correlation of the Activity of Pulmonary Venous Genes with Pulmonary Arterial Tissue Remodeling. Thus far we have considered the association of arterial genes with arterial physiology in the pulmonary circulation. How about the relationship between the pulmonary venous genes and pulmonary arterial physiology? We collected tissue samples of pulmonary arteries and veins at the same time and studied their gene activities the same way. The results are illustrated in Table 4. It is seen that the top genes in the pulmonary veins correlate very well with the physiological variables of the pulmonary arteries, but the top venous genes are not the same as the top arterial genes. Discussion The choice of 9,600 gene probes from a human gene library was arbitrary and limiting, considering that the human genome contains some 35,000 genes (19, 20). Other than the genes in pulmonary veins, we studied only genes and tissues in the pulmonary arteries. The range of questions asked was incomplete. Most likely, other methods to quantify the activity of a gene will come in the future. As new methods are developed, it would be important to apply them and compare the results with the colorimetry method used in this study. The essence of this article is to demonstrate the power of matching the dynamics (i.e., the continuous changes with respect to time) of gene activities with the dynamics of physiological functions. The correlation coefficient is a convenient index of the matching. In dealing with the physiological data that were measured at unevenly distributed instants of time, we converted the integrals in Eq. 1 to summations, so that the R(x i, y j )is x i t k y j t k t k x i 2 t k t k y j 2 t k t k 1/2, [8] where the summation is over k 1, 2,..., N, t k s are the instants of time, x i (t k ) and y j (t k ) are the mean values of the measured data at time t k, t k is the interval of time associated with t k, and N is the total number of data points. Because x i (t 1 ) and y j (t 1 ) are zero for the changes of x i and y j from the steady state, there are two sets of N-1 nonvanishing data points x i, y j, and N-1 time intervals. If we took t k t N (N 1) for all k from1ton, we obtain the results of R ij presented in this article. If we took t k t k t k 1, we obtain a set of different R ij s, which are fairly similar to the numbers presented here. Other mathematical methods to identify correlation, and the documentation of the effects of many details in the protocol of collecting tissue specimens for gene activity measurements, especially the temperature, the bubbling with a gas mixture of 95% O 2 5% CO 2 during tissue collection, and the length of tissue collection time are still under investigation. Concluding Remarks The correlation between gene activity with physiological functions has put some order to the great army of genes in a panoramic view. We believe that the method described here is an effective way to study the relationship among genes, physiology, and biomechanics. Future Goal. The present article is designed to identify the genes in blood vessels that are closely associated with the mechanical stress induced by the blood pressure in an animal. To link the mechanics of the genes with the mechanics of the tissues is the ultimate goal of our research. 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