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Regulation of gene expression by Goodwin’s loop with many genes

Regulation of gene expression by Goodwin’s loop with many genes Our consideration is much simpler and includes the case with many genes with repressed or induced transcription. We hope that it will be more intelligible to undergraduate students. 2. THE SCHEME OF THE MODEL We wish to analyze dynamical features of the loops containing both repressed and induced genes, whose products (proteins) undergo many changes before their interaction with the next gene. Let us assume that the loop contains a number of genes. Some of those genes are down regulated and some of them are up regulated by proteins encoded in the loop. The structure of the system is sketched in Figure 1. For the rate of transcription we accept the usual formulae of cooperative kinetics [1]. We assume that the rates of repressed and induced transcriptions are respectively proportional to or to ; x is a concentration of the transcription factor ( protein), a and b are constants and n is the Hill coef¿cient. Fig. 1. Goodwin's loop with a few up- and down- regulated genes. Symbols X1, Xp+1, Xq+1 refer to mRNA. The rest of Xi denote proteins, gj are regulated genes 3. GOODWIN'S LOOP CONTAINING SINGLE REPRESSED GENE Goodwin's loop containing single repressed gene can be represented by a set of ordinary differential equations (1): x1 xi a x1 , 1 xsl hi xi 1 ki xi , i 2, 3,..., s (1) where dimensionless variables xi are used with unit concentration adjusted to a concentration of xs , which makes the rate of transcription equal to the half of its maximum value a. The unit time is equal to the mean lifetime of mRNA (x1). In equations (1), l is the Hill coef¿cient of cooperativity. The symbols hi and ki refer to rate constants of synthesis (hi) and decay (ki) of the corresponding species. The system (1) has a single equilibrium point determined by the relations: xi j i 1 s j i 1 hj xs , i 1,..., s 1. a 1 xsl D xs , k h j 2 j 2 s (2) where xi denotes the values of the corresponding variable xi in equilibrium points. The derivative of the nonlinear term in the ¿rst equation of (1) with respect to xs is: alxsl 1 , 1 xsl 2 (3) where f1 replaces the function on the right of the ¿rst equation (1). In the equilibrium point the value of this derivative (see (2)) can be expressed as: . (4) The diagonal of the Jacobian matrix of the system (1) consists of the species decay rate constants (kj). Positions (j, j-1) cover the rate constants for the production of corresponding species (hj). The position (1, s) is occupied by the derivative g. The structure of the Jacobian determines the following equation for its eigenvalues ( ): . (5) At what values of parameters does the equilibrium become unstable? Is it possible to have a Hopf bifurcation in this system? Suf¿ciently simple analytical solution of (5) can be found only for s=1 or s=2. One could ¿nd an answer to the second question, assuming for (5) pure imaginary and searching for conditions consistent with such an assumption. This way is effective for s not higher than 6. For the higher orders, equation (5) can be solved analytically only under severe simpli¿cation that the system has very high symmetry. If we assume that all k j 1 , then (5) can be written as: 1 O s It follows from (6) that Ok g h j cos S i sin S j 2 (6) s ·s § 2k 1 S i § g s h · s sin 2k 1 S , ¨ j¸ 1 ¨ g h j ¸ cos ¸ ¨ ¸ ¨ s s © j 2 ¹ © j 2 ¹ 0,1..., s 1. (7 ) (7) The equilibrium will be unstable if the real part of any is positive. So far as we are looking for the minimum conditions of the equilibrium destabilization, we should consider those roots (7), which have the largest real parts. The two complex conjugate roots 0 and s-1 have the highest real parts. They will be positive and consistent with (4), if . (8) Taking into account the de¿nition of the (2) and the assumption , one can conclude that double inequality (8) can be satis¿ed only under the condition . (9) So, Hopf bifurcation in system (1) is possible, if the Hill coef¿cient for cooperative repression is high enough and s 3. Let us note two features of the eigenvalues (7). All of them are placed in a complex plane on the circle with the radius equal to (10) centered at the point (-1, 0). In the particular case . At the bifurcation, when Re( 0)=0, and . (11) (10a) The imaginary part of this eigenvalue, or the cyclic frequency of small oscillations around the equilibrium, is given by (11). 4. GOODWIN'S LOOP CONTAINING A SINGLE INDUCED GENE The system of equations describing such a loop differs from the system (1) solely in the ¿rst equation referring to transcription. x1 xi a xsl x1 1 xsl hi xi 1 ki xi , i 2, 3,..., s . (12) We can repeat calculations similarly to those done in the previous section and obtain corresponding relations determining the coordinates of equilibrium points: (13) where is de¿ned exactly in the same way as in (2). As it follows from (13), the equilibrium is found in the origin of coordinates. Other equilibrium points can be found at xs0. . (14) The left hand side of (14) has a maximum at x sl l 1 . The additional points of equilibrium exist, when the maximum value of this function is higher than /a. In the opposite case, graphs of the functions of xs on the left and on the right sides of (14) will have no common points and equation (14) will have no real solutions: . (15) The derivative of the nonlinear term in (12) is alxsl 1 . 1 xsl 2 (16) The derivative (16) has the structure which is fully analogical to that of (3). They differ only by the sign. Let us note that the derivative (16) is equal to zero in the origin of coordinates. The value of this derivative in other equilibrium points can be obtained by inserting a from (14) into (16). This results in formula: . (17) The Jacobian structure of the system (12) is exactly the same as it was in (1). In consequence, characteristic equation of eigenvalues will also have the same structure: . (18) This time, however, the right hand side of the equation is not negative. It follows immediately from (16) that equilibrium in the origin of coordinates is stable for any physically meaningful values of the rate constants. For g=0 the solution of (18) becomes . All of the eigenvalues are real and negative and the equilibrium is stable. Polynomial in on the left of (18) is a monotonously increasing function of its argument for positive . For =0, the polynomial has the value eigenvalues appear only for . (19) . Positive real The case of both sides of (19) equal to each other corresponds to a saddle-node bifurcation. Taking into account that g satis¿es at the same time (17), we arrive at the conclusion that bifurcation takes place for equilibrium value of xs,, when x sl l 1 . So, the condition for the existence of nonzero equilibrium points and the condition for unstable equilibrium are the same. Let us shortly consider the very symmetrical case with all kj=1. In such a case characteristic equation (18) acquires the shape: . (20) And eigenvalues are: . There is at least one real eigenvalue ( 0). It will be positive when the inequality (22) is satis¿ed. . (22) In this case (kj=1) inequalities (22) and (19) are equivalent. At a maximum transcription rate (a) which is high enough in comparison with the overall rate of proteins decay (15), the loop under consideration is a bistable trigger with one of the stable equilibrium points in the origin of coordinates and the second at . In the opposite case the system has only one equilibrium with all of the variables equal to zero. 5. GOODWIN'S LOOP CONTAINING MANY REGULATED GENES We will not present in detail the systems with two regulated genes. It can be shown that a loop containing two repressed genes behaves as a bistable trigger at high enough maximum rate of transcription. Two stable equilibrium points correspond to high concentrations of one of the regulating proteins and low concentrations of the other. There is no equilibrium in the origin of coordinates. There are no oscillations, either. At the low rate of transcription, a loop containing two induced genes behaves like the loop with a single induced gene. It can have an equilibrium point for all the variables equal to zero. Alternatively, it can have two additional points of equilibrium outside the origin of coordinates. One of these points is unstable and the second one is stable. The loop behaves then as a bistable trigger. As an example of the loop with many regulated genes we are going to discuss the loop containing three regulated genes. One of these genes is repressed and two of them are induced. The system can be represented by a set of ordinary differential equations (23): (23) Nonlinear expressions in equations 1, q+1 and r+1 refer to gene transcription with a repression or induction. Dynamical variables with the mentioned indices refer to mRNA concentration. The rest of variables denotes concentrations of proteins. Equations (23) have been written in a dimensionless form. The unit time is equal to the mean lifetime of the mRNA, whose concentration is denoted by x1 in equations (23). Parameters R2 and R3 denote the ratio of the decay constants between xq+1, xr+1 and x1 respectively. Concentrations units are not the same for all variables. It follows from equations 1, q+1 and r+1 in (23) that unit values of xs, xq and xr make the rates of corresponding transcriptions equal to halves of their maximum values. The variables x1...xq are expressed in the same units as xq. The variables xq+1 ...xr have the same concentration units as xr. Similarly, variables xr+1 ...xs have the same concentration units as xs. Equating the right sides of linear equations in (23), we can relate equilibrium values of all the variables to those of xq , xr and xs . (24) (25) (26) Equilibrium values of xq, xr and xs can be obtained from equations (27): . (27) To linearize the system (23), we will calculate derivatives of the nonlinear terms with respect to their arguments: C1l xsl 1 , 1 xsl 2 wf q 1 wxq m C2 m xq 1 1 x m q , 2 wf r 1 wxr C3n xrn 1 . 1 xrn 2 (28) The symbols f1 , fq+1 and fr+1 refer to functions in the right sides of the corresponding equations in (23). If we put the values of C1, C2 and C3 obtained from (27) into (28), we will arrive to expressions for the values of the derivatives in the equilibrium point: (29) (30) (31) Let us notice that the absolute value of the product (32) has an upper bound equal to the limit value at x s o f and x q x r 0 (32). Now, we ought to ¿nd eigenvalues of the Jacobian matrix of the system. (33) We have to transform the determinant (33) into polynomial and solve the resulting equation. The Jacobian contains two nonzero terms in each row. One of these terms is situated on the main diagonal, in the position (j, j) and is equal to the rate constant of decay of the respective substance. The other nonzero terms, the rate synthesis constants of the respective substances, have positions (j, j-1). In the rows 1, q+1 and r+1, the values of derivatives of the terms describing transcription are placed in these positions instead of rate constants of synthesis. Derivatives are positive in the case of induction and they are negative in the case of repression. In the ¿rst row the derivative g1 is situated in the position (1, s). Equation (33) is equivalent to polynomial equation (34). (34) Because of the repression, there is no equilibrium in the origin of coordinates. Neither of derivatives g is equal to zero, but their product can be positive or negative. When the right hand side of (34) is positive, the system has a phase portrait of a trigger. When it is negative, the system evolves similar to that with one repressed gene. If parameters satisfy some additional condition, the system will be able to show auto-oscillations. These conditions consist in high enough maximum rates of transcription C1 , C 2 , C 3 and decay constants R2 , R3 , k j not too much different from 1. In the particular case of the system (23) the product of gi and the right side of (34) is negative. Equation (34) has no simple solution in general case. But it is solvable under quite severe conditions that (35) The conditions (35) are equivalent to the assumption that all reagents of the system have the same mean rate constants of their decay. If it is so, then equation (34) can be written as (36) Let us to notice that under condition (35) the absolute value of g1g2g3 (32) (37) Taking into account that, according to (24­26 and 35) (38) 44 we obtain the relation (39): (39) The insertion of (39) into (36) results in the equation for eigenvalues . . (40) It is evident that the absolute value of the right hand part of (40) satis¿es the inequality: . (41) In the case of the negative product of derivatives, at odd number of repressions, the equation (40) can be written as: . and . (43) 1 s (42) All of the eigenvalues (43) are located on a circle with the radius R with the centre in the point (-1, 0) on the complex plane. and Im Re Fig. 2. Eigenvalues (21): a) R<1, b) R=1, c) R>1 The roots 0 and s-1 have the same and the largest real part. Equilibrium of our system will be destabilized through Hopf bifurcation when the real part of these eigenvalues becomes positive. It happens when . (44) At the same time R should satisfy the inequality (41). It is possible, if . (45) Relation (45) and a negative product of nonlinear term derivatives in (23) constitute the necessary conditions of Hopf bifurcation in the system with a single repressed and two induced genes. This result can be generalized on a long loop with an odd number of repressed genes and with any number of induced genes. So, an oscillatory solution is possible in a Goodwin's loop when it contains an odd number of repressed genes and has the product of Hill coef¿cients higher than , where s is the number of chemical species participating in the loop or the order of the system of ordinary differential equations describing the loop. The main requirement for the existence of oscillatory solutions in the loops under consideration is an odd number of repressed genes. However, the presence of cooperative induction makes the product of Hill coef¿cients higher. On the other hand, at the presence of many protein interconvertions, even described by linear equations, s is higher and the extreme left side of (45) is lower. Both factors make oscillatory solution more probable. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Physica de Gruyter

Regulation of gene expression by Goodwin’s loop with many genes

Sielewiesiuk, Jan; Łopaciuk, Agata
Annales UMCS, Physica , Volume 67 (1) – Jan 1, 2012
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de Gruyter
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Copyright © 2012 by the
ISSN
0137-6861
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10.2478/v10246-012-0013-6
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Abstract

Our consideration is much simpler and includes the case with many genes with repressed or induced transcription. We hope that it will be more intelligible to undergraduate students. 2. THE SCHEME OF THE MODEL We wish to analyze dynamical features of the loops containing both repressed and induced genes, whose products (proteins) undergo many changes before their interaction with the next gene. Let us assume that the loop contains a number of genes. Some of those genes are down regulated and some of them are up regulated by proteins encoded in the loop. The structure of the system is sketched in Figure 1. For the rate of transcription we accept the usual formulae of cooperative kinetics [1]. We assume that the rates of repressed and induced transcriptions are respectively proportional to or to ; x is a concentration of the transcription factor ( protein), a and b are constants and n is the Hill coef¿cient. Fig. 1. Goodwin's loop with a few up- and down- regulated genes. Symbols X1, Xp+1, Xq+1 refer to mRNA. The rest of Xi denote proteins, gj are regulated genes 3. GOODWIN'S LOOP CONTAINING SINGLE REPRESSED GENE Goodwin's loop containing single repressed gene can be represented by a set of ordinary differential equations (1): x1 xi a x1 , 1 xsl hi xi 1 ki xi , i 2, 3,..., s (1) where dimensionless variables xi are used with unit concentration adjusted to a concentration of xs , which makes the rate of transcription equal to the half of its maximum value a. The unit time is equal to the mean lifetime of mRNA (x1). In equations (1), l is the Hill coef¿cient of cooperativity. The symbols hi and ki refer to rate constants of synthesis (hi) and decay (ki) of the corresponding species. The system (1) has a single equilibrium point determined by the relations: xi j i 1 s j i 1 hj xs , i 1,..., s 1. a 1 xsl D xs , k h j 2 j 2 s (2) where xi denotes the values of the corresponding variable xi in equilibrium points. The derivative of the nonlinear term in the ¿rst equation of (1) with respect to xs is: alxsl 1 , 1 xsl 2 (3) where f1 replaces the function on the right of the ¿rst equation (1). In the equilibrium point the value of this derivative (see (2)) can be expressed as: . (4) The diagonal of the Jacobian matrix of the system (1) consists of the species decay rate constants (kj). Positions (j, j-1) cover the rate constants for the production of corresponding species (hj). The position (1, s) is occupied by the derivative g. The structure of the Jacobian determines the following equation for its eigenvalues ( ): . (5) At what values of parameters does the equilibrium become unstable? Is it possible to have a Hopf bifurcation in this system? Suf¿ciently simple analytical solution of (5) can be found only for s=1 or s=2. One could ¿nd an answer to the second question, assuming for (5) pure imaginary and searching for conditions consistent with such an assumption. This way is effective for s not higher than 6. For the higher orders, equation (5) can be solved analytically only under severe simpli¿cation that the system has very high symmetry. If we assume that all k j 1 , then (5) can be written as: 1 O s It follows from (6) that Ok g h j cos S i sin S j 2 (6) s ·s § 2k 1 S i § g s h · s sin 2k 1 S , ¨ j¸ 1 ¨ g h j ¸ cos ¸ ¨ ¸ ¨ s s © j 2 ¹ © j 2 ¹ 0,1..., s 1. (7 ) (7) The equilibrium will be unstable if the real part of any is positive. So far as we are looking for the minimum conditions of the equilibrium destabilization, we should consider those roots (7), which have the largest real parts. The two complex conjugate roots 0 and s-1 have the highest real parts. They will be positive and consistent with (4), if . (8) Taking into account the de¿nition of the (2) and the assumption , one can conclude that double inequality (8) can be satis¿ed only under the condition . (9) So, Hopf bifurcation in system (1) is possible, if the Hill coef¿cient for cooperative repression is high enough and s 3. Let us note two features of the eigenvalues (7). All of them are placed in a complex plane on the circle with the radius equal to (10) centered at the point (-1, 0). In the particular case . At the bifurcation, when Re( 0)=0, and . (11) (10a) The imaginary part of this eigenvalue, or the cyclic frequency of small oscillations around the equilibrium, is given by (11). 4. GOODWIN'S LOOP CONTAINING A SINGLE INDUCED GENE The system of equations describing such a loop differs from the system (1) solely in the ¿rst equation referring to transcription. x1 xi a xsl x1 1 xsl hi xi 1 ki xi , i 2, 3,..., s . (12) We can repeat calculations similarly to those done in the previous section and obtain corresponding relations determining the coordinates of equilibrium points: (13) where is de¿ned exactly in the same way as in (2). As it follows from (13), the equilibrium is found in the origin of coordinates. Other equilibrium points can be found at xs0. . (14) The left hand side of (14) has a maximum at x sl l 1 . The additional points of equilibrium exist, when the maximum value of this function is higher than /a. In the opposite case, graphs of the functions of xs on the left and on the right sides of (14) will have no common points and equation (14) will have no real solutions: . (15) The derivative of the nonlinear term in (12) is alxsl 1 . 1 xsl 2 (16) The derivative (16) has the structure which is fully analogical to that of (3). They differ only by the sign. Let us note that the derivative (16) is equal to zero in the origin of coordinates. The value of this derivative in other equilibrium points can be obtained by inserting a from (14) into (16). This results in formula: . (17) The Jacobian structure of the system (12) is exactly the same as it was in (1). In consequence, characteristic equation of eigenvalues will also have the same structure: . (18) This time, however, the right hand side of the equation is not negative. It follows immediately from (16) that equilibrium in the origin of coordinates is stable for any physically meaningful values of the rate constants. For g=0 the solution of (18) becomes . All of the eigenvalues are real and negative and the equilibrium is stable. Polynomial in on the left of (18) is a monotonously increasing function of its argument for positive . For =0, the polynomial has the value eigenvalues appear only for . (19) . Positive real The case of both sides of (19) equal to each other corresponds to a saddle-node bifurcation. Taking into account that g satis¿es at the same time (17), we arrive at the conclusion that bifurcation takes place for equilibrium value of xs,, when x sl l 1 . So, the condition for the existence of nonzero equilibrium points and the condition for unstable equilibrium are the same. Let us shortly consider the very symmetrical case with all kj=1. In such a case characteristic equation (18) acquires the shape: . (20) And eigenvalues are: . There is at least one real eigenvalue ( 0). It will be positive when the inequality (22) is satis¿ed. . (22) In this case (kj=1) inequalities (22) and (19) are equivalent. At a maximum transcription rate (a) which is high enough in comparison with the overall rate of proteins decay (15), the loop under consideration is a bistable trigger with one of the stable equilibrium points in the origin of coordinates and the second at . In the opposite case the system has only one equilibrium with all of the variables equal to zero. 5. GOODWIN'S LOOP CONTAINING MANY REGULATED GENES We will not present in detail the systems with two regulated genes. It can be shown that a loop containing two repressed genes behaves as a bistable trigger at high enough maximum rate of transcription. Two stable equilibrium points correspond to high concentrations of one of the regulating proteins and low concentrations of the other. There is no equilibrium in the origin of coordinates. There are no oscillations, either. At the low rate of transcription, a loop containing two induced genes behaves like the loop with a single induced gene. It can have an equilibrium point for all the variables equal to zero. Alternatively, it can have two additional points of equilibrium outside the origin of coordinates. One of these points is unstable and the second one is stable. The loop behaves then as a bistable trigger. As an example of the loop with many regulated genes we are going to discuss the loop containing three regulated genes. One of these genes is repressed and two of them are induced. The system can be represented by a set of ordinary differential equations (23): (23) Nonlinear expressions in equations 1, q+1 and r+1 refer to gene transcription with a repression or induction. Dynamical variables with the mentioned indices refer to mRNA concentration. The rest of variables denotes concentrations of proteins. Equations (23) have been written in a dimensionless form. The unit time is equal to the mean lifetime of the mRNA, whose concentration is denoted by x1 in equations (23). Parameters R2 and R3 denote the ratio of the decay constants between xq+1, xr+1 and x1 respectively. Concentrations units are not the same for all variables. It follows from equations 1, q+1 and r+1 in (23) that unit values of xs, xq and xr make the rates of corresponding transcriptions equal to halves of their maximum values. The variables x1...xq are expressed in the same units as xq. The variables xq+1 ...xr have the same concentration units as xr. Similarly, variables xr+1 ...xs have the same concentration units as xs. Equating the right sides of linear equations in (23), we can relate equilibrium values of all the variables to those of xq , xr and xs . (24) (25) (26) Equilibrium values of xq, xr and xs can be obtained from equations (27): . (27) To linearize the system (23), we will calculate derivatives of the nonlinear terms with respect to their arguments: C1l xsl 1 , 1 xsl 2 wf q 1 wxq m C2 m xq 1 1 x m q , 2 wf r 1 wxr C3n xrn 1 . 1 xrn 2 (28) The symbols f1 , fq+1 and fr+1 refer to functions in the right sides of the corresponding equations in (23). If we put the values of C1, C2 and C3 obtained from (27) into (28), we will arrive to expressions for the values of the derivatives in the equilibrium point: (29) (30) (31) Let us notice that the absolute value of the product (32) has an upper bound equal to the limit value at x s o f and x q x r 0 (32). Now, we ought to ¿nd eigenvalues of the Jacobian matrix of the system. (33) We have to transform the determinant (33) into polynomial and solve the resulting equation. The Jacobian contains two nonzero terms in each row. One of these terms is situated on the main diagonal, in the position (j, j) and is equal to the rate constant of decay of the respective substance. The other nonzero terms, the rate synthesis constants of the respective substances, have positions (j, j-1). In the rows 1, q+1 and r+1, the values of derivatives of the terms describing transcription are placed in these positions instead of rate constants of synthesis. Derivatives are positive in the case of induction and they are negative in the case of repression. In the ¿rst row the derivative g1 is situated in the position (1, s). Equation (33) is equivalent to polynomial equation (34). (34) Because of the repression, there is no equilibrium in the origin of coordinates. Neither of derivatives g is equal to zero, but their product can be positive or negative. When the right hand side of (34) is positive, the system has a phase portrait of a trigger. When it is negative, the system evolves similar to that with one repressed gene. If parameters satisfy some additional condition, the system will be able to show auto-oscillations. These conditions consist in high enough maximum rates of transcription C1 , C 2 , C 3 and decay constants R2 , R3 , k j not too much different from 1. In the particular case of the system (23) the product of gi and the right side of (34) is negative. Equation (34) has no simple solution in general case. But it is solvable under quite severe conditions that (35) The conditions (35) are equivalent to the assumption that all reagents of the system have the same mean rate constants of their decay. If it is so, then equation (34) can be written as (36) Let us to notice that under condition (35) the absolute value of g1g2g3 (32) (37) Taking into account that, according to (24­26 and 35) (38) 44 we obtain the relation (39): (39) The insertion of (39) into (36) results in the equation for eigenvalues . . (40) It is evident that the absolute value of the right hand part of (40) satis¿es the inequality: . (41) In the case of the negative product of derivatives, at odd number of repressions, the equation (40) can be written as: . and . (43) 1 s (42) All of the eigenvalues (43) are located on a circle with the radius R with the centre in the point (-1, 0) on the complex plane. and Im Re Fig. 2. Eigenvalues (21): a) R<1, b) R=1, c) R>1 The roots 0 and s-1 have the same and the largest real part. Equilibrium of our system will be destabilized through Hopf bifurcation when the real part of these eigenvalues becomes positive. It happens when . (44) At the same time R should satisfy the inequality (41). It is possible, if . (45) Relation (45) and a negative product of nonlinear term derivatives in (23) constitute the necessary conditions of Hopf bifurcation in the system with a single repressed and two induced genes. This result can be generalized on a long loop with an odd number of repressed genes and with any number of induced genes. So, an oscillatory solution is possible in a Goodwin's loop when it contains an odd number of repressed genes and has the product of Hill coef¿cients higher than , where s is the number of chemical species participating in the loop or the order of the system of ordinary differential equations describing the loop. The main requirement for the existence of oscillatory solutions in the loops under consideration is an odd number of repressed genes. However, the presence of cooperative induction makes the product of Hill coef¿cients higher. On the other hand, at the presence of many protein interconvertions, even described by linear equations, s is higher and the extreme left side of (45) is lower. Both factors make oscillatory solution more probable.

Journal

Annales UMCS, Physica – de Gruyter

Published: Jan 1, 2012

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