This article was downloaded by: [Rutgers University] On: 07 April 2015, At: 07:49 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Biomolecular Structure and Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tbsd20
An Effective Enzyme Interacting with Poly (dTdA)· Poly (dT-dA): A Dynamic Enhancer-Repressor Action a
R. Beger & E. W. Prohofsky
a
a
Department of Physics , Purdue University , West Lafayette , IN , 47907 Published online: 21 May 2012.
To cite this article: R. Beger & E. W. Prohofsky (1991) An Effective Enzyme Interacting with Poly (dT-dA)· Poly (dT-dA): A Dynamic Enhancer-Repressor Action, Journal of Biomolecular Structure and Dynamics, 9:2, 239-249, DOI: 10.1080/07391102.1991.10507910 To link to this article: http://dx.doi.org/10.1080/07391102.1991.10507910
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Downloaded by [Rutgers University] at 07:49 07 April 2015
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102 Volume 9, Issue Number 2 (1991), "'Adenine Press (1991).
An Effective Enzyme Interacting with Poly (dT-dA) · Poly (dT-dA): A Dynamic Enhancer-Repressor Action
Downloaded by [Rutgers University] at 07:49 07 April 2015
R. Beger and E.W. Prohofsky Department of Physics Purdue University West Lafayette, IN 47907 Abstract The Green function technique is used to study the open hydrogen bond probability of poly(dT-d.A) · poly(dT-d.A) when an effective enzyme is attached to the helix. The DNAinterstrand hydrogen bond mean motion and probability of fluctuating to an open state depends on the internal vibrational frequency of the enzyme. An enzyme with internal frequency of 80 em -I reduces hydrogen bond motion and the resulting probability of hydrogen bond fluctuational opening. An enzyme with internal frequency of72 em -I increases hydrogen bond motion and the probability of hydrogen bond breaking.
Introduction There has been considerable interest recently in relating enzyme-DNA dynamics to biological activity. Computer dynamics calculations indicate that proteins may sustain large scale fluctuations [1 ). How these fluctuations influence biological activity is not clear. Considerable effort has also been expended by a number groups in the study of the dynamics of the DNA double helix [2-9). The dynamics of extracted DNA is in itself a very interesting problem. However, much of the motivation for the study of DNA dynamics comes from a desire to understand the mechanisms of DNA function in biological processes. In these processes the DNA is in contact with one to several enzymes. The biologically significant dynamics is then more complicated; it is the dynamics of a DNA-enzyme complex. It is at present not possible to carry out the direct analysis of a DNA-enzyme complex by molecular dynamics simulations. Current simulation deals with ~ 10 base pair segments and can run these for~ 10 picoseconds. Melting of DNA takes place on a time scale of milliseconds not picoseconds. The combined system would also be much larger than the 10 base pair segment as well. The number of coordinates would be 3X(NDNA+Nenzyme>Normal mode analysis ofDNA takes advantage of the screw axis symmetry of DNA to reduce the number of coordinates from 3XNDNA to three times the number of heavy atoms inside a unit cell. The method we develop reduces the dimensionality of the enzyme which has few internal vibrational modes and several points of attachment to the DNA helix. We then solve for the effect this effective system has on
239
240
Beger and Prohofsky
Downloaded by [Rutgers University] at 07:49 07 April 2015
DNA degrees of freedom for the combined system. The particular DNA coordinates studied are the stretch of the interbase hydrogen bonds. The motion in these coordinates can be directly related to the enhanced probability ofbond disruption or transition to an open state (melted base pair) configuration. Normal mode analysis is a natural approach to the dynamics of DNA as relevant resonant modes are observed in DNA by both Raman scattering [3] and infrared absorption (4]. The linewidths have been analyzed and modes are expected to be resonant down to ~ em - 1 frequencies (1OJ. The low frequency modes of importance in strand separation are ~ 85 em - 1 [11] and are not overdamped. Modes at this frequency cannot be resolved by picosecond molecular dynamic simulations. Some form of mode analysis is essential to explore this region of DNA dynamics. Normal mode analysis also has significant advantages over simulations for statistical analysis. The mode analysis used is not simple harmonic mode analysis but makes use of a modified self-consistent phonon approximation (MSPA). The true interaction across the H-bonds is represented by Morse potentials and the MSPA formalism then is used to find an effective harmonic force constant for H-bond interaction to use in mode analysis. The force constant is temperature dependent and softens with increased thermal motion corresponding to the system being at farther out weaker regions of the Morse potential. In a system made of a helix and an effective enzyme the dynamics would be that of an interconnected system of two parts. The important element in causing changes in helix dynamics for that case is the relative motion of the atoms on either side of the helix-effective enzyme attachments. The green function method allows one to find the combined dynamics based on the motion these enzyme atoms would have in the free unconnected enzyme. This motion can be expressed by the vibrational modes of the free enzyme in the Einstein approximation, that is, we replace vibrational bands by a single vibrational frequency with an appropriate amplitude. Any true enzyme would have many frequencies and the effects we probe would have to be integrated over the enzyme spectral density. Enhancement or repression by enzyme ofDNA activity can be caused by a number of mechanisms. The helix secondary and tertiary structure may be altered. Specific docking sites for one enzyme may be covered by a second etc. There is the probability however that dynamic effects, as described here play a role in achieving enhancement or repression. We find that there is a coupling between dynamic factors associated with an enzyme and the tendency for the helix strand separation to occur. We therefore explore this particular avenue for enzyme effects on helix opening. All the attachments we have studied have shown that the thermal hydrogen bond stretch is dependent on resonant frequency of the enzymes. That is some frequencies of a particular enzyme which has amplitude on the attachment atoms enhance strand separation and others frequencies repress strand separation. The distribution density of frequencies in an enzyme can then determine its effect on DNA Biological processes such as transcription and replication requires strand separation. Careful analysis of the use of nucleoside triphosphates in such processes imply that
A Dynamic Enhancer-Repressor Action
241
Downloaded by [Rutgers University] at 07:49 07 April 2015
the actual strand separation processes occur at a net energy loss. Therefore the actual strand separation process such as the elongation of the open state and the initiation of the open state are likely facilitated by thermal fluctuational opening of the interbase hydrogen bonds [12). The energy is supplied by thermal fluctuations. Enhancing these opening probabilities can then enhance all the processes needing strand separation. To study the effects of different places where an enzyme may attach we first study a limited number of attachments locations. We assume that an effective system of some kind attaches over a region of three base pairs of a DNA double helix. We chose poly(dT-dA) · poly(dT-dA) as alternating AT sequences are often found in sites associated with initiation. We study the DNA polymer for three cases. In one case a two point attachment is to the N3 atom of the Adenine base and to the 02 atom of the Thymine base in a base pair which is two base pairs away. This emulates an enzyme attaching through the minor groove and will be called the minor groove attachment. In case two the two point attachment is to the N7 atom of the Adenine base and the methyl group of the Thymine base in a pair two base pairs away. This emulates an enzyme or a drug that attaches to the bases from the major groove side and will be called the major groove attachment. These choices are crudely in accord with some observations of enzyme and drug attachments [13-17). We also study single point attachments which are all the individual attachments in the two point attachments discussed above. For the numerical calculations we assume the attachments are by . mdyne hydrogen bondmg and use a force constant of0.115 ~for all attachments. The distances between a vibrating atom and the atom it attaches to is set at 1. 73 Angstroms. This is appropriate to assuming hydrogen bond attachment. The force constant was varied at some frequencies to see if there was force constant dependence for the resulting dynamics of the DNA helix. The larger the force constant was, the more the resulting modes were altered. This is not a systematic study of the possible enzyme features but it does show that some features ofinteraction can have a major effect on the DNA degrees of freedom. Formalism Normal mode analysis is carried out by setting up a matrix of force constants which is diagonalized to give normal modes. The combined helix enzyme system has a force constant matrix which includes more elements than the separate parts and has the eigen equation. (F - w 2I - C)q
=
0
[1)
where F is the force constant matrix for the separate helix and enzyme with no attachments. w and q are the new eigenvalues and eigenvectors. w is a frequency expressed in em -I, q gives the atom displacements in a mass weighted corrdinate system. When multiplied by an occupation number, or excitation level, q can be converted to atomic displacements in angstroms. The perturbation matrix, C, is the change in F necessary to bring about the attachments of enzyme to the infinite helix. The new structure breaks the helical symmetry and normal mode solutions can only
242
Beger and Prohofsky
be calculated by Green function. Since C is a matrix of relatively low dimensionality, Eq. [1) is solved in terms of the Green function of the unperturbed system. The Green function like q describes the response of the system: see eqn. [3] below. The Green function for the unperturbed and the perturbed systems are defined as 2
g = (w I- F)
-I
Downloaded by [Rutgers University] at 07:49 07 April 2015
and
respectively. They are related by the following Dyson equation [18), G
=
g
+ gTg
[2)
and where T
=
C(l- gC)- 1•
We are interested in the breathing modes which are relatively low frequency ( ~ 85 em -I) vibrations characterized by a large average stretch in the hydrogen bonds and shown to play a significant role in DNA melting. A melting coordinate, s, is defined as the average hydrogen bond stretch amplitude. A thermal mean-square displacement amplitude for the melting coordinate in cell m is then obtained from [19-22]. [3)
which represent the contribution to the hydrogen bond fluctuation from an in band mode of frequency w. In Eq. [3) Tis the temperature. A temperature of293K is used in this particular calculation. The coth term essentially puts in the thermal activation of the Bose-Einstein system. The imaginary part of the Green function is seen to be the response of the system per quantized exicitation i.e. per phonon. The Green function for the unperturbed system can be written in terms of the solution of the isolated helix eigen value equation i.e. Eq. [1] with C = 0. [19). In the internal coordinates, mn 2 g .. (w) ,,
1 "" fn
=- L 7T f...
0
Re(s; (0) l,.• (0) ei(m-n)6) de -----+·-----..,,.,-::-:-(1)
-
~
(0)
[4)
where st..(e), normalized within a unit cell, is the eigenvector belonging to the eigenvalue w~-..(e) for a phase angle ein the internal coordinates. Eq. [4) represents the contributions from all the vibrational modes of the poly(dT-d.A) · poly(dT-d.A) strands to the motion at frequency w. Ifw lies within one or more phonon dispersion branches of the perfect helix then the perfect helix Green function involves an intergration over one or more singularities.
243
A Dynamic Enhancer-Repressor Action
This difficulty may be circumvented by moving this pole off the real axis by giving w a small imaginary part i/r.. Physically, this corresponds to an exponential decay of the normal modes caused by, for example anharmonic interactions causing loss of energy to nearby modes or by dissipative forces such that the molecule would feel in a solution environment. We replace w2 by
Downloaded by [Rutgers University] at 07:49 07 April 2015
(w + ild
=
w2 + 2iw/r: - l/t.2
~
w2
+ zy 0
forsmalll/r.. The use ofw 2 above in Eq. [4] follows that shown byFengetal. [23). The Green's function G is temperature independent and needs to be calculated only once from the initial helix system. The inband thermal mean-squared vibrational amplitude for the m1h hydrogen bond may be obtained by integrating over w in Eq. [3). The probability of a single hydrogen bond, being disrupted in the pre melting state is then given by Y. Z. Chen eta/. [24) as P;
=
C;
f
inf L
[5)
du exp(-(u-R/!W;)
ax
7
where C-L ;-
f
inf
d u e -u 2/2D, .
[6)
h,
Here C; is the normalization factor for the lh hydrogen bond. Lm;x is the maximum value of stretch before the lh hydrogen bond is considered in the open state. D; is the mean square stretch of the th hydrogen bond. R; is the thermal equilibrium length of the lh hydrogen bond. h; is determined from the relation given below V(R;- h;) = V(inf) = 0.
[7)
Table I The Morse Parmeters of AT Base Pairs Bond
a A~l
N(l)-H-N(3) N(6)-H-0(4)
2.402 2.713
vo
mdynA
Lmax
A
2.782 2.698
0.01702 0.01959
3.158 3.146
Ro
A
and the Morse potential is given by
[8) We define a mean open base pair probability that can be calculated by multiplying all the open hydrogen bond probabilities in a certain base pair. The open base pair
244
Beger and Prohofsky .090.---------------------------------~
Downloaded by [Rutgers University] at 07:49 07 April 2015
075
>r-
xx
.060
1-i
_j
+
1-i
~
a
. OLtS
!Xl
0
~ Q_
+
~
** +
X
+
X
030
(!)~(!)~ ~~~ l!{:t
0~
X
**
$(!)(!) C!>(!)
~~~~.~~ f
015
0.00+-----.----.----~----~----~--~
TATATATATATATATATATATA ATATATATATAIATATATATAT
HYDROGEN-BOND Figure 1: Major groove open hydrogen probability ofpoly(dT-d4) · poly(dT-d4) with enzyme attachment at 72.43(cm -I) and an amplitude of0.26 A. The circles are for an N7 of Adenine attachment. Crosses are for a methyl group attachment ofThymine the same base pair. A plus sign is for the N7 attachment in one base and the methyl attachment in a base pair two pairs away. The solid line shows the unattached hydrogen bond opening probability. The underlined bases are where the attachments take place.
probability is then approximately the square (or cube if GC is used) of the single open hydrogen bond probabilities in the base pair. As in ref. 24 we assume a Morse potential as the true potential for the interbase hydrogen bonds. The Morse parameters and maximum length of all the hydrogen bonds used in this calculation are shown in Table I for Adenosine-Thymine.
245
A Dynamic Enhancer-Repressor Action
090.-----------------------------------,
.075
.060 Downloaded by [Rutgers University] at 07:49 07 April 2015
}-
I1-1
_J 1-1
t:Q
0Lt5
Journal of Biomolecular Structure and Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tbsd20
An Effective Enzyme Interacting with Poly (dTdA)· Poly (dT-dA): A Dynamic Enhancer-Repressor Action a
R. Beger & E. W. Prohofsky
a
a
Department of Physics , Purdue University , West Lafayette , IN , 47907 Published online: 21 May 2012.
To cite this article: R. Beger & E. W. Prohofsky (1991) An Effective Enzyme Interacting with Poly (dT-dA)· Poly (dT-dA): A Dynamic Enhancer-Repressor Action, Journal of Biomolecular Structure and Dynamics, 9:2, 239-249, DOI: 10.1080/07391102.1991.10507910 To link to this article: http://dx.doi.org/10.1080/07391102.1991.10507910
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
Downloaded by [Rutgers University] at 07:49 07 April 2015
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102 Volume 9, Issue Number 2 (1991), "'Adenine Press (1991).
An Effective Enzyme Interacting with Poly (dT-dA) · Poly (dT-dA): A Dynamic Enhancer-Repressor Action
Downloaded by [Rutgers University] at 07:49 07 April 2015
R. Beger and E.W. Prohofsky Department of Physics Purdue University West Lafayette, IN 47907 Abstract The Green function technique is used to study the open hydrogen bond probability of poly(dT-d.A) · poly(dT-d.A) when an effective enzyme is attached to the helix. The DNAinterstrand hydrogen bond mean motion and probability of fluctuating to an open state depends on the internal vibrational frequency of the enzyme. An enzyme with internal frequency of 80 em -I reduces hydrogen bond motion and the resulting probability of hydrogen bond fluctuational opening. An enzyme with internal frequency of72 em -I increases hydrogen bond motion and the probability of hydrogen bond breaking.
Introduction There has been considerable interest recently in relating enzyme-DNA dynamics to biological activity. Computer dynamics calculations indicate that proteins may sustain large scale fluctuations [1 ). How these fluctuations influence biological activity is not clear. Considerable effort has also been expended by a number groups in the study of the dynamics of the DNA double helix [2-9). The dynamics of extracted DNA is in itself a very interesting problem. However, much of the motivation for the study of DNA dynamics comes from a desire to understand the mechanisms of DNA function in biological processes. In these processes the DNA is in contact with one to several enzymes. The biologically significant dynamics is then more complicated; it is the dynamics of a DNA-enzyme complex. It is at present not possible to carry out the direct analysis of a DNA-enzyme complex by molecular dynamics simulations. Current simulation deals with ~ 10 base pair segments and can run these for~ 10 picoseconds. Melting of DNA takes place on a time scale of milliseconds not picoseconds. The combined system would also be much larger than the 10 base pair segment as well. The number of coordinates would be 3X(NDNA+Nenzyme>Normal mode analysis ofDNA takes advantage of the screw axis symmetry of DNA to reduce the number of coordinates from 3XNDNA to three times the number of heavy atoms inside a unit cell. The method we develop reduces the dimensionality of the enzyme which has few internal vibrational modes and several points of attachment to the DNA helix. We then solve for the effect this effective system has on
239
240
Beger and Prohofsky
Downloaded by [Rutgers University] at 07:49 07 April 2015
DNA degrees of freedom for the combined system. The particular DNA coordinates studied are the stretch of the interbase hydrogen bonds. The motion in these coordinates can be directly related to the enhanced probability ofbond disruption or transition to an open state (melted base pair) configuration. Normal mode analysis is a natural approach to the dynamics of DNA as relevant resonant modes are observed in DNA by both Raman scattering [3] and infrared absorption (4]. The linewidths have been analyzed and modes are expected to be resonant down to ~ em - 1 frequencies (1OJ. The low frequency modes of importance in strand separation are ~ 85 em - 1 [11] and are not overdamped. Modes at this frequency cannot be resolved by picosecond molecular dynamic simulations. Some form of mode analysis is essential to explore this region of DNA dynamics. Normal mode analysis also has significant advantages over simulations for statistical analysis. The mode analysis used is not simple harmonic mode analysis but makes use of a modified self-consistent phonon approximation (MSPA). The true interaction across the H-bonds is represented by Morse potentials and the MSPA formalism then is used to find an effective harmonic force constant for H-bond interaction to use in mode analysis. The force constant is temperature dependent and softens with increased thermal motion corresponding to the system being at farther out weaker regions of the Morse potential. In a system made of a helix and an effective enzyme the dynamics would be that of an interconnected system of two parts. The important element in causing changes in helix dynamics for that case is the relative motion of the atoms on either side of the helix-effective enzyme attachments. The green function method allows one to find the combined dynamics based on the motion these enzyme atoms would have in the free unconnected enzyme. This motion can be expressed by the vibrational modes of the free enzyme in the Einstein approximation, that is, we replace vibrational bands by a single vibrational frequency with an appropriate amplitude. Any true enzyme would have many frequencies and the effects we probe would have to be integrated over the enzyme spectral density. Enhancement or repression by enzyme ofDNA activity can be caused by a number of mechanisms. The helix secondary and tertiary structure may be altered. Specific docking sites for one enzyme may be covered by a second etc. There is the probability however that dynamic effects, as described here play a role in achieving enhancement or repression. We find that there is a coupling between dynamic factors associated with an enzyme and the tendency for the helix strand separation to occur. We therefore explore this particular avenue for enzyme effects on helix opening. All the attachments we have studied have shown that the thermal hydrogen bond stretch is dependent on resonant frequency of the enzymes. That is some frequencies of a particular enzyme which has amplitude on the attachment atoms enhance strand separation and others frequencies repress strand separation. The distribution density of frequencies in an enzyme can then determine its effect on DNA Biological processes such as transcription and replication requires strand separation. Careful analysis of the use of nucleoside triphosphates in such processes imply that
A Dynamic Enhancer-Repressor Action
241
Downloaded by [Rutgers University] at 07:49 07 April 2015
the actual strand separation processes occur at a net energy loss. Therefore the actual strand separation process such as the elongation of the open state and the initiation of the open state are likely facilitated by thermal fluctuational opening of the interbase hydrogen bonds [12). The energy is supplied by thermal fluctuations. Enhancing these opening probabilities can then enhance all the processes needing strand separation. To study the effects of different places where an enzyme may attach we first study a limited number of attachments locations. We assume that an effective system of some kind attaches over a region of three base pairs of a DNA double helix. We chose poly(dT-dA) · poly(dT-dA) as alternating AT sequences are often found in sites associated with initiation. We study the DNA polymer for three cases. In one case a two point attachment is to the N3 atom of the Adenine base and to the 02 atom of the Thymine base in a base pair which is two base pairs away. This emulates an enzyme attaching through the minor groove and will be called the minor groove attachment. In case two the two point attachment is to the N7 atom of the Adenine base and the methyl group of the Thymine base in a pair two base pairs away. This emulates an enzyme or a drug that attaches to the bases from the major groove side and will be called the major groove attachment. These choices are crudely in accord with some observations of enzyme and drug attachments [13-17). We also study single point attachments which are all the individual attachments in the two point attachments discussed above. For the numerical calculations we assume the attachments are by . mdyne hydrogen bondmg and use a force constant of0.115 ~for all attachments. The distances between a vibrating atom and the atom it attaches to is set at 1. 73 Angstroms. This is appropriate to assuming hydrogen bond attachment. The force constant was varied at some frequencies to see if there was force constant dependence for the resulting dynamics of the DNA helix. The larger the force constant was, the more the resulting modes were altered. This is not a systematic study of the possible enzyme features but it does show that some features ofinteraction can have a major effect on the DNA degrees of freedom. Formalism Normal mode analysis is carried out by setting up a matrix of force constants which is diagonalized to give normal modes. The combined helix enzyme system has a force constant matrix which includes more elements than the separate parts and has the eigen equation. (F - w 2I - C)q
=
0
[1)
where F is the force constant matrix for the separate helix and enzyme with no attachments. w and q are the new eigenvalues and eigenvectors. w is a frequency expressed in em -I, q gives the atom displacements in a mass weighted corrdinate system. When multiplied by an occupation number, or excitation level, q can be converted to atomic displacements in angstroms. The perturbation matrix, C, is the change in F necessary to bring about the attachments of enzyme to the infinite helix. The new structure breaks the helical symmetry and normal mode solutions can only
242
Beger and Prohofsky
be calculated by Green function. Since C is a matrix of relatively low dimensionality, Eq. [1) is solved in terms of the Green function of the unperturbed system. The Green function like q describes the response of the system: see eqn. [3] below. The Green function for the unperturbed and the perturbed systems are defined as 2
g = (w I- F)
-I
Downloaded by [Rutgers University] at 07:49 07 April 2015
and
respectively. They are related by the following Dyson equation [18), G
=
g
+ gTg
[2)
and where T
=
C(l- gC)- 1•
We are interested in the breathing modes which are relatively low frequency ( ~ 85 em -I) vibrations characterized by a large average stretch in the hydrogen bonds and shown to play a significant role in DNA melting. A melting coordinate, s, is defined as the average hydrogen bond stretch amplitude. A thermal mean-square displacement amplitude for the melting coordinate in cell m is then obtained from [19-22]. [3)
which represent the contribution to the hydrogen bond fluctuation from an in band mode of frequency w. In Eq. [3) Tis the temperature. A temperature of293K is used in this particular calculation. The coth term essentially puts in the thermal activation of the Bose-Einstein system. The imaginary part of the Green function is seen to be the response of the system per quantized exicitation i.e. per phonon. The Green function for the unperturbed system can be written in terms of the solution of the isolated helix eigen value equation i.e. Eq. [1] with C = 0. [19). In the internal coordinates, mn 2 g .. (w) ,,
1 "" fn
=- L 7T f...
0
Re(s; (0) l,.• (0) ei(m-n)6) de -----+·-----..,,.,-::-:-(1)
-
~
(0)
[4)
where st..(e), normalized within a unit cell, is the eigenvector belonging to the eigenvalue w~-..(e) for a phase angle ein the internal coordinates. Eq. [4) represents the contributions from all the vibrational modes of the poly(dT-d.A) · poly(dT-d.A) strands to the motion at frequency w. Ifw lies within one or more phonon dispersion branches of the perfect helix then the perfect helix Green function involves an intergration over one or more singularities.
243
A Dynamic Enhancer-Repressor Action
This difficulty may be circumvented by moving this pole off the real axis by giving w a small imaginary part i/r.. Physically, this corresponds to an exponential decay of the normal modes caused by, for example anharmonic interactions causing loss of energy to nearby modes or by dissipative forces such that the molecule would feel in a solution environment. We replace w2 by
Downloaded by [Rutgers University] at 07:49 07 April 2015
(w + ild
=
w2 + 2iw/r: - l/t.2
~
w2
+ zy 0
forsmalll/r.. The use ofw 2 above in Eq. [4] follows that shown byFengetal. [23). The Green's function G is temperature independent and needs to be calculated only once from the initial helix system. The inband thermal mean-squared vibrational amplitude for the m1h hydrogen bond may be obtained by integrating over w in Eq. [3). The probability of a single hydrogen bond, being disrupted in the pre melting state is then given by Y. Z. Chen eta/. [24) as P;
=
C;
f
inf L
[5)
du exp(-(u-R/!W;)
ax
7
where C-L ;-
f
inf
d u e -u 2/2D, .
[6)
h,
Here C; is the normalization factor for the lh hydrogen bond. Lm;x is the maximum value of stretch before the lh hydrogen bond is considered in the open state. D; is the mean square stretch of the th hydrogen bond. R; is the thermal equilibrium length of the lh hydrogen bond. h; is determined from the relation given below V(R;- h;) = V(inf) = 0.
[7)
Table I The Morse Parmeters of AT Base Pairs Bond
a A~l
N(l)-H-N(3) N(6)-H-0(4)
2.402 2.713
vo
mdynA
Lmax
A
2.782 2.698
0.01702 0.01959
3.158 3.146
Ro
A
and the Morse potential is given by
[8) We define a mean open base pair probability that can be calculated by multiplying all the open hydrogen bond probabilities in a certain base pair. The open base pair
244
Beger and Prohofsky .090.---------------------------------~
Downloaded by [Rutgers University] at 07:49 07 April 2015
075
>r-
xx
.060
1-i
_j
+
1-i
~
a
. OLtS
!Xl
0
~ Q_
+
~
** +
X
+
X
030
(!)~(!)~ ~~~ l!{:t
0~
X
**
$(!)(!) C!>(!)
~~~~.~~ f
015
0.00+-----.----.----~----~----~--~
TATATATATATATATATATATA ATATATATATAIATATATATAT
HYDROGEN-BOND Figure 1: Major groove open hydrogen probability ofpoly(dT-d4) · poly(dT-d4) with enzyme attachment at 72.43(cm -I) and an amplitude of0.26 A. The circles are for an N7 of Adenine attachment. Crosses are for a methyl group attachment ofThymine the same base pair. A plus sign is for the N7 attachment in one base and the methyl attachment in a base pair two pairs away. The solid line shows the unattached hydrogen bond opening probability. The underlined bases are where the attachments take place.
probability is then approximately the square (or cube if GC is used) of the single open hydrogen bond probabilities in the base pair. As in ref. 24 we assume a Morse potential as the true potential for the interbase hydrogen bonds. The Morse parameters and maximum length of all the hydrogen bonds used in this calculation are shown in Table I for Adenosine-Thymine.
245
A Dynamic Enhancer-Repressor Action
090.-----------------------------------,
.075
.060 Downloaded by [Rutgers University] at 07:49 07 April 2015
}-
I1-1
_J 1-1
t:Q
0Lt5
