BioSystems 7 (1975) 196--205 © North-Holland Publishing Company, Amsterdam -- Printed in The Netherlands
A COMPUTER SIMULATION MODEL FOR ANALYSIS OF CONFORMATION OF NUCLEAR CHROMATIN AND OF THE TRANSCRIPTION PROCESS K. BELLMANN 1, R. LINDIGKEIT2 and R SCHULZ3 Akademie der Wissenschaften der DDR; 1Zentralinstitut fitr Kybernetik und Informationsprozesse, 1199 Berlin-Adlershof, Rudower Chaussee 5; 2Zentralinstitut fitr MolekularbioIogie, 115 Berlin-Buch, Lindenberger Weg; 3Zentralinstitut fiir Mathematik und Mechanik, 108 Berlin, Mohrenstrasse 39, DDR
Based on experimental data (see Lindigkeit et al., 1974) an algorithmic computer model was devised with the following parameters: (a) nbr. and position of hypothetical blocker sites at the DNA which inhibit transcription in such a manner that distinct RNA chain lengths arise; (2) time depending probability functions that a polymerase molecule (p.m.) is able to pass a blocker site; (3) time depending rate of viability of synthetizing p.m. and (4) distribution of the p.m. on the template at the time t o . For comparison between observed and computed results 3 parameters are used: (1) distribution of chain lengths of RNA molecules in vitro synthetized; (2) their total amount and (3) the number of active p.m. By means of comparisons between computed and experimental results it is possible to test hypotheses about internal structural and functional parameters of the system under investigation, e.g. estimation of the rel. influence of template vs. p.m. characters in the transcription process, hypotheses about the type of distribution of p.m. at time to, their initiation and salt depending removal probability of the blocker structures.
1. I n t r o d u c t i o n As s h o w n by Lindigkeit et al. ( 1 9 7 4 ) the R N A s y n t h e t i z e d in vitro in isolated nuclei o f 50-day-old Wistar rats b y the B - e n z y m e consists o f a c o l l e c t i o n o f distinct R N A fractions with d i f f e r e n t chain lengths. The results o f chase e x p e r i m e n t s p e r f o r m e d at 0.3 M or 0.1 M a m m o n i u m sulfate (AS) exclude the possibility t h a t t h e distinct R N A fractions arose f r o m a s u b s e q u e n t cleavage o f original longer chains. Chase e x p e r i m e n t s as well as a series o f time e x p e r i m e n t s s h o w e d clearly t h a t t h e R N A molecules were enlarged during t h e in vitro assay via t h e discrete chain lengths. It was f u r t h e r s h o w n t h a t the relative a m o u n t s o f the distinct R N A fractions d e p e n d on the salt c o n c e n t r a t i o n in such a w a y t h a t the m o r e rapidly s e d i m e n t i n g R N A f r a c t i o n s increased with increasing AS c o n c e n t r a t i o n bet w e e n 0.1 and 0.4 M AS as well as with the time. T h e following distinct chain lengths as t h e y are characteristic for t h e in vitro R N A synthesis p r o d u c t of R N A p o l y m e r a s e B in
t e r m s o f s e d i m e n t a t i o n coefficients (S) are: 5, 7, 9, 11, 12.5, 14, 15.5, 17, 19, 21, 22.5, 24, 25.5, 27, 29, 32. The R N A s y n t h e s i z e d in vitro in the nuclei is, o f course, t h e result o f the t r a n s c r i p t i o n o f a large n u m b e r o f d i f f e r e n t t r a n s c r i p t i o n units. We suggest t h a t t h e discrete length o f the R N A molecules reflect a general o r g a n i z a t i o n principle o f t h e c h r o m a t i n . The simplest w a y t o explain t h e fact is t h e assumpt i o n t h a t at least o n parts o f the t r a n s c r i p t i o n units of the isolated nuclei " t r a n s c r i p t i o n b l o c k e r s " are l o c a l i z e d . T h e y a p p e a r to be arranged at distinct places and give rise to a t e m p o r a r y or final stop o f t h e travelling R N A p o l y m e r a s e molecules. These h y p o t h e t i c a l t r a n s c r i p t i o n blockers m a y be p a r t l y r e m o v e d or inactivated b y increasing salt c o n c e n t r a t i o n s as well as with time. B e t w e e n t w o blockers the R N A p o l y m e r a s e m a y travel m o r e or less unc h e c k e d (see Fig. 1). In o r d e r to get m o r e insight into the process o f the in vitro R N A synthesis a a l g o r i t h m i c m o d e l for c o m p u t e r s i m u l a t i o n was c o n s t r u c t e d .
197 / /m
Polymerose
Blockers
P/ •
BO
B1
(-,_C] . . . .
•
•
•
•
•
[]
•
[]
•
•
[]
State of T1 (DNA) Qt t i m e t 1 : [ BO, B2, B4,B7REMOVED I
•
[]
s t o r e of T1 (DNA) at time t2:
B3
_[3 . . . . . ~ . . . .
B4
E]
RNA chGilq
0
0
Stote of T1 (DNA) ot t i m e to: , , I NO BLOCKER REMOVED I
•
B2
',:',~--~llm RNA J
E]
/
0
B5
~m
B6
B7
J
[]
[]
.-.F]
[ B1,B3 REMOVED]
[]
I-]~
"-'---
RNA ClqQrn of full length
J
StGte of T1 (DNA) Gt time t n I ALL BLOCKERS REMOVED] L
Fig. 1. Scheme of time dependent RNA synthesis on a single template in connection with the blocker removal process.
2. T h e m o d e l 2.1. A i m and basic idea T h e aim o f this m o d e l is t o test t h e p h e n o m e nological e f f e c t s o f t h e a s s u m e d t r a n s c r i p t i o n inhibiting b l o c k e r s . By m e a n s o f t h e c o m p a r i son b e t w e e n t h e e x p e r i m e n t a l results a n d t h e s i m u l a t i o n results o f t h e c o m p u t e r m o d e l it is possible t o verify h y p o t h e s e s a b o u t t h e internal system parameters. T h e basic idea o f t h e t r a n s c r i p t i o n m o d e l is as follows. We d e f i n e a t r a n s c r i p t i o n s y s t e m fir consisting o f t r a n s c r i p t i o n units Wk e fir. O n e a c h unit T k (k = 1, 2, ..., K) o p e n f o r t r a n s c r i p t i o n d u r i n g t h e e x p e r i m e n t b l o c k e r s Br are localized at t e m p l a t e sites A (Br) o n e a c h t e m p l a t e T k . This is suggested i m m e d i a t e l y f r o m e x p e r i m e n tal results ( n e c e s s a r y c o n c l u s i o n ) . Be Mrk a R N A m o l e c u l e o f length xr s y n t h e t i z e d on t h e k-th t e m p l a t e (Fig. 2). We get in fir t h e subsets r w i t h e l e m e n t s Mrk. P (Mr) is t h e p r o b a b i l i t y t h a t a r a n d o m c h o s e n R N A m o l e c u l e M is an e l e m e n t o f 9ff ~ (see Fig. 1 a n d t h e f o l l o w i n g scheme)
M1,
M2, ...,
Mr, ...,
MN
T1 T2
M11 M1 2
M21
Mrl
MN 1
Tk
MI k
Mrk ..-
Ms
TK
M1 K
Mrk "'"
Ms K
--.
---
k
L e t n~k be t h e n u m b e r o f Mrk w i t h i n crg r nr = ~K = 1 nrk is t h e n u m b e r o f R N A m o l e cules o f length xr in t h e w h o l e s y s t e m a n d n = zN= 1 nr t h e t o t a l n u m b e r o f R N A m o l e c u l e s . We d e f i n e n r / n = p (xr), i.e. t h e p r o b a b i l i t y p (xr) t h a t a r a n d o m c h o s e n R N A m o l e c u l e f r o m is o f length xi is e q u a l t o t h e relative a m o u n t o f all R N A m o l e c u l e s o f length x i . T h e m o d e l is b a s e d on t h e s e p r o b a b i l i t i e s p(xi) d e f i n e d as relative a m o u n t s . This a p p r o a c h is n e e d e d because in t h e real e x p e r i m e n t n a n d t h e ni are n e v e r e s t i m a b l e b u t t h e p(xi). As an e x a m p l e let us a s s u m e 10 t r a n s c r i p t i o n units T ] , ..., T1 0 (see Fig. 3). O n e a c h o f t h e m a
198 BO
Br
61
Bb
T~ - T2 - -
Brk
7"
Tk -~.Mr k With l e n g t h Xr
lot o f b l o c k e r s are r e m o v e d by salt ions ( B 0 , B1 .... , Bs : b l o c k e r sites). In T 1 the p o l y m e r a s e m o l e c u l e is able to travel till B5 p r o d u c i n g a R N A m o l e c u l e with a length o f 5 intersections. In T7 n o p o l y m e r a s e travel is possible. As result w e o b t a i n the f o l l o w i n g c o m p o s i t i o n o f t h e R N A pool: i
Mr
xr
0
1
2
3
4
15
6
7
8
1
T1 TK - -
T2 Mrk : m o l e c u l e of the k-th t r o n s c r i p t i o n Llnit Tk stopped by t h e r - t h b l o c k e r B r of t h e k - t h t r o n s c r i p t i o n u n i t w i t h l e n g t h Xr ( e x p r e s s e d in i n t e r b l o c k e r - s e c t i o n s ) r=
0,1,
.,b;
T3 T4
b: n u m b e r of b l o c k e r s
Fig. 2. Formalization o f a transcription s y s t e m consisting on K transcription units, each w i t h b blockers at the same t e m p l a t e site.
T5 T6
T7
] 1
T8 Bo 61
B2
B3
B4
B5 B6
T1
,.')A-)--E;-9--~m
T2
,.-.,[] j . . . . .[]
T3
%r
•
B7
•
J
_..,o-,, m •
[] [] [] [] ~[]"m
•
[] [] [] [] ~DAm ,/
[] []
Ts
~)~mm
%
( : .--./ ,~m
T7
|1
•
T8
.-,D
[]
,--,•
[]
[]
[]
T9 TIO
[]
[]
n
[]
[]
[]
[]
[]
[]
•
[]
~D" m /
[]
[]
[]
[]
[]
[]
[]
[]
-~•¢m /
0
I
0
0,1 0,] 0,4
1
4
0
0
1
p(xr)
0,1 0,2
0
0
0,1
8
n=E
nr=10
r=O
•
[]
2
1
[]
[] [] [] [] [] []
1
[] nr
J
T4
1
T10 [] •
Z
I
T9
[]
[] IqA
"4W ,/ []
M r : R N A m o l e c u l e s o f length xr (in i n t e r b l o c k er units); r = 0, 2, ..., 8. T h e t o t a l a m o u n t o f s y n t h e t i z e d R N A is 8
Y = ~
p ( x r ) x r = 3,7.
r=0
B
non removed
13 : r e m o v e d
Fig. 3. A possible state o f the transcription s y s t e m (K = 10) w i t h a stochastic pattern of salt removed blockers (B 8 as the terminal blocker is o m i t t e d ) .
B e c a u s e t h e e x p e r i m e n t a l results d o n o t c o n cern a single t e m p l a t e ' s b e h a v i o r but t h e y are integrative results c o n c e r n i n g a set o f transcriptable D N A - s e q u e n c e s as a w h o l e , this prob-
199
abilistic approach is suitable. According to the nature of the real process the model is a discrete one.
the blocker sites A(Bj) on the S-scale in form of a table of the following type Bj
2. 2 Parameters
The real system to be mapped is the transcription apparatus with its structure and function elements: (1) DNS --sequence, (2) blocker sites inhibiting RNA-synthesis in such a manner that R N A chains with distinct lengths arise; each blocker with a time depending probability to be removed at a given salt concentration; (3) polymerase molecules synthetizing RNA with a certain velocity and with a time depending probability to be unable to synthesize further RNA, i.e. to have :;he velocity zero, and with a certain distribution at the start of the experiment (start distribution). As a first step the model template T was discretisized in sections of equal length. T represents a S-scala (or transformed a nucleotide scala) with equidistant, discrete points Ao, A 1 , A 2 , ... , A i . . . . A. N . It holds the relation A0, A1, A2, ..., Ai, ... AN 0S,
1S,
2S .... , iS, ... NS
N is the maximal length of the model template or the maximal length of the R N A molecule synthetized. The interval is a tact and the time needed by a polymerase molecule to synthetize a R N A molecule oi' length 1 S too. Internal basis parameters (characterizing the internal system features) B 1 : number of blockers: b On the basis of experimental results the number b of blockers Bj is available (j = 0,2 .... , b). Bo is identical with repressor like region. B 2 : b l o c k e r position function: A(Bi) Analysing the ::esults of density gradient centrifugation it is possible to estimate the blocker positions (assuming all polymerases are in front of blocker Bo, i.e. all polymerase molecules are starting from the extreme left site). The blocker position function designates
0 I 2
A(Bi)
3
4
5
6
7
8
9 10 11 12
0 4 9 11 14 17 19 21 24 27 29 32 34
j = 0 , 1 ..... b.
[S]
B 3 : time depending blocker removal probability function: PB j(t). To each point A(Bj) on T a binary time depending value Dj(t) corresponds designating whether a polymerase arriving Bj at time t is able to continue its run or not. Dj(t) is defined for t = to, ..., tr, ..., tz (tz : end of experiment) and designates the blocker state Z(Bj). i if Bj is able to stop the polymerase travel Z(B~ )t
=
if Bj is n o t able to stop the polymerase travel It is 0 if polymerase is stopped by Bj Dj(t) = 1 if polymerase is n o t stopped by Bj PB j(t) is the probability that Dj(t) = 1. It is the passing-through-probability. If PB,(t~+ 1 ) /> PBj(tr) and Dj(tr ) = 0 it holds Dj(t~+l~ = 1 and a polymerase localized in front of Bj is able n o w to continue its run. If PBi(t~+l) < PBi(t~) and D~(tr) = 0 it holds Dj(tr+l) = 0 and a polymerase localized in front of Bj is not able to continue its run. In Fig. 4 examples for PBj(t) are given. The function PBj(t) is settled in form of a table using a small number of points on the time axis. B 4 : polymerase start distribution on the template: C ( t o ) = ( C o ( t o ) , C1 ( t o ) . . . . . C i ( t o ), ..., CN ( t o ) ) .
Ci(t0 ) is the probability that at the begin of the experiment a random chosen polymerase molecule is found on template point Ai. In many cases of the simulation runs we have assumed
200 11 / b if i designating (Bi) Ci(to)
/o if i
binary f u n c t i o n t h a t indicate the state Z(t) o f a p o l y m e r a s e m o l e c u l e at t i m e t
n o t designating A(Bj)
i.e. equal p r o b a b i l i t y , t h a t a p o l y m e r a s e is localized in f r o n t o f each b l o c k e r Bj. A m o r e a p p r o p r i a t e a s s u m p t i o n (settled on e x p e r i m e n t a l results) is Ci(t0)=
/10 i f i = 0 ifi~
0
i.e. at the begin o f the in vitro e x p e r i m e n t all p o l y m e r a s e molecules are localized in f r o n t o f the B0 (at the p r o m o t o r region). A n o t h e r possibility is to assume Poisson distribution o f p o l y m e r a s e molecules. B 5 : velocity o f p o l y m e r a s e : v=
distance (Ai, A i + 1) t~ + 1 -- t~
[ S-units per sec ]
o n the basis o f a settled v e l o c i t y in nucleotides per second. B 6: time depending polymerase f u n c t i o n Pl(t). During the e x p e r i m e n t we have to the ratio of active p o l y m e r a s e and
terms of lethality consider define a
/ 1, if a p o l y m e r a s e m o l e c u l e is lethal at Z(t) = / t i m e t | 0, else (i.e. p o l y m e r a s e m o l e c u l e living) P1 (t) is the p r o b a b i l i t y t h a t Z(t) = 1 at t i m e t, i.e. t h a t a r a n d o m c h o s e n p o l y m e r a s e m o l e c u l e is of lethal state unable to c o n t i n u e its run. In every case it is P I ( t T ÷ I ) ~ PI(tT) (V = 0,1,2, ..., Z). If Z(tT) = 1 it is Z(tr+ 1 ) = 1 t o o . An e x a m p l e for P1 (t) is given in Fig. 5. T h e function P l ( t ) is settled in f o r m o f a table using a small n u m b e r o f points o n the t i m e axis. P h e n o m e n o l o g i c a l c h e c k p a r a m e t e r s (for c o m parison o f model's and real system's behavior) P 1 : t i m e d e p e n d i n g d i s t r i b u t i o n Pxi(t) o f RNA-chain lengths xi. Let xi (i = 0,1,2, ..., N) be the lengths of a de novo s y n t h e t i z e d RNA-chain and px i is the relative a m o u n t o f R N A molecules o f length x i expressed in units o f the s e d i m e n t a t i o n constant S (or t r a n s f o r m e d in n u c l e o t i d e units). Pxi can be calculated in real as well as in c o m p u t e r e x p e r i m e n t s for a given t i m e t. Px i (t) = 0 is possible. N o t e : pxi(t) can be calculated b y the algor i t h m in the case o f v is finite. In the case of
PL(t)
P~(t) 1
0
m t(rnin) tz
Fig. 4. Examples for the time dependent probability function PBj(t) that a blocker Bi is removed at time t by salt ions.
0
k t(min) tz
Fig. 5. Examples for the time dependent probability function Pl(t), that a random chosen polymerase molecule is lethal at time t.
201
v = oo, px i can be calculated by an analytical formula. This approach is helpful to reduce the number of parameter sets. The basis of this analytical approach is as follows. The probability P(A(Bv ) IA(Bu )) that a polymerase molecule is to be found at blocker position A(Bv) under the condition that it was moving from blocker position A(B u ) is
P(A(Bv)/A(Bu)) =
PBp "PBp+ 1 "(1 -- P B v _ I )
ifv:>p
(1 -- PBv=p )
if v = p
0
ifv
A COMPUTER SIMULATION MODEL FOR ANALYSIS OF CONFORMATION OF NUCLEAR CHROMATIN AND OF THE TRANSCRIPTION PROCESS K. BELLMANN 1, R. LINDIGKEIT2 and R SCHULZ3 Akademie der Wissenschaften der DDR; 1Zentralinstitut fitr Kybernetik und Informationsprozesse, 1199 Berlin-Adlershof, Rudower Chaussee 5; 2Zentralinstitut fitr MolekularbioIogie, 115 Berlin-Buch, Lindenberger Weg; 3Zentralinstitut fiir Mathematik und Mechanik, 108 Berlin, Mohrenstrasse 39, DDR
Based on experimental data (see Lindigkeit et al., 1974) an algorithmic computer model was devised with the following parameters: (a) nbr. and position of hypothetical blocker sites at the DNA which inhibit transcription in such a manner that distinct RNA chain lengths arise; (2) time depending probability functions that a polymerase molecule (p.m.) is able to pass a blocker site; (3) time depending rate of viability of synthetizing p.m. and (4) distribution of the p.m. on the template at the time t o . For comparison between observed and computed results 3 parameters are used: (1) distribution of chain lengths of RNA molecules in vitro synthetized; (2) their total amount and (3) the number of active p.m. By means of comparisons between computed and experimental results it is possible to test hypotheses about internal structural and functional parameters of the system under investigation, e.g. estimation of the rel. influence of template vs. p.m. characters in the transcription process, hypotheses about the type of distribution of p.m. at time to, their initiation and salt depending removal probability of the blocker structures.
1. I n t r o d u c t i o n As s h o w n by Lindigkeit et al. ( 1 9 7 4 ) the R N A s y n t h e t i z e d in vitro in isolated nuclei o f 50-day-old Wistar rats b y the B - e n z y m e consists o f a c o l l e c t i o n o f distinct R N A fractions with d i f f e r e n t chain lengths. The results o f chase e x p e r i m e n t s p e r f o r m e d at 0.3 M or 0.1 M a m m o n i u m sulfate (AS) exclude the possibility t h a t t h e distinct R N A fractions arose f r o m a s u b s e q u e n t cleavage o f original longer chains. Chase e x p e r i m e n t s as well as a series o f time e x p e r i m e n t s s h o w e d clearly t h a t t h e R N A molecules were enlarged during t h e in vitro assay via t h e discrete chain lengths. It was f u r t h e r s h o w n t h a t the relative a m o u n t s o f the distinct R N A fractions d e p e n d on the salt c o n c e n t r a t i o n in such a w a y t h a t the m o r e rapidly s e d i m e n t i n g R N A f r a c t i o n s increased with increasing AS c o n c e n t r a t i o n bet w e e n 0.1 and 0.4 M AS as well as with the time. T h e following distinct chain lengths as t h e y are characteristic for t h e in vitro R N A synthesis p r o d u c t of R N A p o l y m e r a s e B in
t e r m s o f s e d i m e n t a t i o n coefficients (S) are: 5, 7, 9, 11, 12.5, 14, 15.5, 17, 19, 21, 22.5, 24, 25.5, 27, 29, 32. The R N A s y n t h e s i z e d in vitro in the nuclei is, o f course, t h e result o f the t r a n s c r i p t i o n o f a large n u m b e r o f d i f f e r e n t t r a n s c r i p t i o n units. We suggest t h a t t h e discrete length o f the R N A molecules reflect a general o r g a n i z a t i o n principle o f t h e c h r o m a t i n . The simplest w a y t o explain t h e fact is t h e assumpt i o n t h a t at least o n parts o f the t r a n s c r i p t i o n units of the isolated nuclei " t r a n s c r i p t i o n b l o c k e r s " are l o c a l i z e d . T h e y a p p e a r to be arranged at distinct places and give rise to a t e m p o r a r y or final stop o f t h e travelling R N A p o l y m e r a s e molecules. These h y p o t h e t i c a l t r a n s c r i p t i o n blockers m a y be p a r t l y r e m o v e d or inactivated b y increasing salt c o n c e n t r a t i o n s as well as with time. B e t w e e n t w o blockers the R N A p o l y m e r a s e m a y travel m o r e or less unc h e c k e d (see Fig. 1). In o r d e r to get m o r e insight into the process o f the in vitro R N A synthesis a a l g o r i t h m i c m o d e l for c o m p u t e r s i m u l a t i o n was c o n s t r u c t e d .
197 / /m
Polymerose
Blockers
P/ •
BO
B1
(-,_C] . . . .
•
•
•
•
•
[]
•
[]
•
•
[]
State of T1 (DNA) Qt t i m e t 1 : [ BO, B2, B4,B7REMOVED I
•
[]
s t o r e of T1 (DNA) at time t2:
B3
_[3 . . . . . ~ . . . .
B4
E]
RNA chGilq
0
0
Stote of T1 (DNA) ot t i m e to: , , I NO BLOCKER REMOVED I
•
B2
',:',~--~llm RNA J
E]
/
0
B5
~m
B6
B7
J
[]
[]
.-.F]
[ B1,B3 REMOVED]
[]
I-]~
"-'---
RNA ClqQrn of full length
J
StGte of T1 (DNA) Gt time t n I ALL BLOCKERS REMOVED] L
Fig. 1. Scheme of time dependent RNA synthesis on a single template in connection with the blocker removal process.
2. T h e m o d e l 2.1. A i m and basic idea T h e aim o f this m o d e l is t o test t h e p h e n o m e nological e f f e c t s o f t h e a s s u m e d t r a n s c r i p t i o n inhibiting b l o c k e r s . By m e a n s o f t h e c o m p a r i son b e t w e e n t h e e x p e r i m e n t a l results a n d t h e s i m u l a t i o n results o f t h e c o m p u t e r m o d e l it is possible t o verify h y p o t h e s e s a b o u t t h e internal system parameters. T h e basic idea o f t h e t r a n s c r i p t i o n m o d e l is as follows. We d e f i n e a t r a n s c r i p t i o n s y s t e m fir consisting o f t r a n s c r i p t i o n units Wk e fir. O n e a c h unit T k (k = 1, 2, ..., K) o p e n f o r t r a n s c r i p t i o n d u r i n g t h e e x p e r i m e n t b l o c k e r s Br are localized at t e m p l a t e sites A (Br) o n e a c h t e m p l a t e T k . This is suggested i m m e d i a t e l y f r o m e x p e r i m e n tal results ( n e c e s s a r y c o n c l u s i o n ) . Be Mrk a R N A m o l e c u l e o f length xr s y n t h e t i z e d on t h e k-th t e m p l a t e (Fig. 2). We get in fir t h e subsets r w i t h e l e m e n t s Mrk. P (Mr) is t h e p r o b a b i l i t y t h a t a r a n d o m c h o s e n R N A m o l e c u l e M is an e l e m e n t o f 9ff ~ (see Fig. 1 a n d t h e f o l l o w i n g scheme)
M1,
M2, ...,
Mr, ...,
MN
T1 T2
M11 M1 2
M21
Mrl
MN 1
Tk
MI k
Mrk ..-
Ms
TK
M1 K
Mrk "'"
Ms K
--.
---
k
L e t n~k be t h e n u m b e r o f Mrk w i t h i n crg r nr = ~K = 1 nrk is t h e n u m b e r o f R N A m o l e cules o f length xr in t h e w h o l e s y s t e m a n d n = zN= 1 nr t h e t o t a l n u m b e r o f R N A m o l e c u l e s . We d e f i n e n r / n = p (xr), i.e. t h e p r o b a b i l i t y p (xr) t h a t a r a n d o m c h o s e n R N A m o l e c u l e f r o m is o f length xi is e q u a l t o t h e relative a m o u n t o f all R N A m o l e c u l e s o f length x i . T h e m o d e l is b a s e d on t h e s e p r o b a b i l i t i e s p(xi) d e f i n e d as relative a m o u n t s . This a p p r o a c h is n e e d e d because in t h e real e x p e r i m e n t n a n d t h e ni are n e v e r e s t i m a b l e b u t t h e p(xi). As an e x a m p l e let us a s s u m e 10 t r a n s c r i p t i o n units T ] , ..., T1 0 (see Fig. 3). O n e a c h o f t h e m a
198 BO
Br
61
Bb
T~ - T2 - -
Brk
7"
Tk -~.Mr k With l e n g t h Xr
lot o f b l o c k e r s are r e m o v e d by salt ions ( B 0 , B1 .... , Bs : b l o c k e r sites). In T 1 the p o l y m e r a s e m o l e c u l e is able to travel till B5 p r o d u c i n g a R N A m o l e c u l e with a length o f 5 intersections. In T7 n o p o l y m e r a s e travel is possible. As result w e o b t a i n the f o l l o w i n g c o m p o s i t i o n o f t h e R N A pool: i
Mr
xr
0
1
2
3
4
15
6
7
8
1
T1 TK - -
T2 Mrk : m o l e c u l e of the k-th t r o n s c r i p t i o n Llnit Tk stopped by t h e r - t h b l o c k e r B r of t h e k - t h t r o n s c r i p t i o n u n i t w i t h l e n g t h Xr ( e x p r e s s e d in i n t e r b l o c k e r - s e c t i o n s ) r=
0,1,
.,b;
T3 T4
b: n u m b e r of b l o c k e r s
Fig. 2. Formalization o f a transcription s y s t e m consisting on K transcription units, each w i t h b blockers at the same t e m p l a t e site.
T5 T6
T7
] 1
T8 Bo 61
B2
B3
B4
B5 B6
T1
,.')A-)--E;-9--~m
T2
,.-.,[] j . . . . .[]
T3
%r
•
B7
•
J
_..,o-,, m •
[] [] [] [] ~[]"m
•
[] [] [] [] ~DAm ,/
[] []
Ts
~)~mm
%
( : .--./ ,~m
T7
|1
•
T8
.-,D
[]
,--,•
[]
[]
[]
T9 TIO
[]
[]
n
[]
[]
[]
[]
[]
[]
•
[]
~D" m /
[]
[]
[]
[]
[]
[]
[]
[]
-~•¢m /
0
I
0
0,1 0,] 0,4
1
4
0
0
1
p(xr)
0,1 0,2
0
0
0,1
8
n=E
nr=10
r=O
•
[]
2
1
[]
[] [] [] [] [] []
1
[] nr
J
T4
1
T10 [] •
Z
I
T9
[]
[] IqA
"4W ,/ []
M r : R N A m o l e c u l e s o f length xr (in i n t e r b l o c k er units); r = 0, 2, ..., 8. T h e t o t a l a m o u n t o f s y n t h e t i z e d R N A is 8
Y = ~
p ( x r ) x r = 3,7.
r=0
B
non removed
13 : r e m o v e d
Fig. 3. A possible state o f the transcription s y s t e m (K = 10) w i t h a stochastic pattern of salt removed blockers (B 8 as the terminal blocker is o m i t t e d ) .
B e c a u s e t h e e x p e r i m e n t a l results d o n o t c o n cern a single t e m p l a t e ' s b e h a v i o r but t h e y are integrative results c o n c e r n i n g a set o f transcriptable D N A - s e q u e n c e s as a w h o l e , this prob-
199
abilistic approach is suitable. According to the nature of the real process the model is a discrete one.
the blocker sites A(Bj) on the S-scale in form of a table of the following type Bj
2. 2 Parameters
The real system to be mapped is the transcription apparatus with its structure and function elements: (1) DNS --sequence, (2) blocker sites inhibiting RNA-synthesis in such a manner that R N A chains with distinct lengths arise; each blocker with a time depending probability to be removed at a given salt concentration; (3) polymerase molecules synthetizing RNA with a certain velocity and with a time depending probability to be unable to synthesize further RNA, i.e. to have :;he velocity zero, and with a certain distribution at the start of the experiment (start distribution). As a first step the model template T was discretisized in sections of equal length. T represents a S-scala (or transformed a nucleotide scala) with equidistant, discrete points Ao, A 1 , A 2 , ... , A i . . . . A. N . It holds the relation A0, A1, A2, ..., Ai, ... AN 0S,
1S,
2S .... , iS, ... NS
N is the maximal length of the model template or the maximal length of the R N A molecule synthetized. The interval is a tact and the time needed by a polymerase molecule to synthetize a R N A molecule oi' length 1 S too. Internal basis parameters (characterizing the internal system features) B 1 : number of blockers: b On the basis of experimental results the number b of blockers Bj is available (j = 0,2 .... , b). Bo is identical with repressor like region. B 2 : b l o c k e r position function: A(Bi) Analysing the ::esults of density gradient centrifugation it is possible to estimate the blocker positions (assuming all polymerases are in front of blocker Bo, i.e. all polymerase molecules are starting from the extreme left site). The blocker position function designates
0 I 2
A(Bi)
3
4
5
6
7
8
9 10 11 12
0 4 9 11 14 17 19 21 24 27 29 32 34
j = 0 , 1 ..... b.
[S]
B 3 : time depending blocker removal probability function: PB j(t). To each point A(Bj) on T a binary time depending value Dj(t) corresponds designating whether a polymerase arriving Bj at time t is able to continue its run or not. Dj(t) is defined for t = to, ..., tr, ..., tz (tz : end of experiment) and designates the blocker state Z(Bj). i if Bj is able to stop the polymerase travel Z(B~ )t
=
if Bj is n o t able to stop the polymerase travel It is 0 if polymerase is stopped by Bj Dj(t) = 1 if polymerase is n o t stopped by Bj PB j(t) is the probability that Dj(t) = 1. It is the passing-through-probability. If PB,(t~+ 1 ) /> PBj(tr) and Dj(tr ) = 0 it holds Dj(t~+l~ = 1 and a polymerase localized in front of Bj is able n o w to continue its run. If PBi(t~+l) < PBi(t~) and D~(tr) = 0 it holds Dj(tr+l) = 0 and a polymerase localized in front of Bj is not able to continue its run. In Fig. 4 examples for PBj(t) are given. The function PBj(t) is settled in form of a table using a small number of points on the time axis. B 4 : polymerase start distribution on the template: C ( t o ) = ( C o ( t o ) , C1 ( t o ) . . . . . C i ( t o ), ..., CN ( t o ) ) .
Ci(t0 ) is the probability that at the begin of the experiment a random chosen polymerase molecule is found on template point Ai. In many cases of the simulation runs we have assumed
200 11 / b if i designating (Bi) Ci(to)
/o if i
binary f u n c t i o n t h a t indicate the state Z(t) o f a p o l y m e r a s e m o l e c u l e at t i m e t
n o t designating A(Bj)
i.e. equal p r o b a b i l i t y , t h a t a p o l y m e r a s e is localized in f r o n t o f each b l o c k e r Bj. A m o r e a p p r o p r i a t e a s s u m p t i o n (settled on e x p e r i m e n t a l results) is Ci(t0)=
/10 i f i = 0 ifi~
0
i.e. at the begin o f the in vitro e x p e r i m e n t all p o l y m e r a s e molecules are localized in f r o n t o f the B0 (at the p r o m o t o r region). A n o t h e r possibility is to assume Poisson distribution o f p o l y m e r a s e molecules. B 5 : velocity o f p o l y m e r a s e : v=
distance (Ai, A i + 1) t~ + 1 -- t~
[ S-units per sec ]
o n the basis o f a settled v e l o c i t y in nucleotides per second. B 6: time depending polymerase f u n c t i o n Pl(t). During the e x p e r i m e n t we have to the ratio of active p o l y m e r a s e and
terms of lethality consider define a
/ 1, if a p o l y m e r a s e m o l e c u l e is lethal at Z(t) = / t i m e t | 0, else (i.e. p o l y m e r a s e m o l e c u l e living) P1 (t) is the p r o b a b i l i t y t h a t Z(t) = 1 at t i m e t, i.e. t h a t a r a n d o m c h o s e n p o l y m e r a s e m o l e c u l e is of lethal state unable to c o n t i n u e its run. In every case it is P I ( t T ÷ I ) ~ PI(tT) (V = 0,1,2, ..., Z). If Z(tT) = 1 it is Z(tr+ 1 ) = 1 t o o . An e x a m p l e for P1 (t) is given in Fig. 5. T h e function P l ( t ) is settled in f o r m o f a table using a small n u m b e r o f points o n the t i m e axis. P h e n o m e n o l o g i c a l c h e c k p a r a m e t e r s (for c o m parison o f model's and real system's behavior) P 1 : t i m e d e p e n d i n g d i s t r i b u t i o n Pxi(t) o f RNA-chain lengths xi. Let xi (i = 0,1,2, ..., N) be the lengths of a de novo s y n t h e t i z e d RNA-chain and px i is the relative a m o u n t o f R N A molecules o f length x i expressed in units o f the s e d i m e n t a t i o n constant S (or t r a n s f o r m e d in n u c l e o t i d e units). Pxi can be calculated in real as well as in c o m p u t e r e x p e r i m e n t s for a given t i m e t. Px i (t) = 0 is possible. N o t e : pxi(t) can be calculated b y the algor i t h m in the case o f v is finite. In the case of
PL(t)
P~(t) 1
0
m t(rnin) tz
Fig. 4. Examples for the time dependent probability function PBj(t) that a blocker Bi is removed at time t by salt ions.
0
k t(min) tz
Fig. 5. Examples for the time dependent probability function Pl(t), that a random chosen polymerase molecule is lethal at time t.
201
v = oo, px i can be calculated by an analytical formula. This approach is helpful to reduce the number of parameter sets. The basis of this analytical approach is as follows. The probability P(A(Bv ) IA(Bu )) that a polymerase molecule is to be found at blocker position A(Bv) under the condition that it was moving from blocker position A(B u ) is
P(A(Bv)/A(Bu)) =
PBp "PBp+ 1 "(1 -- P B v _ I )
ifv:>p
(1 -- PBv=p )
if v = p
0
ifv
