Mathematical Method for Determining the Distribution of Substituted DNA Zone Lengths with Reference to Equilibrium Density Profiles C. DELCROIX and F. R. L. CANTRAINE, Institute of Interdisciplinary Research, Laboratory o f Medical Statistics, School of Medicine, Free University of Brussels, 1000-Brussels, Belgium; and N. HARDT, Department of Molecular Biology, Free University of Brussels, 1000-Brussels, Belgium Synopsis A method of calculation to determine the profile in density-gradient sedimentation of bromodeoxyuridine-substituted DNA lengths prior to sonication is developed. The method may he summarized as follows: (1)A description of a formulation of random cleavage of a substituted zone i in fragments of length m; (2) generalization and application of this formulation to all substituted zone lengths simultaneously present in a given DNA extract, such generalization being performed by means of a matrix calculation; (3) adjustment of this formulation for the limits of the experimental method, i.e., number of gradient fractions which can he recovered, and for the evolution of the DNA fragment lengths which drop in the course of successive sonications; (4) simulation of the radioactive profiles. By this method it is possible to obtain certain indications concerning the types of initial (i.e., presonication) distributions that are compatible with observed profiles and to test the validity of assumptions relating to the initial distribution. This method has been applied in analyzing DNA synthesis in lymphocytes stimulated by Concanavalin and allowed us to determine the significance of the discontinuities appearing in the elongation with a suboptimal stimulating dose. The proposed method could also he applied to the analysis of all random populations of mixed sequences.
INTRODUCTION
A method commonly used to determine the existence and the length of DNA segments newly synthesized either during m i t ~ s i s l or - ~ by repair4 is based on replacing thymidine in the culture medium by bromodeoxyuridine (BrdUrd). The DNA is subsequently extracted and analyzed by densitygradient centrifugation. The results are difficult to analyze because DNA extraction introduces random breaks both in the substituted and nonsubstituted segments. The end result is a population of molecules, generally of homogeneous size but with varying BrdUrd-substituted zone fractions. Thus, it is difficult to determine the initial lengths of the substituted zones based on the characteristics of the population of observed molecules.1,2 In general, in order to check the assumed initial distribution of substituted lengths, different sedimentation profiles of the same material are Biopolymers, Vol. 17, 2865-2883 (1978) 62 1978 John Wiley & Sons, Inc.
0006-3525/78/0017-2865$01.OO
2866
DELCROIX, CANTRAINE, AND HARDT
established after successive sonications of the DNA fragment^.^,^,^ This procedure provides material for a qualitative interpretation. In order to improve the interpretation, we propose a simple method of comparing the results of mathematical simulations with the experimental data. This method allows one to predict with a satisfactory approximation the properties and characteristics of the initial distribution of the substituted lengths.
METHODS The model is based on four assumptions: (1) The DNA fragments obtained after sonication are all of the same length (see Discussion). The same applies for the DNA fragments obtained during extraction. (2) Each phosphodiester bond between neighboring bases has an equal probability of being broken during extraction or sonication. Each base can serve as an “end” following strand breakage. (3) The minimum operational unit of breakage is the substitution unit, i.e., the smallest average sequence of nucleotides containing a single thymidine capable of being replaced by BrdUrd. (4) The substituted zones are cleaved independently, either because the distance between two substituted zones is greater than the size of fragments produced by sonication or because the cleaved DNA strand contains only one substituted zone.
Introduction to the Mathematical Formulation
A double-stranded DNA is cut in fragments of the same length m during sonication, i.e., drawing m substitutable units. Depending on the random cutting during sonication, the substituted zones i are separated or not. 1 doubie-
stranded .,
,
f : 4
k
I
,
,
I
DNA tiber
4
cleaved tragment rn
We adopt the following terminology for the conditions before extraction and sonication: i = substituted zone length (y77nrm), S; = number of substituted zones of length i within a population of double-stranded DNA, and ,,i = maximum substituted zone length within a population of double-stranded DNA containing a population of substituted zone lengths; and for conditions after extraction and sonication: m = cleaved fragment length, k = substituted zone length within a cleaved fragment m , and Nk = number of cleaved fragments m containing a substituted zone of length k.
POPULATION ANALYSIS OF MIXED SEQUENCES
2867
Let us consider the case where the substituted zone length, i, is shorter than the cleaved fragment of size, m. For example, in Fig. 1all the possibilities are obtained by taking every substitutable unit as a starting point. It is possible to cut the double-stranded DNA in m different ways such that in m - i 1 fragments the substituted zone is not cut and that all other fragments furnish two labeled pieces of a shorter length than i. Part of Fig. 1 presents the distribution of these fragments Nk as a function of the number of substituted units. This distribution is equivalent to the optical density profile of a density gradient. In Fig. l(c), Nk-k, the analog of radioactivity profile, has been obtained by multiplying the DNA content of each class by the fraction number of the class. Under the inverse conditions, that is, when i, the substituted zone length, is longer than the substitutable zone m, it is sufficient to interchange m and i. This is illustrated in Fig. 2. To sum up, in every case the number of fragments containing maximum substitution is given by
+
N ; = li
- ml
+1
Fig. 1. (a) All the possible cleavages of a double-stranded DNA fiber containing a substituted zone of length i in fragments of length m , where m > i, e.g., m = 8 units, i = 5 units, s, = 8. (b) Nk = f ( h ) . Distribution of these fragments as a function of the number of substituted units (analogous t o the optical density profile of a gradient). (c) T h e simulation of the radioactivity profile calculated by multiplying t h e DNA content of each class by t h e fraction number of the class, Nk.k = g ( k ) .
DELCROIX, CANTRAINE, AND HARDT
2868
TT, 8 7 6 5 4 3 2 1 0 k
8 7 6 5 4 3 2 1 0 k
Fig. 2. Same illustration as in Fig. 1 applied to the case where m 10 units, S; = 8.
< i, e.g., m
= 8 units, i =
Each fragment of substituted length k appears twice:
Nk = 2
forO
INTRODUCTION
A method commonly used to determine the existence and the length of DNA segments newly synthesized either during m i t ~ s i s l or - ~ by repair4 is based on replacing thymidine in the culture medium by bromodeoxyuridine (BrdUrd). The DNA is subsequently extracted and analyzed by densitygradient centrifugation. The results are difficult to analyze because DNA extraction introduces random breaks both in the substituted and nonsubstituted segments. The end result is a population of molecules, generally of homogeneous size but with varying BrdUrd-substituted zone fractions. Thus, it is difficult to determine the initial lengths of the substituted zones based on the characteristics of the population of observed molecules.1,2 In general, in order to check the assumed initial distribution of substituted lengths, different sedimentation profiles of the same material are Biopolymers, Vol. 17, 2865-2883 (1978) 62 1978 John Wiley & Sons, Inc.
0006-3525/78/0017-2865$01.OO
2866
DELCROIX, CANTRAINE, AND HARDT
established after successive sonications of the DNA fragment^.^,^,^ This procedure provides material for a qualitative interpretation. In order to improve the interpretation, we propose a simple method of comparing the results of mathematical simulations with the experimental data. This method allows one to predict with a satisfactory approximation the properties and characteristics of the initial distribution of the substituted lengths.
METHODS The model is based on four assumptions: (1) The DNA fragments obtained after sonication are all of the same length (see Discussion). The same applies for the DNA fragments obtained during extraction. (2) Each phosphodiester bond between neighboring bases has an equal probability of being broken during extraction or sonication. Each base can serve as an “end” following strand breakage. (3) The minimum operational unit of breakage is the substitution unit, i.e., the smallest average sequence of nucleotides containing a single thymidine capable of being replaced by BrdUrd. (4) The substituted zones are cleaved independently, either because the distance between two substituted zones is greater than the size of fragments produced by sonication or because the cleaved DNA strand contains only one substituted zone.
Introduction to the Mathematical Formulation
A double-stranded DNA is cut in fragments of the same length m during sonication, i.e., drawing m substitutable units. Depending on the random cutting during sonication, the substituted zones i are separated or not. 1 doubie-
stranded .,
,
f : 4
k
I
,
,
I
DNA tiber
4
cleaved tragment rn
We adopt the following terminology for the conditions before extraction and sonication: i = substituted zone length (y77nrm), S; = number of substituted zones of length i within a population of double-stranded DNA, and ,,i = maximum substituted zone length within a population of double-stranded DNA containing a population of substituted zone lengths; and for conditions after extraction and sonication: m = cleaved fragment length, k = substituted zone length within a cleaved fragment m , and Nk = number of cleaved fragments m containing a substituted zone of length k.
POPULATION ANALYSIS OF MIXED SEQUENCES
2867
Let us consider the case where the substituted zone length, i, is shorter than the cleaved fragment of size, m. For example, in Fig. 1all the possibilities are obtained by taking every substitutable unit as a starting point. It is possible to cut the double-stranded DNA in m different ways such that in m - i 1 fragments the substituted zone is not cut and that all other fragments furnish two labeled pieces of a shorter length than i. Part of Fig. 1 presents the distribution of these fragments Nk as a function of the number of substituted units. This distribution is equivalent to the optical density profile of a density gradient. In Fig. l(c), Nk-k, the analog of radioactivity profile, has been obtained by multiplying the DNA content of each class by the fraction number of the class. Under the inverse conditions, that is, when i, the substituted zone length, is longer than the substitutable zone m, it is sufficient to interchange m and i. This is illustrated in Fig. 2. To sum up, in every case the number of fragments containing maximum substitution is given by
+
N ; = li
- ml
+1
Fig. 1. (a) All the possible cleavages of a double-stranded DNA fiber containing a substituted zone of length i in fragments of length m , where m > i, e.g., m = 8 units, i = 5 units, s, = 8. (b) Nk = f ( h ) . Distribution of these fragments as a function of the number of substituted units (analogous t o the optical density profile of a gradient). (c) T h e simulation of the radioactivity profile calculated by multiplying t h e DNA content of each class by t h e fraction number of the class, Nk.k = g ( k ) .
DELCROIX, CANTRAINE, AND HARDT
2868
TT, 8 7 6 5 4 3 2 1 0 k
8 7 6 5 4 3 2 1 0 k
Fig. 2. Same illustration as in Fig. 1 applied to the case where m 10 units, S; = 8.
< i, e.g., m
= 8 units, i =
Each fragment of substituted length k appears twice:
Nk = 2
forO
