The stochastic model of non-equilibrium mutagen-induced alterations of DNA: Implication to genetic instability in cancer

Abstract

Genetic alterations such as point mutations, chromosomal rearrangements, modification of DNA methylation and chromosome aberrations accumulate during the lifetime of an organism. They can be caused by intrinsic errors in the DNA replication and repair as well as by external factors such as exposure to mutagenic substances or radiation. The main purpose of the present work is to begin an exploration of the stochastic nature of non-equilibrium DNA alteration caused by events such as tautomeric shifts. This is done by modeling the genetic DNA code chain as a sequence of DNA-bit values (‘1’ for normal bases and ‘−1’ for abnormal bases). We observe the number of DNA-bit changes resulting from the random point mutation process which, in the model, is being induced by a stochastic Brownian mutagen (BM) as it diffuses through the DNA-bit systems. Using both an analytical and Monte Carlo (MC) simulation techniques, we observe the local and global number of DNA-bit changes. It is found that in 1D, the local DNA-bit density behaves like $1/\sqrt{t}$, the global total number of the switched (abnormal) DNA-bit increases as $\sqrt{t}$. The probability distribution P(b, 0, t) of b(0, t) is log–normal. It is also found that when the number of mutagens is increased, the number of the total abnormal DNA-bits does not grow linearly with the number of mutagens. All analytic results are in good agreement with the simulation results.

Introduction

Genetic alterations such as point mutations, chromosomal rearrangements, unequal crossing over, loss of heterozygosity, modification of DNA methylation and chromosome aberrations accumulate during the lifetime of the organism. They are caused by intrinsic errors in the DNA replication and repair as well as by external factors such as exposure to mutagenic substances or radiation. Since the discovery that the configuration of a DNA or RNA molecule is a double helix (Watson et al., 1988), molecular biologists and geneticists have been studying the crucial role of DNA in the genome organization. Once it was recognized that DNA is the informational active chemical component of essentially all genetic materials, it was assumed that this macromolecule must be extraordinarily stable in order to maintain the degree of fidelity required of a master blueprint.

It was something of a surprise to learn that the primary structure of DNA is quite dynamic and subject to constant changes. For example, gene transposition is a well-established phenomenon in prokaryotic and eukaryotic cells (Finnegan, 1990, Kleckner, 1981). In addition, DNA is subject to alteration in the chemistry or sequence of individual nucleotides (Lindahl, 1993, Roberts, 1978, Singer and Kusmierek, 1982). Many of these changes arise as a consequence of errors introduced during replication, recombination and repairing itself. Other basic alterations arise from the inherent instability of specific chemical bonds that constitute the normal chemistry of nucleotides under physiological conditions of temperature and pH. Finally, the DNA of living cells reacts to a variety of chemical compounds and a smaller number of physical agents, many of which are present in the environment. Each of these modifications of the molecular structure of genetic material is appropriately considered to be a DNA damage. DNA damages can be classified into two major classes, spontaneous and environmental. However, in some cases the actual chemical changes in DNA that occur “spontaneously” are indistinguishable from those brought about through interaction with certain environmental agents. The term “spontaneous” may merely imply that we have not identified a particular environmental culprit. Changes in the DNA sequence may result from processes such as insertion, deletion, transversion and transition. For example, the genetic instability characteristic of cancer cells may be due, in part, to mutations in genes whose products normally function to ensure DNA integrity. DNA replication in normal human cells is an extremely accurate process. During the cell division cycle, copy errors occur with probabilities less than 10⁻⁹ to 10⁻¹⁰ per nucleotide. In contrast, the malignant cells that constitute cancer tissues are markedly heterogeneous and exhibit alterations in nucleotide sequence of DNA.

As initially proposed by Delbruck et al. (1935) and Watson and Crick (1953), spontaneous mutations are initiated by quantum jump events such as tautomeric shifts in single protons of DNA bases. Even what may be the most common of spontaneous mutations involves a chemical mechanism which must involve quantum uncertainty, since it occurs when individual electrons shift their positions to produce “tautomers”.

Specifically, nucleotide transitions can be induced by exposure to endogeneous and exogeneous mutagens (agents causing genetic changes) such as chemical carcinogens. However, not all mutagens are carcinogenic. The nucleotide transitions are the interchange of bases of the same shape, e.g., the purine bases transition, C(cytosine) ↔ T(thymine) or the pyrimidine bases transitions, A(adenine) ↔ G(guanine). One of the mechanisms that can cause the transition is the shift of the positions of the electrons for the bases to become a transient form (known in organic chemistry as a tautomeric shift).

In standard complementary pairing, G pairs with C and A with T. Keto-enol tautomeric shift leads to non-standard form of G: G ↔ G^* resulting in G^* pairing with T. Amino-imino tautomeric shift leads to a non-standard form of A: A ↔ A^* resulting in A^* pairing with C. Non-standard bases alter the pairing specificity, i.e., modified purine pairs with the wrong pyrimidine and modified pyrimidine pairs with the wrong purine. Fig. 1 shows an example of the keto-enol tautomeric shift that results in a transition mutation of the complementary strand. Consider the pairing of ATGC with TACG: Let G in the first strand undergo a tautomeric shift to G^*. The complementary strand that is generated would be TATG, not TACG. This would be a transition from C ↔ T. To complete the process of producing a mutation, a tautomeric shift must take place during replication, either in the template chain, or in the deoxyribonucleotide being added by the DNA polymerase. Since the shifted form retains its rare mis-matching structure for only a brief period, the next replication cycle will most likely find itself reverted to its normal form, and the polymerase will pair it with its normal mate. Thus, in two cycles of replication, an A–T pair is changed to a G–C pair, or vice versa. This, in turn, can often result in a change in a triplet code, leading to an amino acid substitution in a protein, and a modification of some visible phenotypic property of the organism.

Although it has never been demonstrated experimentally that rare tautomers are responsible for spontaneous mutations, subsequent experimental and theoretical investigations (Leszczynski, 1999; Radchenko et al., 1983) seem to confirm the essential correctness of this postulate. It should be remarked that Neo–Darwinian evolutionary theory is founded on the principle that mutations occur randomly, and the direction of evolutionary change is provided by selection for advantageous mutations. However, the central tenet, that mutations occur randomly, has recently been challenged by the finding of the phenomenon termed adaptive or directed mutation.

There have been a few approaches used to investigate this mutation complex process ranging from wet lab research to highly complicated computational calculations. Theoretical models fall into two very broad classes: deterministic and stochastic models. Deterministic models attempt to model or predict the average behavior of systems according to some precise rules. In contrast, stochastic models describe the probability of very specific behaviors of individuals rather than average behavior of the population. Stochasticity has been recognized in the biology field of research and modeling as the description of life systems (Kurakin, 2006). It had appeared as general principles underlying the dynamics and organization of biological systems at all scales: gene expression (Kurakin, 2005), enzymes (Xie and Lu, 1999), self-organization of macromolecular complexes mediating transcription (Dundr et al., 2002, Kimura et al., 2002), and DNA repair (Essers et al., 2002, Hoogstraten et al., 2002).

Because a gene or DNA is a molecule, the statistical fluctuations of atomic or molecular scale cannot be avoided. Mathematical modeling of genetic instability has led to considerable insight into human tumorigenesis. One study of the mutational spectrum gave the type, location and frequency of DNA changes in a particular gene (Hussain and Harris, 1999). Claytong and Robertson (1955) proposed a random walk mutation model as a model for genetic analysis. It was later proposed explicitly by Crowj and Kimura (1964), by Kimura (1965), and subsequently popularized by Lander (1975). Zeng and Cockerham (1993) proposed a more general mutation model, called the regression mutation model. This model regards the regression coefficient of the effect of an allele after mutation on the effect of the allele before mutation as a parameter.

In 1989, Nowak and Schuster (1989) investigated error thresholds in finite populations. They determined that, at error rates above the critical value, the quasispecies ceases to be localized in sequence space and start to drift randomly. Sole’ and Deisboeck (2004) used a quasispecies model to investigate the error threshold in cancer cells. They demonstrated that, once the threshold is reached, the highly unstable cancer cells become unable to maintain their genetic information, leading to a decrease in the velocity of tumor growth. The original quasispecies model assumes that genomes replicate conservatively, i.e., each single-stranded genome replicates by producing a new, possibly error-prone, single stranded copy without affecting the original. In this form, the quasispecies model predicts the existence of an error catastrophe or “error threshold”, a threshold mutation rate above which no viable species can exist. This threshold depends on the replication rate of the fittest sequence, the master sequence (Komarova et al., 2002) utilize a stochastic model to evaluate the rate of formation of dysplastic crypts by chromosomal instability (CIN) and microsatellite instability (MIN) mechanisms in sporadic colon cancer to obtain broad qualitative agreement with the relative importance of CIN and MIN and the number of polyps generated under these conditions.

The main purpose of the present work is to begin an exploration of the stochastic nature of non-equilibrium DNA alteration caused by events such as tautomeric shifts in a theoretical DNA-bit alteration model This is done by modeling the genetic DNA (or RNA) code chain as a sequence of DNA-bit values (‘1’ for normal bases and ‘−1’ for abnormal bases). This is similar to what is used in computers or electronics. We observe the number of DNA-bit changes resulting from the random point mutation process (to mimic tautomeric shifts) which is being induced by a stochastic Brownian mutagen (BM) as it diffuses through the DNA-bit systems. We will make analytic predictions and simulate the non-equilibrium process using the Monte Carlo (MC) method. To the best of our knowledge, there has not been a stochastic approach to investigate the non-equilibrium stochastic kinetics of DNA-alteration. This work therefore represents a new avenue for studying non-equilibrium mutation.