Topological Parameters of DNA and Genome Complexity

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February 5, 2025

Introduction

This lecture revisits the topological parameters of DNA, focusing on linking number, twist, and writhe, and their interrelationships. Visual aids, including animations, will illustrate these concepts and their relevance to DNA supercoiling. We will explore how cells utilize DNA supercoiling and examine the biological and medical implications of DNA topology, particularly the role of topoisomerases as therapeutic targets in cancer therapy. Subsequently, the lecture will address the informational content of the genome, covering genome size, gene density, and the C-value paradox, with a specific focus on the organization and evolution of the human genome.

Topological Parameters of DNA

Linking Number, Twist, and Writhe

The topology of DNA, particularly in covalently closed circular DNA (cccDNA), is described by three interrelated parameters: linking number ($\text{Lk}$), twist ($\text{Tw}$), and writhe ($\text{Wr}$). Understanding these parameters is essential for comprehending the spatial organization and functional properties of DNA.

Definition 1.

Linking Number ($\text{Lk}$). The linking number ($\text{Lk}$) is a topological invariant that quantifies the number of times two DNA strands are intertwined. Mathematically, it represents the number of times one strand would have to pass through the other to achieve complete separation, without breaking any covalent bonds. For cccDNA, $\text{Lk}$ is always an integer and remains constant unless a strand is cleaved.

Definition 2.

Twist ($\text{Tw}$). Twist ($\text{Tw}$) measures the helical winding of the DNA strands around each other. It is the number of helical turns of one strand about the axis of the double helix. In relaxed, planar DNA, the linking number is equivalent to the twist. By convention, right-handed helices, like standard B-DNA, are assigned a positive twist value.

Definition 3.

Writhe ($\text{Wr}$). Writhe ($\text{Wr}$) describes the supercoiling of the DNA double helix axis in three-dimensional space. It represents the coiling of the helix axis upon itself and quantifies the DNA’s tertiary structure beyond the simple double helix. Writhe can manifest in two primary forms:

Plectonemic Writhe (Interwound): The DNA helix is wound around itself in a helical manner, resembling a twisted rope.
Toroidal Writhe (Spiral): The DNA helix is wound around a central axis, often associated with DNA wrapping around proteins, such as histones in nucleosomes.

These three parameters are related by the fundamental topological equation: \[\label{eq:linking_equation}\text{Lk}= \text{Tw}+ \text{Wr}\] For a relaxed, planar cccDNA molecule, $\text{Wr}= 0$, and therefore $\text{Lk}= \text{Tw}$. However, if the DNA is not constrained to a plane or is supercoiled, writhe becomes non-zero, and changes in twist must be compensated by changes in writhe to maintain a constant linking number in topologically closed systems.

DNA Supercoiling

DNA supercoiling is the coiling of the DNA double helix upon itself, resulting in a more compact structure and introducing torsional stress. Supercoiling is crucial for DNA function and can be categorized as negative or positive.

Negative Supercoiling

Negative supercoiling occurs when the DNA is underwound, meaning it has fewer helical turns than relaxed DNA.

Mechanism: Negative supercoiling arises from a decrease in the helical twist ($\text{Tw}$) relative to the relaxed state. To maintain a constant linking number ($\text{Lk}$) in cccDNA, this reduction in twist is compensated by the introduction of negative writhe ($\text{Wr}< 0$). This can be achieved by reducing the number of base pairs per helical turn, effectively tightening the helix.
Effect on Twist and Writhe: An increase in twist (numerically, making it less negative or more positive, e.g., from -10 to -5, or by reducing base pairs per turn) while holding $\text{Lk}$ constant necessitates a compensatory negative writhe to satisfy Equation [eq:linking_equation].
Biological Significance: Negative supercoiling is the predominant form in vivo and is essential for various DNA-dependent processes:
- Facilitation of DNA Denaturation: Negative supercoiling stores free energy, making it easier to separate DNA strands. This is crucial for processes like replication, transcription, and repair, which require local DNA denaturation.
- Chromatin Condensation: In eukaryotes, negative supercoiling is intrinsically linked to chromatin structure. The wrapping of DNA around histone octamers to form nucleosomes induces toroidal, left-handed supercoils, contributing significantly to overall negative supercoiling and DNA compaction. Histones H3 and H4 play a key role in this process.

Positive Supercoiling

Positive supercoiling occurs when the DNA is overwound, having more helical turns than relaxed DNA.

Mechanism: Positive supercoiling results from an increase in helical twist ($\text{Tw}$) relative to the relaxed state. To maintain a constant $\text{Lk}$, this increase in twist is balanced by positive writhe ($\text{Wr}> 0$). This can be induced by increasing the number of base pairs per turn, effectively unwinding the helix.
Effect on Twist and Writhe: A decrease in twist (numerically, making it more negative or less positive, e.g., from +10 to +5, or by increasing base pairs per turn) while keeping $\text{Lk}$ constant results in a compensatory positive writhe.
Biological Context: Positive supercoiling is generated transiently during DNA replication and transcription ahead of the replication fork or transcription bubble as the DNA helix is unwound. However, excessive positive supercoiling is generally detrimental as it impedes these processes.
Positive Supercoiling in Extremophiles: Interestingly, some extremophilic bacteria, particularly thermophiles, employ positive supercoiling as a strategy for DNA stabilization. Positive supercoiling increases the melting temperature of DNA, protecting it from thermal denaturation in extreme environments. This adaptation, along with high GC content in their genomes, enhances DNA stability at high temperatures. Enzymes like gyrases in these organisms are responsible for introducing and maintaining positive supercoiling.

Biological Implications of DNA Topology

DNA topology, particularly the degree and type of supercoiling, has profound biological consequences, influencing DNA accessibility, stability, and interactions with cellular machinery.

Negative Supercoiling in Vivo

In most living organisms, cellular DNA is predominantly maintained in a negatively supercoiled state. This is quantified by the superhelical density, $\sigma$, which is a measure of the number of supercoils relative to the relaxed state of the DNA.

\[\label{eq:superhelical_density}\sigma = \frac{\Delta \text{Lk}}{\text{Lk}_0} = \frac{\text{Lk}- \text{Lk}_0}{\text{Lk}_0}\] where $\Delta \text{Lk}$ is the difference between the linking number of the DNA ($\text{Lk}$) and the linking number of the relaxed DNA ($\text{Lk}_0$). A negative value of $\sigma$ indicates negative supercoiling.

Facilitation of Essential DNA Processes: Negative supercoiling is thermodynamically favorable for DNA strand separation, a prerequisite for fundamental processes such as:
- Replication: Unwinding of the DNA double helix at the replication fork is facilitated by the stored energy in negatively supercoiled DNA.
- Transcription: Promoter regions and gene regulatory elements become more accessible due to the reduced helical stress, enhancing transcription initiation.
- DNA Repair: Damaged DNA regions requiring strand separation for repair processes are more readily accessible in negatively supercoiled domains.
Chromatin Organization in Eukaryotes: In eukaryotic cells, negative supercoiling is intrinsically linked to chromatin structure. The hierarchical organization of DNA into nucleosomes and higher-order structures involves:
- Toroidal Supercoiling by Histones: The wrapping of DNA around histone octamers introduces left-handed toroidal supercoils. Histones, particularly H3 and H4, are instrumental in inducing this negative supercoiling, contributing to DNA compaction and regulation. This toroidal form is the primary mode of negative supercoiling in eukaryotic chromatin.
Plectonemic and Toroidal Forms of Supercoiling: Negative supercoiling can be represented in two topologically equivalent forms, as depicted in 1:
- Plectonemic (Interwound): Supercoils are interwound, resembling a twisted telephone cord.
- Toroidal (Spiral): Supercoils are arranged in a spiral, often around a central axis, as seen in DNA wrapped around nucleosomes.
While topologically interconvertible, the toroidal form is biologically relevant in the context of chromatin structure and histone interactions in vivo.

Plectonemic and Toroidal representations of negative supercoiling, both exhibiting a linking number difference of -1 relative to relaxed DNA. These forms are topologically equivalent and represent the same degree of supercoiling.

Positive Supercoiling in Extremophiles

In contrast to the prevalence of negative supercoiling in most organisms, certain extremophiles, notably thermophilic bacteria, exhibit positive supercoiling. This adaptation is crucial for survival in extreme environments.

Protection Against Thermal Denaturation: Positive supercoiling enhances the thermal stability of DNA. By overwinding the double helix, it becomes more resistant to denaturation at elevated temperatures, a critical adaptation for thermophiles thriving in high-temperature habitats.
Role of DNA Gyrases in Prokaryotes: In bacteria, including extremophiles, the introduction and maintenance of positive supercoiling are mediated by specialized enzymes called DNA gyrases.
- Gyrase Activity: DNA gyrases are a subclass of type II topoisomerases that uniquely introduce negative supercoils into DNA, which can be conceptually inverted to understand their role in preventing positive supercoiling in extremophiles. In essence, by actively managing supercoiling, gyrases in thermophiles may prevent the DNA from becoming excessively negatively supercoiled, thus favoring a more positively supercoiled or less negatively supercoiled state that is more stable at high temperatures.
- Prokaryotic Supercoiling Control: Gyrases in prokaryotes fulfill a role analogous to that of histones in eukaryotes, albeit in the opposite direction of supercoiling in extremophiles, by controlling the overall level of DNA supercoiling to suit environmental demands.
Synergistic Effect with High GC Content: The genomes of thermophilic organisms often exhibit a higher guanine-cytosine (GC) content. GC base pairs have three hydrogen bonds compared to the two in adenine-thymine (AT) pairs, contributing to greater thermal stability. This intrinsic stability, combined with positive supercoiling, provides a dual mechanism for protecting DNA integrity in extreme thermal conditions.

Topoisomerases

Topoisomerases are a class of essential enzymes that manage DNA topology by catalyzing changes in the linking number ($\text{Lk}$). These enzymes are critical for alleviating torsional stress during fundamental cellular processes such as DNA replication, transcription, chromosome segregation, and DNA recombination and repair. Topoisomerases act by transiently breaking and rejoining DNA strands, allowing for the relaxation of supercoils and the disentanglement of DNA molecules.

Classification of Topoisomerases

Topoisomerases are broadly classified into two main types, Type I and Type II, based on their structure, mechanism of action, and the change they induce in the linking number.

Type I Topoisomerases

Type I topoisomerases alter the linking number in increments of one.

Mechanism: Type I topoisomerases function by creating a transient single-strand break (nick) in the DNA backbone. This nick allows for the controlled rotation of one DNA strand around the other, relieving torsional stress. After relaxation, the enzyme reseals the break. The process is illustrated in 2.
Change in Linking Number: Each catalytic cycle of a Type I topoisomerase changes the linking number ($\text{Lk}$) by $\pm 1$.
ATP Dependence: Type I topoisomerases are ATP-independent, meaning they do not require ATP hydrolysis to perform their function. The energy stored in the supercoiled DNA is sufficient to drive the reaction.
Biological Roles: Type I topoisomerases primarily function to:
- Relax Supercoiled DNA: They remove both positive and negative supercoils, although they are generally more efficient at relaxing negative supercoils. This is crucial for maintaining DNA in a state accessible for replication and transcription.
- Decatenation and Catenation: They can resolve DNA catenanes (interlinked DNA rings) and, under certain conditions, catalyze catenation, particularly involving single-strand breaks. This activity is important in disentangling newly synthesized DNA molecules.

Mechanism of Type I Topoisomerase. Type I enzymes introduce a transient nick in one DNA strand, allowing relaxation of supercoils and subsequent resealing of the strand.

Type II Topoisomerases

Type II topoisomerases alter the linking number in increments of two.

Mechanism: Type II topoisomerases operate by creating a transient double-strand break in the DNA. They then pass another double-stranded DNA helix through this break before resealing it. This double-strand passage mechanism is more complex than that of Type I enzymes and is depicted in 3.
Change in Linking Number: Each catalytic cycle of a Type II topoisomerase changes the linking number ($\text{Lk}$) by $\pm 2$.
ATP Dependence: Type II topoisomerases are ATP-dependent, requiring ATP hydrolysis to drive the conformational changes necessary for double-strand passage and DNA religation.
Biological Roles: Type II topoisomerases perform a variety of crucial functions:
- Relaxation of Supercoils: Similar to Type I enzymes, Type II topoisomerases can relax both positive and negative supercoils, playing a vital role in managing torsional stress during DNA replication and transcription.
- Decatenation of Interlinked DNA: They are essential for decatenating intertwined DNA molecules, particularly newly replicated sister chromatids or mitochondrial DNA, ensuring proper chromosome segregation during cell division.
- Condensation and Decondensation of Chromosomes: Type II topoisomerases are involved in chromosome condensation and decondensation processes during mitosis and meiosis.
- Introduction of Negative Supercoils (Gyrases): In bacteria, a subclass of Type II topoisomerases known as gyrases has the unique ability to introduce negative supercoils into DNA, utilizing ATP hydrolysis. This is crucial for maintaining the negative superhelical density of bacterial genomes.

Mechanism of Type II Topoisomerase. Type II enzymes introduce a transient double-strand break, allowing another DNA duplex to pass through before resealing the break, changing the linking number by ().

Molecular Mechanism of Topoisomerase Action

Both Type I and Type II topoisomerases share a common mechanistic feature: the formation of a covalent intermediate between the enzyme and the DNA via a phosphotyrosine linkage. This intermediate is crucial for the transient breakage and subsequent rejoining of DNA strands.

Active Site Tyrosine: The active site of topoisomerases contains a conserved tyrosine residue. The hydroxyl group of this tyrosine acts as a nucleophile in the DNA cleavage reaction.
Phosphotyrosine Intermediate Formation: The mechanism proceeds as follows (illustrated in 4):
1. Nucleophilic Attack: The tyrosine hydroxyl group attacks the phosphodiester bond of a DNA strand. This attack is typically facilitated by a nearby basic residue (e.g., lysine or arginine) in the active site, which deprotonates the tyrosine hydroxyl, enhancing its nucleophilicity.
2. DNA Cleavage and Covalent Linkage: This nucleophilic attack results in the cleavage of the phosphodiester bond and the formation of a covalent phosphotyrosine linkage between the 3’-phosphate of the cleaved DNA strand and the tyrosine residue of the enzyme. This creates a transient DNA break (single-strand break for Type I, double-strand break for Type II). The 5’-OH group at the cleavage site is left free.
3. Strand Passage and Religation: The enzyme then facilitates the passage of DNA strand(s) through the break. For Type I, a single strand passes; for Type II, a double helix passes through the double-strand break.
4. Reversal of Phosphotyrosine Linkage: Finally, to complete the catalytic cycle, the phosphotyrosine bond is reversed via nucleophilic attack by the 5’-OH group of the cleaved DNA on the phosphotyrosine linkage. This religates the DNA phosphodiester backbone, releases the enzyme, and restores DNA continuity, albeit with a changed topological state.

Mechanism involving a phosphotyrosine intermediate. Topoisomerases utilize a tyrosine residue in their active site to form a covalent phosphotyrosine linkage with DNA, enabling strand breakage and reunion.

Role of Topoisomerases in DNA Replication

Topoisomerases are indispensable during DNA replication to resolve the topological challenges imposed by the unwinding of the DNA double helix at the replication fork.

Relieving Torsional Stress Ahead of Replication Fork: As DNA helicases unwind the double helix at the replication fork, positive supercoiling is generated ahead of the fork. This accumulation of positive supercoils increases torsional stress, which, if not relieved, would impede the progression of the replication fork and potentially lead to DNA damage.
Function of Topoisomerases at the Replication Fork: Topoisomerases, particularly Type II topoisomerases, are crucial for resolving this topological problem:
- Removal of Positive Supercoils: They act ahead of the replication fork to remove positive supercoils, allowing the DNA to unwind and the replication fork to advance smoothly.
- Preventing Replication Fork Stalling: By reducing torsional stress, topoisomerases prevent replication fork stalling and ensure efficient and continuous DNA synthesis.
Preventing DNA Breakage and Genome Instability: If positive supercoils are not removed, the excessive torsional stress can lead to DNA double-strand breaks. These breaks are highly detrimental to genome stability and can result in mutations, chromosomal aberrations, and cell death. Topoisomerases are thus essential for maintaining genome integrity during replication.

Topoisomerases as Targets for Anti-Cancer Therapy

The critical role of topoisomerases, especially in rapidly proliferating cells, makes them effective targets for anti-cancer drugs. Cancer cells, characterized by their uncontrolled and rapid division, are particularly vulnerable to disruptions in DNA topology management.

Selective Targeting of Rapidly Dividing Cells: Cancer cells rely heavily on topoisomerase activity to manage the topological stress associated with their high rate of DNA replication. This makes topoisomerases attractive targets for therapeutic intervention.
Topoisomerase Inhibitors in Cancer Therapy: Several clinically significant anti-cancer drugs function as topoisomerase inhibitors. These include:
- Camptothecin and Topotecan: These drugs are Type I topoisomerase inhibitors. Camptothecin and its derivatives, like topotecan, trap Type I topoisomerases in their covalent DNA-bound state, forming a ternary complex that interferes with replication fork progression.
- Etoposide and Teniposide: These are Type II topoisomerase inhibitors. Etoposide and teniposide also stabilize the covalent DNA-enzyme complex of Type II topoisomerases, leading to double-strand breaks and replication blockage.
Mechanism of Action in Cancer Cells: Topoisomerase inhibitors exert their cytotoxic effects through the following mechanisms:
- Stabilization of Covalent Complexes: They stabilize the covalent DNA-topoisomerase intermediate, preventing religation of the DNA break.
- Replication Fork Stalling and DNA Breaks: The trapped topoisomerase-DNA complexes obstruct the progression of the replication fork, leading to replication fork stalling and collapse. Collision of the replication fork with these complexes often results in the formation of persistent DNA double-strand breaks (DSBs).
- Activation of DNA Damage Response and Apoptosis: The accumulation of DSBs triggers the DNA damage response (DDR) pathways in cancer cells. If the damage is irreparable, these pathways activate programmed cell death (apoptosis), eliminating the cancer cells.
Side Effects of Topoisomerase Inhibitors: Because these drugs target rapidly dividing cells, they are not entirely specific to cancer cells and can also affect normal proliferating cells in tissues such as:

These side effects are due to the drugs’ impact on healthy, dividing cells and are a common challenge in cancer chemotherapy.
$\gamma$-H2AX as a Biomarker for DNA Damage: The phosphorylation of histone variant H2AX at serine 139, yielding $\gamma$-H2AX, is a sensitive marker for DNA double-strand breaks. The formation of $\gamma$-H2AX foci in cell nuclei, detectable by immunofluorescence microscopy, indicates the presence of DSBs induced by topoisomerase inhibitors or other DNA-damaging agents. Kinases ATM (Ataxia-Telangiectasia Mutated) and ATR (Ataxia-Telangiectasia and Rad3-related) are key players in the DNA damage response pathway and are responsible for phosphorylating H2AX to form $\gamma$-H2AX in response to DSBs.

Remark. Remark 1.

Topoisomerase inhibitors are a cornerstone of cancer chemotherapy, effective against a wide range of malignancies. However, their use is carefully managed due to the potential for significant side effects arising from their action on normal proliferating cells. Ongoing research aims to develop more selective topoisomerase inhibitors or combination therapies to enhance efficacy and reduce toxicity.

Genome Size and Complexity

This section explores the informational content of genomes, focusing on the relationship between genome size, gene density, and organismal complexity. We will address the C-value paradox and examine how genome organization varies across different organisms, with a particular emphasis on the human genome.

The C-Value Paradox: Genome Size Variation and Complexity

The C-value refers to the total amount of DNA contained within a haploid genome. A long-standing puzzle in genomics is the C-value paradox, which highlights the counterintuitive observation that genome size does not correlate with the perceived complexity of an organism.

Extensive Variation in Genome Size: Genome size varies dramatically across the tree of life, spanning several orders of magnitude, even among closely related species. This variation is evident across different taxonomic groups:
Paradox of Non-Coding DNA: The paradox arises because organismal complexity was initially expected to correlate with the number of protein-coding genes and, consequently, genome size. However, observations revealed that:
Role of Non-Coding DNA: The C-value paradox is largely explained by the varying amounts of non-coding DNA in different genomes. This non-coding DNA includes:

Gene Density and the Dilution of Coding Information

Gene density, defined as the proportion of a genome that is composed of protein-coding sequences, provides another perspective on genome organization and evolution.

Decreasing Gene Density Across Phylogeny: A general trend observed across the phylogeny of life is a decrease in gene density with increasing genome size and organismal complexity.
Dilution of Coding Information: The evolutionary trend towards larger genomes in eukaryotes is accompanied by a "dilution" of coding information. The expansion of genome size is not primarily driven by the multiplication of genes but by the accumulation of non-coding DNA. This suggests that the increased complexity of eukaryotes is not simply a matter of having more genes but rather involves more sophisticated regulation and utilization of genomic information.
Regulatory DNA and Complexity: While non-coding DNA was once considered "junk DNA," it is now recognized to play crucial roles, particularly in gene regulation. The expansion of non-coding regions, especially regulatory sequences, is thought to be a key factor in the evolution of increased regulatory complexity in eukaryotes. This regulatory complexity allows for finer control over gene expression in space and time, contributing to the development of complex multicellular life forms.

Introns: A Major Component of Eukaryotic Gene Structure

Introns are non-coding DNA sequences located within eukaryotic genes that are transcribed into precursor mRNA but are subsequently removed by RNA splicing to produce mature mRNA. Introns significantly contribute to the size and complexity of eukaryotic genes, particularly in humans.

Introns as a Major Determinant of Gene Size in Humans: Human genes are notably larger than prokaryotic genes, primarily due to the extensive presence of introns.
Functional Significance of Introns: Despite being non-coding in the sense that they are not translated into protein, introns are not merely "junk DNA" and are increasingly recognized to have diverse functional roles:

In summary, the organization of genomes, particularly in eukaryotes, is characterized by a complex interplay between coding and non-coding DNA. The C-value paradox highlights that genome size is not a direct measure of organismal complexity. Instead, the expansion of genomes in complex organisms is largely due to the accumulation of non-coding DNA, which plays critical roles in gene regulation, genome evolution, and the generation of biological diversity. The lower gene density and the presence of introns in eukaryotic genomes reflect a shift from genome compactness in prokaryotes to regulatory and structural complexity in higher organisms.

Conclusion

This lecture has provided a comprehensive overview of DNA topology and genome complexity. We have examined the fundamental topological parameters—linking number, twist, and writhe—and their critical role in DNA supercoiling. We discussed the biological significance of supercoiling, contrasting the prevalence of negative supercoiling in most organisms with the adaptive positive supercoiling found in extremophiles. A detailed exploration of topoisomerases, essential enzymes for managing DNA topology, included their classification, mechanisms of action, and their therapeutic relevance as anti-cancer targets. Finally, we addressed the concept of genome size and complexity, elucidating the C-value paradox, the evolutionary trend of decreasing gene density, and the substantial contribution of non-coding introns to the structure of eukaryotic genes, particularly in the human genome.

Key Takeaways:

DNA topology, defined by linking number, twist, and writhe, is a fundamental determinant of DNA structure and function in vivo.
DNA supercoiling, especially negative supercoiling, is essential for facilitating critical cellular processes such as replication, transcription, and DNA repair.
Topoisomerases are indispensable enzymes that maintain proper DNA topology by altering the linking number and are critical targets for anti-cancer therapeutics.
Genome size does not directly correlate with organismal complexity, as exemplified by the C-value paradox, highlighting the significant role of non-coding DNA in genome organization and regulation.

Further Studies: To expand on the topics covered in this lecture, future studies could delve into:

--- title: "Topological Parameters of DNA and Genome Complexity" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture revisits the topological parameters of DNA, focusing on linking number, twist, and writhe, and their interrelationships. Visual aids, including animations, will illustrate these concepts and their relevance to DNA supercoiling. We will explore how cells utilize DNA supercoiling and examine the biological and medical implications of DNA topology, particularly the role of topoisomerases as therapeutic targets in cancer therapy. Subsequently, the lecture will address the informational content of the genome, covering genome size, gene density, and the C-value paradox, with a specific focus on the organization and evolution of the human genome. # Topological Parameters of DNA ## Linking Number, Twist, and Writhe The topology of DNA, particularly in covalently closed circular DNA (cccDNA), is described by three interrelated parameters: linking number ($\text{Lk}$), twist ($\text{Tw}$), and writhe ($\text{Wr}$). Understanding these parameters is essential for comprehending the spatial organization and functional properties of DNA. :::: definition **Definition 1**. ::: tcolorbox ***Linking Number ($\text{Lk}$)**. The linking number ($\text{Lk}$) is a topological invariant that quantifies the number of times two DNA strands are intertwined. Mathematically, it represents the number of times one strand would have to pass through the other to achieve complete separation, without breaking any covalent bonds. For cccDNA, $\text{Lk}$ is always an integer and remains constant unless a strand is cleaved.* ::: :::: :::: definition **Definition 2**. ::: tcolorbox ***Twist ($\text{Tw}$)**. Twist ($\text{Tw}$) measures the helical winding of the DNA strands around each other. It is the number of helical turns of one strand about the axis of the double helix. In relaxed, planar DNA, the linking number is equivalent to the twist. By convention, right-handed helices, like standard B-DNA, are assigned a positive twist value.* ::: :::: :::: definition **Definition 3**. ::: tcolorbox ***Writhe ($\text{Wr}$)**. Writhe ($\text{Wr}$) describes the supercoiling of the DNA double helix axis in three-dimensional space. It represents the coiling of the helix axis upon itself and quantifies the DNA's tertiary structure beyond the simple double helix. Writhe can manifest in two primary forms:* - ***Plectonemic Writhe (Interwound):** The DNA helix is wound around itself in a helical manner, resembling a twisted rope.* - ***Toroidal Writhe (Spiral):** The DNA helix is wound around a central axis, often associated with DNA wrapping around proteins, such as histones in nucleosomes.* ::: :::: These three parameters are related by the fundamental topological equation: $$\label{eq:linking_equation}\text{Lk}= \text{Tw}+ \text{Wr}$$ For a relaxed, planar cccDNA molecule, $\text{Wr}= 0$, and therefore $\text{Lk}= \text{Tw}$. However, if the DNA is not constrained to a plane or is supercoiled, writhe becomes non-zero, and changes in twist must be compensated by changes in writhe to maintain a constant linking number in topologically closed systems. ## DNA Supercoiling DNA supercoiling is the coiling of the DNA double helix upon itself, resulting in a more compact structure and introducing torsional stress. Supercoiling is crucial for DNA function and can be categorized as negative or positive. ### Negative Supercoiling Negative supercoiling occurs when the DNA is underwound, meaning it has fewer helical turns than relaxed DNA. - **Mechanism:** Negative supercoiling arises from a decrease in the helical twist ($\text{Tw}$) relative to the relaxed state. To maintain a constant linking number ($\text{Lk}$) in cccDNA, this reduction in twist is compensated by the introduction of negative writhe ($\text{Wr}< 0$). This can be achieved by reducing the number of base pairs per helical turn, effectively tightening the helix. - **Effect on Twist and Writhe:** An increase in twist (numerically, making it less negative or more positive, e.g., from -10 to -5, or by reducing base pairs per turn) while holding $\text{Lk}$ constant necessitates a compensatory negative writhe to satisfy Equation [\[eq:linking_equation\]](#eq:linking_equation){reference-type="eqref" reference="eq:linking_equation"}. - **Biological Significance:** Negative supercoiling is the predominant form in vivo and is essential for various DNA-dependent processes: - **Facilitation of DNA Denaturation:** Negative supercoiling stores free energy, making it easier to separate DNA strands. This is crucial for processes like replication, transcription, and repair, which require local DNA denaturation. - **Chromatin Condensation:** In eukaryotes, negative supercoiling is intrinsically linked to chromatin structure. The wrapping of DNA around histone octamers to form nucleosomes induces toroidal, left-handed supercoils, contributing significantly to overall negative supercoiling and DNA compaction. Histones H3 and H4 play a key role in this process. ### Positive Supercoiling Positive supercoiling occurs when the DNA is overwound, having more helical turns than relaxed DNA. - **Mechanism:** Positive supercoiling results from an increase in helical twist ($\text{Tw}$) relative to the relaxed state. To maintain a constant $\text{Lk}$, this increase in twist is balanced by positive writhe ($\text{Wr}> 0$). This can be induced by increasing the number of base pairs per turn, effectively unwinding the helix. - **Effect on Twist and Writhe:** A decrease in twist (numerically, making it more negative or less positive, e.g., from +10 to +5, or by increasing base pairs per turn) while keeping $\text{Lk}$ constant results in a compensatory positive writhe. - **Biological Context:** Positive supercoiling is generated transiently during DNA replication and transcription ahead of the replication fork or transcription bubble as the DNA helix is unwound. However, excessive positive supercoiling is generally detrimental as it impedes these processes. - **Positive Supercoiling in Extremophiles:** Interestingly, some extremophilic bacteria, particularly thermophiles, employ positive supercoiling as a strategy for DNA stabilization. Positive supercoiling increases the melting temperature of DNA, protecting it from thermal denaturation in extreme environments. This adaptation, along with high GC content in their genomes, enhances DNA stability at high temperatures. Enzymes like gyrases in these organisms are responsible for introducing and maintaining positive supercoiling. # Biological Implications of DNA Topology DNA topology, particularly the degree and type of supercoiling, has profound biological consequences, influencing DNA accessibility, stability, and interactions with cellular machinery. ## Negative Supercoiling in Vivo In most living organisms, cellular DNA is predominantly maintained in a negatively supercoiled state. This is quantified by the superhelical density, $\sigma$, which is a measure of the number of supercoils relative to the relaxed state of the DNA. $$\label{eq:superhelical_density}\sigma = \frac{\Delta \text{Lk}}{\text{Lk}_0} = \frac{\text{Lk}- \text{Lk}_0}{\text{Lk}_0}$$ where $\Delta \text{Lk}$ is the difference between the linking number of the DNA ($\text{Lk}$) and the linking number of the relaxed DNA ($\text{Lk}_0$). A negative value of $\sigma$ indicates negative supercoiling. - **Facilitation of Essential DNA Processes:** Negative supercoiling is thermodynamically favorable for DNA strand separation, a prerequisite for fundamental processes such as: - **Replication:** Unwinding of the DNA double helix at the replication fork is facilitated by the stored energy in negatively supercoiled DNA. - **Transcription:** Promoter regions and gene regulatory elements become more accessible due to the reduced helical stress, enhancing transcription initiation. - **DNA Repair:** Damaged DNA regions requiring strand separation for repair processes are more readily accessible in negatively supercoiled domains. - **Chromatin Organization in Eukaryotes:** In eukaryotic cells, negative supercoiling is intrinsically linked to chromatin structure. The hierarchical organization of DNA into nucleosomes and higher-order structures involves: - **Toroidal Supercoiling by Histones:** The wrapping of DNA around histone octamers introduces left-handed toroidal supercoils. Histones, particularly H3 and H4, are instrumental in inducing this negative supercoiling, contributing to DNA compaction and regulation. This toroidal form is the primary mode of negative supercoiling in eukaryotic chromatin. - **Plectonemic and Toroidal Forms of Supercoiling:** Negative supercoiling can be represented in two topologically equivalent forms, as depicted in [1](#fig:supercoiling_forms){reference-type="ref+label" reference="fig:supercoiling_forms"}: - **Plectonemic (Interwound):** Supercoils are interwound, resembling a twisted telephone cord. - **Toroidal (Spiral):** Supercoils are arranged in a spiral, often around a central axis, as seen in DNA wrapped around nucleosomes. While topologically interconvertible, the toroidal form is biologically relevant in the context of chromatin structure and histone interactions in vivo. <figure id="fig:supercoiling_forms"> <figcaption>Plectonemic and Toroidal representations of negative supercoiling, both exhibiting a linking number difference of -1 relative to relaxed DNA. These forms are topologically equivalent and represent the same degree of supercoiling.</figcaption> </figure> ## Positive Supercoiling in Extremophiles In contrast to the prevalence of negative supercoiling in most organisms, certain extremophiles, notably thermophilic bacteria, exhibit positive supercoiling. This adaptation is crucial for survival in extreme environments. - **Protection Against Thermal Denaturation:** Positive supercoiling enhances the thermal stability of DNA. By overwinding the double helix, it becomes more resistant to denaturation at elevated temperatures, a critical adaptation for thermophiles thriving in high-temperature habitats. - **Role of DNA Gyrases in Prokaryotes:** In bacteria, including extremophiles, the introduction and maintenance of positive supercoiling are mediated by specialized enzymes called DNA gyrases. - **Gyrase Activity:** DNA gyrases are a subclass of type II topoisomerases that uniquely introduce negative supercoils into DNA, which can be conceptually inverted to understand their role in preventing positive supercoiling in extremophiles. In essence, by actively managing supercoiling, gyrases in thermophiles may prevent the DNA from becoming excessively negatively supercoiled, thus favoring a more positively supercoiled or less negatively supercoiled state that is more stable at high temperatures. - **Prokaryotic Supercoiling Control:** Gyrases in prokaryotes fulfill a role analogous to that of histones in eukaryotes, albeit in the opposite direction of supercoiling in extremophiles, by controlling the overall level of DNA supercoiling to suit environmental demands. - **Synergistic Effect with High GC Content:** The genomes of thermophilic organisms often exhibit a higher guanine-cytosine (GC) content. GC base pairs have three hydrogen bonds compared to the two in adenine-thymine (AT) pairs, contributing to greater thermal stability. This intrinsic stability, combined with positive supercoiling, provides a dual mechanism for protecting DNA integrity in extreme thermal conditions. # Topoisomerases Topoisomerases are a class of essential enzymes that manage DNA topology by catalyzing changes in the linking number ($\text{Lk}$). These enzymes are critical for alleviating torsional stress during fundamental cellular processes such as DNA replication, transcription, chromosome segregation, and DNA recombination and repair. Topoisomerases act by transiently breaking and rejoining DNA strands, allowing for the relaxation of supercoils and the disentanglement of DNA molecules. ## Classification of Topoisomerases Topoisomerases are broadly classified into two main types, Type I and Type II, based on their structure, mechanism of action, and the change they induce in the linking number. ### Type I Topoisomerases Type I topoisomerases alter the linking number in increments of one. - **Mechanism:** Type I topoisomerases function by creating a transient single-strand break (nick) in the DNA backbone. This nick allows for the controlled rotation of one DNA strand around the other, relieving torsional stress. After relaxation, the enzyme reseals the break. The process is illustrated in [2](#fig:topo1_mechanism){reference-type="ref+label" reference="fig:topo1_mechanism"}. - **Change in Linking Number:** Each catalytic cycle of a Type I topoisomerase changes the linking number ($\text{Lk}$) by $\pm 1$. - **ATP Dependence:** Type I topoisomerases are ATP-independent, meaning they do not require ATP hydrolysis to perform their function. The energy stored in the supercoiled DNA is sufficient to drive the reaction. - **Biological Roles:** Type I topoisomerases primarily function to: - **Relax Supercoiled DNA:** They remove both positive and negative supercoils, although they are generally more efficient at relaxing negative supercoils. This is crucial for maintaining DNA in a state accessible for replication and transcription. - **Decatenation and Catenation:** They can resolve DNA catenanes (interlinked DNA rings) and, under certain conditions, catalyze catenation, particularly involving single-strand breaks. This activity is important in disentangling newly synthesized DNA molecules. <figure id="fig:topo1_mechanism"> <figcaption>Mechanism of Type I Topoisomerase. Type I enzymes introduce a transient nick in one DNA strand, allowing relaxation of supercoils and subsequent resealing of the strand.</figcaption> </figure> ### Type II Topoisomerases Type II topoisomerases alter the linking number in increments of two. - **Mechanism:** Type II topoisomerases operate by creating a transient double-strand break in the DNA. They then pass another double-stranded DNA helix through this break before resealing it. This double-strand passage mechanism is more complex than that of Type I enzymes and is depicted in [3](#fig:topo2_mechanism){reference-type="ref+label" reference="fig:topo2_mechanism"}. - **Change in Linking Number:** Each catalytic cycle of a Type II topoisomerase changes the linking number ($\text{Lk}$) by $\pm 2$. - **ATP Dependence:** Type II topoisomerases are ATP-dependent, requiring ATP hydrolysis to drive the conformational changes necessary for double-strand passage and DNA religation. - **Biological Roles:** Type II topoisomerases perform a variety of crucial functions: - **Relaxation of Supercoils:** Similar to Type I enzymes, Type II topoisomerases can relax both positive and negative supercoils, playing a vital role in managing torsional stress during DNA replication and transcription. - **Decatenation of Interlinked DNA:** They are essential for decatenating intertwined DNA molecules, particularly newly replicated sister chromatids or mitochondrial DNA, ensuring proper chromosome segregation during cell division. - **Condensation and Decondensation of Chromosomes:** Type II topoisomerases are involved in chromosome condensation and decondensation processes during mitosis and meiosis. - **Introduction of Negative Supercoils (Gyrases):** In bacteria, a subclass of Type II topoisomerases known as gyrases has the unique ability to introduce negative supercoils into DNA, utilizing ATP hydrolysis. This is crucial for maintaining the negative superhelical density of bacterial genomes. <figure id="fig:topo2_mechanism"> <figcaption>Mechanism of Type II Topoisomerase. Type II enzymes introduce a transient double-strand break, allowing another DNA duplex to pass through before resealing the break, changing the linking number by <span class="math inline">$\pm 2$</span>.</figcaption> </figure> ## Molecular Mechanism of Topoisomerase Action Both Type I and Type II topoisomerases share a common mechanistic feature: the formation of a covalent intermediate between the enzyme and the DNA via a phosphotyrosine linkage. This intermediate is crucial for the transient breakage and subsequent rejoining of DNA strands. - **Active Site Tyrosine:** The active site of topoisomerases contains a conserved tyrosine residue. The hydroxyl group of this tyrosine acts as a nucleophile in the DNA cleavage reaction. - **Phosphotyrosine Intermediate Formation:** The mechanism proceeds as follows (illustrated in [4](#fig:phosphotyrosine_intermediate){reference-type="ref+label" reference="fig:phosphotyrosine_intermediate"}): 1. **Nucleophilic Attack:** The tyrosine hydroxyl group attacks the phosphodiester bond of a DNA strand. This attack is typically facilitated by a nearby basic residue (e.g., lysine or arginine) in the active site, which deprotonates the tyrosine hydroxyl, enhancing its nucleophilicity. 2. **DNA Cleavage and Covalent Linkage:** This nucleophilic attack results in the cleavage of the phosphodiester bond and the formation of a covalent phosphotyrosine linkage between the 3'-phosphate of the cleaved DNA strand and the tyrosine residue of the enzyme. This creates a transient DNA break (single-strand break for Type I, double-strand break for Type II). The 5'-OH group at the cleavage site is left free. 3. **Strand Passage and Religation:** The enzyme then facilitates the passage of DNA strand(s) through the break. For Type I, a single strand passes; for Type II, a double helix passes through the double-strand break. 4. **Reversal of Phosphotyrosine Linkage:** Finally, to complete the catalytic cycle, the phosphotyrosine bond is reversed via nucleophilic attack by the 5'-OH group of the cleaved DNA on the phosphotyrosine linkage. This religates the DNA phosphodiester backbone, releases the enzyme, and restores DNA continuity, albeit with a changed topological state. <figure id="fig:phosphotyrosine_intermediate"> <figcaption>Mechanism involving a phosphotyrosine intermediate. Topoisomerases utilize a tyrosine residue in their active site to form a covalent phosphotyrosine linkage with DNA, enabling strand breakage and reunion.</figcaption> </figure> ## Role of Topoisomerases in DNA Replication Topoisomerases are indispensable during DNA replication to resolve the topological challenges imposed by the unwinding of the DNA double helix at the replication fork. - **Relieving Torsional Stress Ahead of Replication Fork:** As DNA helicases unwind the double helix at the replication fork, positive supercoiling is generated ahead of the fork. This accumulation of positive supercoils increases torsional stress, which, if not relieved, would impede the progression of the replication fork and potentially lead to DNA damage. - **Function of Topoisomerases at the Replication Fork:** Topoisomerases, particularly Type II topoisomerases, are crucial for resolving this topological problem: - **Removal of Positive Supercoils:** They act ahead of the replication fork to remove positive supercoils, allowing the DNA to unwind and the replication fork to advance smoothly. - **Preventing Replication Fork Stalling:** By reducing torsional stress, topoisomerases prevent replication fork stalling and ensure efficient and continuous DNA synthesis. - **Preventing DNA Breakage and Genome Instability:** If positive supercoils are not removed, the excessive torsional stress can lead to DNA double-strand breaks. These breaks are highly detrimental to genome stability and can result in mutations, chromosomal aberrations, and cell death. Topoisomerases are thus essential for maintaining genome integrity during replication. ## Topoisomerases as Targets for Anti-Cancer Therapy The critical role of topoisomerases, especially in rapidly proliferating cells, makes them effective targets for anti-cancer drugs. Cancer cells, characterized by their uncontrolled and rapid division, are particularly vulnerable to disruptions in DNA topology management. - **Selective Targeting of Rapidly Dividing Cells:** Cancer cells rely heavily on topoisomerase activity to manage the topological stress associated with their high rate of DNA replication. This makes topoisomerases attractive targets for therapeutic intervention. - **Topoisomerase Inhibitors in Cancer Therapy:** Several clinically significant anti-cancer drugs function as topoisomerase inhibitors. These include: - **Camptothecin and Topotecan:** These drugs are Type I topoisomerase inhibitors. Camptothecin and its derivatives, like topotecan, trap Type I topoisomerases in their covalent DNA-bound state, forming a ternary complex that interferes with replication fork progression. - **Etoposide and Teniposide:** These are Type II topoisomerase inhibitors. Etoposide and teniposide also stabilize the covalent DNA-enzyme complex of Type II topoisomerases, leading to double-strand breaks and replication blockage. - **Mechanism of Action in Cancer Cells:** Topoisomerase inhibitors exert their cytotoxic effects through the following mechanisms: - **Stabilization of Covalent Complexes:** They stabilize the covalent DNA-topoisomerase intermediate, preventing religation of the DNA break. - **Replication Fork Stalling and DNA Breaks:** The trapped topoisomerase-DNA complexes obstruct the progression of the replication fork, leading to replication fork stalling and collapse. Collision of the replication fork with these complexes often results in the formation of persistent DNA double-strand breaks (DSBs). - **Activation of DNA Damage Response and Apoptosis:** The accumulation of DSBs triggers the DNA damage response (DDR) pathways in cancer cells. If the damage is irreparable, these pathways activate programmed cell death (apoptosis), eliminating the cancer cells. - **Side Effects of Topoisomerase Inhibitors:** Because these drugs target rapidly dividing cells, they are not entirely specific to cancer cells and can also affect normal proliferating cells in tissues such as: These side effects are due to the drugs' impact on healthy, dividing cells and are a common challenge in cancer chemotherapy. - **$\gamma$-H2AX as a Biomarker for DNA Damage:** The phosphorylation of histone variant H2AX at serine 139, yielding $\gamma$-H2AX, is a sensitive marker for DNA double-strand breaks. The formation of $\gamma$-H2AX foci in cell nuclei, detectable by immunofluorescence microscopy, indicates the presence of DSBs induced by topoisomerase inhibitors or other DNA-damaging agents. Kinases ATM (Ataxia-Telangiectasia Mutated) and ATR (Ataxia-Telangiectasia and Rad3-related) are key players in the DNA damage response pathway and are responsible for phosphorylating H2AX to form $\gamma$-H2AX in response to DSBs. :::: remark **Remark 1**. ::: tcolorbox *Topoisomerase inhibitors are a cornerstone of cancer chemotherapy, effective against a wide range of malignancies. However, their use is carefully managed due to the potential for significant side effects arising from their action on normal proliferating cells. Ongoing research aims to develop more selective topoisomerase inhibitors or combination therapies to enhance efficacy and reduce toxicity.* ::: :::: # Genome Size and Complexity This section explores the informational content of genomes, focusing on the relationship between genome size, gene density, and organismal complexity. We will address the C-value paradox and examine how genome organization varies across different organisms, with a particular emphasis on the human genome. ## The C-Value Paradox: Genome Size Variation and Complexity The C-value refers to the total amount of DNA contained within a haploid genome. A long-standing puzzle in genomics is the C-value paradox, which highlights the counterintuitive observation that genome size does not correlate with the perceived complexity of an organism. - **Extensive Variation in Genome Size:** Genome size varies dramatically across the tree of life, spanning several orders of magnitude, even among closely related species. This variation is evident across different taxonomic groups: - **Paradox of Non-Coding DNA:** The paradox arises because organismal complexity was initially expected to correlate with the number of protein-coding genes and, consequently, genome size. However, observations revealed that: - **Role of Non-Coding DNA:** The C-value paradox is largely explained by the varying amounts of non-coding DNA in different genomes. This non-coding DNA includes: ## Gene Density and the Dilution of Coding Information Gene density, defined as the proportion of a genome that is composed of protein-coding sequences, provides another perspective on genome organization and evolution. - **Decreasing Gene Density Across Phylogeny:** A general trend observed across the phylogeny of life is a decrease in gene density with increasing genome size and organismal complexity. - **Dilution of Coding Information:** The evolutionary trend towards larger genomes in eukaryotes is accompanied by a \"dilution\" of coding information. The expansion of genome size is not primarily driven by the multiplication of genes but by the accumulation of non-coding DNA. This suggests that the increased complexity of eukaryotes is not simply a matter of having more genes but rather involves more sophisticated regulation and utilization of genomic information. - **Regulatory DNA and Complexity:** While non-coding DNA was once considered \"junk DNA,\" it is now recognized to play crucial roles, particularly in gene regulation. The expansion of non-coding regions, especially regulatory sequences, is thought to be a key factor in the evolution of increased regulatory complexity in eukaryotes. This regulatory complexity allows for finer control over gene expression in space and time, contributing to the development of complex multicellular life forms. ## Introns: A Major Component of Eukaryotic Gene Structure Introns are non-coding DNA sequences located within eukaryotic genes that are transcribed into precursor mRNA but are subsequently removed by RNA splicing to produce mature mRNA. Introns significantly contribute to the size and complexity of eukaryotic genes, particularly in humans. - **Introns as a Major Determinant of Gene Size in Humans:** Human genes are notably larger than prokaryotic genes, primarily due to the extensive presence of introns. - **Functional Significance of Introns:** Despite being non-coding in the sense that they are not translated into protein, introns are not merely \"junk DNA\" and are increasingly recognized to have diverse functional roles: In summary, the organization of genomes, particularly in eukaryotes, is characterized by a complex interplay between coding and non-coding DNA. The C-value paradox highlights that genome size is not a direct measure of organismal complexity. Instead, the expansion of genomes in complex organisms is largely due to the accumulation of non-coding DNA, which plays critical roles in gene regulation, genome evolution, and the generation of biological diversity. The lower gene density and the presence of introns in eukaryotic genomes reflect a shift from genome compactness in prokaryotes to regulatory and structural complexity in higher organisms. # Conclusion This lecture has provided a comprehensive overview of DNA topology and genome complexity. We have examined the fundamental topological parameters---linking number, twist, and writhe---and their critical role in DNA supercoiling. We discussed the biological significance of supercoiling, contrasting the prevalence of negative supercoiling in most organisms with the adaptive positive supercoiling found in extremophiles. A detailed exploration of topoisomerases, essential enzymes for managing DNA topology, included their classification, mechanisms of action, and their therapeutic relevance as anti-cancer targets. Finally, we addressed the concept of genome size and complexity, elucidating the C-value paradox, the evolutionary trend of decreasing gene density, and the substantial contribution of non-coding introns to the structure of eukaryotic genes, particularly in the human genome. **Key Takeaways:** - DNA topology, defined by linking number, twist, and writhe, is a fundamental determinant of DNA structure and function in vivo. - DNA supercoiling, especially negative supercoiling, is essential for facilitating critical cellular processes such as replication, transcription, and DNA repair. - Topoisomerases are indispensable enzymes that maintain proper DNA topology by altering the linking number and are critical targets for anti-cancer therapeutics. - Genome size does not directly correlate with organismal complexity, as exemplified by the C-value paradox, highlighting the significant role of non-coding DNA in genome organization and regulation. **Further Studies:** To expand on the topics covered in this lecture, future studies could delve into: