Biological physics

Spindle Assembly Checkpoint

Leif Matsson’s (LM) primary research in biological physics is focused on DNA replication [B1-B5, B8] and segregation of replicated chromosomes, in particular on the dynamics underlying the spindle assembly checkpoint (SAC) and the higher order folding of DNA, i.e., condensation of chromatin by consensin [B11-B12]. SAC is the mechanism by which a cell decides to inhibit or allow anaphase entry and separation of replicated chromosomes into two identical genomes. For the cell to enter anaphase the kinetochore protein complexes on all sister chromatid (SC) pairs must be first stably attached by microtubules (MT) from the two spindle poles (bi-orientated). Before anaphase entry can take place, in higher eukaryotes all SC pairs must then also line up on the metaphase plate. This indicates that strong spatial correlations between all SC pairs and position dependent forces are at work. However, segregation is well synchronized also in budding yeast cells, indicating that bi-oriented SC pairs are strongly correlated in space also in that case. Except for a general interest in how the living cell works, a deeper understanding of this process in normal cell division would almost certainly also lead to a better insight in cancerous cell division.

The currently asked question in molecular biology in this context is how events at the kinetochore complex, which contains some 80 proteins with different functions, are converted to inhibit the anaphase promoting complex (APC) and its co-activator Cdc20. However, at the same time it is clear that a cell’s decision to allow anaphase entry and divide is taken collectively by all stably attached kinetochores (or by all bioriented SC pairs). Similarly, this decision is collectively inhibited as long as there are unattached kinetochores, i.e., while the number of stably attached kinetochore complexes still increases. This non-equilibrium machinery works like a clock-work that counts the numbers of stably attached kinetochores. From a many-body physical aspect, except for the non-equilibrium dynamical conditions, the dividing cell is not different from any other condensed matter system with strong spatial correlations between the constituent particles. It is a well-known fact in physics that the collective effects of a system cannot be explained by the properties and functions of the individual constituent particle (the stably attached kinetochore complex). As described by LM [B12], the SAC mechanism is controlled by a non-equilibrium collective dynamics of one variable only, i.e., the number of stably attached kinetochores (the order parameter). This is fortunate also because it reduces the number of things that can go wrong and thereby increases the fidelity of the segregation process.

The SAC problem thus boils down to formulating the non-equilibrium collective potential energy of the system. In contrast to inanimate condensed matter systems driven by temperature variations under chemical equilibrium conditions, the dynamics in a dividing cell is driven by the increase in the number of stably attached kinetochores, i.e., by the attachment rate equation, implying that the chemical potential increases, the temperature being constant. It was then demonstrated by LM that the position dependent integrated poleward and antipoleward forces and the in vivo integrated tension in kinetochores and chromatin can be derived from the collective potential energy as functions of the order parameter [B12]. The second derivative of the collective potential energy yields the spectra of both metaphase and anaphase oscillations, which appear to be key to the synchronizations of anaphase entry and of chromosome separation.

It may look as if the last stable kinetochore attachment triggers anaphase entry. However, the relaxation time after each stable attachment, needed for the potential energy to be redistributed and shared equally between all previously stably attached kinetochores and SC pairs, shows that the transition into anaphase is a collective event. As was shown by LM [B12], the collective binding energy for all SC pairs can attain its minimum, which is required for a synchronized and faithful (normal) proteolytic separation of all SC pairs, only after relaxation of the turmoil (redistribution of the collective energy) created by the last stable attachment.

Chromatin condensation

A faithful segregation of replicated chromosomes also requires reorganization and higher order folding of DNA into well compacted chromatids. This condensation process, which proceeds in different steps in the cell cycle, is in prophase mediated by condensin II and in prometaphase and metaphase by condensin I. However, it is also linked to the spindle assembly process and depends on the release of cohesin and a large turnover of histones. Different studies indicate that condensin forms pairwise topological links between different parts of the rod-like chromosomal fiber and that this can take place in a rather irregular manner. As was shown by LM [B12], the compaction of chromatin becomes completed only after bi-orientation of all SC pairs and after a certain finite number of topological links. He also showed that the integrated MT mediated poleward forces, together with the chromatin compaction force induced by condensin I, yield the putative in-vivo tension in kinetochores and centromeric and peri-centromeric chromatin as functions of the order parameter.

Relationship to single DNA molecule studies

Stretching of a single DNA molecule in vitro by an external force has provided a wealth of information about DNA. However, in a living cell the chromatin becomes stretched by opposite microtubule mediated pulling forces which act pairwise on the two sister kinetochores, generating tension in kinetochores and centromeric and pericentromeric chromatin. As was demonstrated by LM [B11, B12], the integrated (collective) force acting on kinetochores in a living cell can be related to the force-extension formula observed in laboratory. This relationship can in turn be employed to translate different results from single molecule force spectroscopy studies into the non-equilibrium conditions that prevail in a living cell. For instance, the putative in vivo tension in kinetochores and centromeric and peri-centromeric chromatin derived from the collective model could be validated indirectly [B12]. Hopefully, alternative experimental methods can now be developed for more direct in vivo assessments.

Non-equilibrium statistical mechanics

Self-organization in non-equilibrium inanimate and biological systems can be described provided that these systems are void of strong spatial correlations between their respective constituent particles. Systems restricted by strong spatial correlations can be described too if the number of constituent particles is constant or fluctuates about a constant number. The long standing problem in physics has been to model systems with an increasing number of constituent particles (or molecules) that become strongly correlated in space. A dividing cell exhibits exactly this type of non-equilibrium dynamics that inhibits the APC in prometaphase and metaphase before the SAC control. To provide a solution to the SAC problem, LM thus first had to solve the non-equilibrium statistical physical problem [B12].

Spontaneous symmetry breakdown

As was described by LM [B12], the rate equation for attachment of microtubules to kinetochores and the strong spatial correlations between stably attached kinetochores, combined with the initial boundary constraints, uniquely define the non-equilibrium dynamics underlying the SAC mechanism. This dynamics happens to be symmetric with respect to change of sign of the order parameter ϕ, which also represents the probability density for stable attachments. To make this probability density positive definite (a physical requirement), a constant number N must be added, ϕ + N = ϕ’ (or ϕ → −N + ϕ),  implying that the actual symmetry becomes broken in living cells too. This operation allows the collective potential energy to also attain minimum in the absence of stable attachments (a second physical requirement). Compare a wine bottle with a hill shaped bottom on which a small ball is placed symmetrically on the top. The ball then spontaneously falls down to an asymmetric position at the circular lower level of the bottom. This is also what happens in a dividing cell once all kinetochores have been stably attached and the system has relaxed after redistribution of the collective potential energy [B12].

Current projects

  1. DNA replication.
  2. Spindle checkpoint and segregation.
  3. Realtionship between DNA dynamics in a living cell and force-extensionformula observed on a single DNA molecule in the laboratory.
  4. Non-equilibrium statistical mechanics.
  5. Motor proteins.
  6. Ion channels.

Fields of interest

Spindle checkpoint

At the spindle checkpoint the physical condensed matter properties of the system of sister-chromatids (SCs) stably attached to the spindle become crucial. The spatial correlations, without which the SCs would not be able to assemble at the metaphase plate [11] (Fig. 1), then become directly observable. Moreover, to reach the threshold for metaphase-anaphase transition, the number of SCs stably attached by microtubules from the two spindle-poles must increase, creating an intriguing problem that has been hitherto overlooked in earlier spindle checkpoint studies.

Spindle checkpoint
Fig. 1

Physics can only handle condensed spatially correlated systems under conditions of chemical equilibrium, allowing in principle only fluctuations in the number of particles. Thus the spindle checkpoint problem has been confined in an unexplored knowledge gap between molecular biology and equilibrium condensed matter physics, like a modern “quinta essentia” type of question.

Non-equilibrium statistic

To solve this problem I had to obtain a non-equilibrium statistic, which could hopefully also serve as a platform for development of a more general non-equilibrium statistical physics. As a first step in this direction, and to explain the molecular physics machinery underlying the spindle checkpoint, I have demonstrated that there exists a nontrivial dynamics that complies both with the actual spatial correlations and chemical non-equilibrium conditions [1-5, 7, 8, 11]. The fact that this non-equilibrium model is formally identical to its corresponding equilibrium model, could lead to assumptions that there is no difference. However, there are. For instance, whereas in the equilibrium model the order parameter is usually temperature dependent (thermotropic system), in the non-equilibrium version the order parameter is reactant concentration dependent (lyotropic system). Similarly, the coefficients that define the non-equilibrium model are functions of the initial reactant concentrations. In addition, the lyotropic order parameter represents a coarse-grained mapping between the reactive molecular level and the system’s macroscopic level, at which important events, such as the metaphase-anaphase transition, take place and can be observed.

G1 restriction point and S phase entry

The switch from mitogen-dependent to mitogen-independent phosphorylation of Rb protein at the G1 restriction point, the so-called R point [1, 2], and the S phase entry, i.e. the initiation of DNA replication, are other examples of checkpoints that I have studied [3-5, 8].

What can force-extension studies tell about native DNA dynamics?

Optical tweezers has been used to obtain a wealth of information about single biological molecules, such as DNA, and how they work at the molecular level. It is therefore in place to ask how information about DNA in such laboratory studies can provide knowledge about DNA and it’s dynamics in living cells. I have derived a relationship between the DNA dynamics in a living cell and the force-extension formula obtained by stretching a single DNA molecule in the laboratory [11]. Hopefully this relationship can be used to interpret results from such laboratory studies and translate them to native DNA dynamics. In any case the relationship shows that native DNA dynamics cannot be directly observed in single molecule stretch experiments. For instance, tension in DNA of a living cell is not generated by an external force to mention just one difference.

Ion channels and motor proteins

Other related subjects of interest to me are ion channels [6, 9] and motor proteins [10].