Brownian dynamics simulation of protofilament relaxation during rapid freezing

Electron cryo-microscopy (Cryo-EM) is a powerful method for visualizing biological objects with up to near-angstrom resolution. Instead of chemical fixation, the method relies on very rapid freezing to immobilize the sample. Under these conditions, crystalline ice does not have time to form and distort structure. For many practical applications, the rate of cooling is fast enough to consider sample immobilization instantaneous, but in some cases, a more rigorous analysis of structure relaxation during freezing could be essential. This difficult yet important problem has been significantly under-reported in the literature, despite spectacular recent developments in Cryo-EM. Here we use Brownian dynamics modeling to examine theoretically the possible effects of cryo-immobilization on the apparent shapes of biological polymers. The main focus of our study is on tubulin protofilaments. These structures are integral parts of microtubules, which in turn are key elements of the cellular skeleton, essential for intracellular transport, maintenance of cell shape, cell division and migration. We theoretically examine the extent of protofilament relaxation within the freezing time as a function of the cooling rate, the filament’s flexural rigidity, and the effect of cooling on water’s viscosity. Our modeling suggests that practically achievable cooling rates are not rapid enough to capture tubulin protofilaments in conformations that are incompletely relaxed, suggesting that structures seen by cryo-EM are good approximations to physiological shapes. This prediction is confirmed by our analysis of curvatures of tubulin protofilaments, using samples, prepared and visualized with a variety of methods. We find, however, that cryofixation may capture incompletely relaxed shapes of more flexible polymers, and it may affect Cryo-EM-based measurements of their persistence lengths. This analysis will be valuable for understanding of structures of different types of biopolymers, observed with Cryo-EM.

Reviewer #1: In this work, the authors attempted to understand the observed microtubule structures after rapid freezing with Brownian dynamics simulation. it is interesting and meaningful for the communities to evaluate the potential artifacts of Cryo-EM based measurements in (bio)polymers. However, as what the authors said in the discussion, 'tubulin protofilament curvature gradients are unlikely to originate solely as an artifact of rapid freezing.' Hence the manuscript has not convince me the general meanings and importance of the manuscript, and if the simulation is applicable to different kinds of (bio)polymers. The results do not agree with the experimental observation.
We have substantially re-written the manuscript and added new simulations of high-pressure freezing (new Fig. 3) to make the logic of the paper more clear. In fact, the simulation results do not contradict experimental observations. Rather, our simulations predict that cryfixation process, happening at the rates, achievable during high pressure or plunge freezing, should not artificially induce any noticeable curvature gradients in rigid polymers like tubulin protofilaments. Therefore, the tubulin curvature gradients, which we previously described in McIntosh et al JCB 2018, must have another origin. As we hope now is now stated clearly in the paper, the gradients are very likely to be a real structural feature of tubulin protofilaments and may play some physiological role. Our modeling prediction is supported by our new comparative analysis of our previously collected data (Fig. 6). In full agreement with the model, it reveals that the tubulin protofilament curvature gradients in different experimental conditions are almost identical, regardless of the cooling rate and fixation method. This confirms the model conclusions that the freezing conditions commonly used for cryofixation are too slow to generate tubulin curvature gradients. The model, however, does predict that freezing might affect softer filaments, like DNA or collagen.
Besides, the authors discussed a lot of the perspective using the case of DNA, which is based on single data point of Fig. 5b. I am doubt if the supportive is solid or if there are any experimental evidences.
The linear relationships between cooling rate and the dynamic persistence length, presented in former Fig 5b (new Fig 7b), are based on 4 points. Each point is based on analysis of 20 independent simulation repeats.
To strengthen our point about the potential value of our Brownian dynamics approach to analysis of cryofixation effects on other polymers, we have carried out additional simulations (new Fig  7B,C), demonstrating apparent change of persistence length of soft polymers, frozen with a set of cooling rates. This suggests that cryofixation effects are not negligible and they should be taken into account in cryo-EM-based measurements of the flexibility of biological polymers.
Reviewer #2: It is an interesting study to use Brownian dynamics modeling to investigate possible cryo-immobilization effects on the apparent curvatures of tubulin protofilaments from straight to curved shapes. It is found that the cooling rate and the flexural rigidity can influence the curvature gradients. This work provides useful information for the cryo-TEM samples preparation and shape analysis. I suggest that this paper can be accepted after a minor revision by addressing following questions: 1. In Figure 1b, a scale bar should be provided for cryo-TEM images.
Fixed. Thank you. Figure 1d, f, why do the curvature far from tip close to 5-10 deg/dimer. Following my understanding, it should close to zero.

In
The reviewer is correct. In the ideal case, one should expect zero, but we traced only the parts of PFs, which were visually 'curved', hence the lowest "observed" curvature is non-zero.
3. From Figure 3c-f, the authors suppose that cooling rate (106K/s) was not enough to produce a curvature gradient in protofilament tip, and higher cooling rate (106K/s) could lead to the formation of gradient (Figure 4c-f). But they concluded that "tubulin protofilament curvature gradients are unlikely to originate solely as an artifact of rapid freezing" from Figure 6. So how do we distinguish which reason results in the formation of curvature gradient when analyze them? Could the authors provide more explanations?
We have substantially re-written the manuscript and added new simulations of high-pressure freezing (Fig 3, Video S1) to make the logic of the paper more clear. Please, also see our response to a similar point of Reviewer 1. This is a very good point. However, our observations of microtubule structure consistently showed protofilaments flaring out in planes that contained the microtubule axis and separated from their neighbors by ~27 o (Mcintosh et al., 2018). This angular separation is great enough to prevent lateral interactions between bending protofilaments, except very near the base of the PF (where it starts to flare) or through hydrodynamics. If some aspect of our observations is faulty (failing to reveal PFs that stick to one another over some fraction of their length), then PF interaction might be a factor. We have added a note about this to the introduction.

The equations in Manuscript should provide references to support.
We have added more references and to explain and support the equations better.
Reviewer #3: In this study, Ulyanov et. al. used Brownian dynamics modeling to theoretically examine possible cryo-immobilization effects on the apparent shapes of tubulin protofilaments, the key elements of the cellular skeleton. After their theoretically examined the extent of protofilament relaxation within the freezing time, they analyzed the microtubule curvatures and flexibilities, they conclude, that tubulin protofilament curvature gradients are unlikely to originate solely as an artifact of rapid freezing. The study is interesting, however, this referee has following comments Major comments, 1. Line 105; The molecules are 3D objects. Thus the energy between two adjacent tubulin monomers should depended on their 3D angles, in which, other than the bending angle, the tilting angle (perpendicular to the bending angle) and twisting angle (rotation along the tilting axis) should be also considered.
Here we considered a 2D case, which was essentially dictated by planar shape of the protofilament curls in cryo- ET [McIntosh et al JCB 2018, Gudimchuk et al., Nat Commun. 2020. Moreover, in many cases, cryo-EM produces 2D projections of polymers, so our 2D analysis will be applicable. We believe that a full account of 3D would not affect the outcome of the analysis, but it would add some new unknown parameters.

Line 106: How B was defined in the simulation
We explained the meaning of B, and varied it in the range of plausible values.
3. Line 115: How k was defined in the simulation Thank you. We have added a definition of k.
4. Line 138: The temperature decreasing was assumed as the linear function during the cryofixation. It is not accurate assumption. In the early state of cryofixation, the sample surrounded ethane could be evaporated and the ethane gas will surround the sample/grid and reduce the sample cooling speed until the sample temperature is dropped below -88C, the liquified temperature of ethane. Author should discuss how the unified temperature influence on the flexibility of measurement.
We state in the text that the linear decrease of temperature is an approximation, which allows using a notion of some 'constant cooling rate' (lines 154-161): Practically, the steep increase of water viscosity with temperature resulted in complete immobilization at a temperature of about ~ -45 o C, well above the temperature of the liquid ethane in plunge freeze experiments (-165 o C) or liquid nitrogen jets (-196 o C) during high pressure freezing. When the sample enters the liquid ethane in drop freezing experiment or when the sample is cooled with a jet of liquid nitrogen, it is moving fast enough relative to the cryogen that it is continuously exposed to fresh cryogen at this temperature. Therefore, while the sample is cooling from 22 o C to ~ -45 o C (still much warmer than the cryogen), we considered the cooling rate, v, to be approximately constant.
This point is additionally illustrated in Fig. R1 below. These plots show that the linear approximation of the exponential curve works fairly well in the range of temperatures between room temperature and the temperature of the solidification (R 2 =0.9914). Reviewer seems to be confusing the persistence length with another characteristic, 'the bending energy', used to describe DNA flexibility in the paper by Zhang et al., Nat. Comm. 2016(DOI: 10.1038. The values cited (~ 116 to ~160, ~150 and ~50) correspond to the estimates of the bending energies of DNA, expressed in kcal/mol in the paper, rather than the persistence length of DNA. In fact, Zhang et al explicitly stated that "the widely used persistence length of 50nm was used to compute the bending energy". This value of persistence length is similar to what we cited in ref 28.