On proper linearization, construction and analysis of the Boyle–van’t Hoff plots and correct calculation of the osmotically inactive volume

Cryobiology 62 (2011) 232–241

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On proper linearization, construction and analysis of the Boyle–van’t Hoff plots and correct calculation of the osmotically inactive volume q Igor I. Katkov ⇑ Stem Cell Center, Sanford-Burnham Medical Reserch Institute, La Jolla, CA 92037, USA CELLTRONIX, San Diego, CA 92126, USA

a r t i c l e

i n f o

Article history: Received 28 September 2010 Accepted 23 February 2011 Available online 2 March 2011 Keywords: Cryopreservation Osmotic modeling Boyle–van’t Hoff Osmotically inactive volume Curve fitting Least square Linearization

a b s t r a c t The Boyle–van’t Hoff (BVH) law of physics has been widely used in cryobiology for calculation of the key osmotic parameters of cells and optimization of cryo-protocols. The proper use of linearization of the Boyle–vant’Hoff relationship for the osmotically inactive volume (vb) has been discussed in a rigorous way in (Katkov, Cryobiology, 2008, 57:142–149). Nevertheless, scientists in the field have been continuing to use inappropriate methods of linearization (and curve fitting) of the BVH data, plotting the BVH line and calculation of vb. Here, we discuss the sources of incorrect linearization of the BVH relationship using concrete examples of recent publications, analyze the properties of the correct BVH line (which is unique for a given vb), provide appropriate statistical formulas for calculation of vb from the experimental data, and propose simplistic instructions (standard operation procedure, SOP) for proper normalization of the data, appropriate linearization and construction of the BVH plots, and correct calculation of vb. The possible sources of non-linear behavior or poor fit of the data to the proper BVH line such as active water and/or solute transports, which can result in large discrepancy between the hyperosmotic and hypoosmotic parts of the BVH plot, are also discussed. Ó 2011 Elsevier Inc. All rights reserved.

Introduction The Boyle–van’t Hoff (BVH) relationship is widely used in osmotic cryobiology and the accurate method of linearization and correct use of that equation is essential for correct calculation of the key osmotic parameters of the cells, namely the fraction of osmotically inactive volume (vb), the water hydraulic conductivity (Lp), and the permeability of the permeable cryoprotective agent (PCPA), determination of osmotic tolerance and for optimization of slow (equilibrium) freezing protocols. The proper use of linearization (plotting) for the Boyle–vant’ Hoff-law in osmotic models has been discussed in our paper published in 2008 [3]. Nevertheless, scientists in the field have been continuing to use inappropriate methods. It is our view that we should re-emphasize these aspects in a more explicit and direct, yet simpler manner. Here, we analyze the BVH equation, setup conditions, which the proper BVH line satisfies, and give the formula for CORRECT calculation of vb. We then analyze the major source if incorrect calculations of vb that we have found in the overwhelming majority of cryobiological literature – the use of ‘‘standard’’ statistical formulas/functions/software for calculation of the intercept, which operates under the assumption that the slope and the intercept of the q

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fitting line are two independent parameters. We then compare the results of proper and incorrect linearization on concrete experimental data. We also briefly discuss the two major ways of normalization of the volumetric data as well as possible sources of deviation of the experimental data from the BVH line. In the latter case, we consider the experimental data plotted separately for hyper- and hypo-osmotic parts of the experiment and hypothesize the possible sources of large discrepancy between two lines (e.g., the active water transport). We then summarize our analysis in a brief Conclusion paragraph and provide a set of instructions (in the form of a standard operation procedure, SOP) which we think should be followed for proper linearization of the BVH plots and calculation of the correct values of vb. Finally, the text will be supplied with a set of Mathematical appendices with rigorous derivations and proofs of the formulas used and statements made in the major text. As an example, we consider here a very recent and otherwise excellent publication on the osmotic response of oocytes by Wang and colleagues [13]. We have selected for illustration the literature that Dr. Wang provided to us in personal communications [2,10,12,14]; however, and unfortunately, the same considerations can be applied to the overwhelming majority of published cryobiological literature (see also our companion Letter to the Editor). We must note though our gratitude to Dr. Wang for providing a template of the actual data shown on the plot on Fig. 2B [13], which has helped us to make precise calculations and comparisons. We

I.I. Katkov / Cryobiology 62 (2011) 232–241

are also grateful to Dr. Lonergan for clarification of the Authors’ position: it has definitely helped us clearly realize the necessity of the analysis and the SOP provided in this paper.

Boyle–van’t Hoff plots The core formula of the Boyle–van’t Hoff (BVH) relationship is the concept of the constant amount of the intracellular impermeable solute Nim, which can be written as follows:

Nim ¼ MðV  v b Þ ¼ Miso ðV iso  v b Þ ¼ const

ð1Þ

where V is the total cell volume at a given intracellular osmolality M, vb is the osmotically inactive cell volume, and subscript ‘‘iso’’ refers to the isosmotic values. The formula is used upon equilibration in an impermeable ideal solute so we can assume that M = Mf, the extracellular osmolality, assuming no active water transport is involved (see also below). We will use M here keeping in mind that it is now the extracellular osmolality. Dividing (1) by MisoViso, and solving the equation for cell volume, we can re-write Eq. (2) in terms of normalized inversed osmolality and total cell volume as follows:

v BVH ¼ ð1  v b Þx þ v b

ð2Þ

where x  Miso/M is the normalized inverse osmolality, vBVH  V/Viso is the normalized cell volume at given M and vb  Vb/Viso is the normalized inactive osmotic volume. Subscript ‘‘BVH’’ emphasizes that this is the ‘‘true’’ and only BVH line at a given vb (see below). The BVH formula (2) is the basis for determination of the portion of osmotically inactive volume vb based on experimental results; note that all the parameters must be normalized to the iso-tonic values.1

1 Note that there is a lot of confusion and inconsistency on this matter in the literature. We have selected several papers that were provided by Dr. Wang [2,10,12,14] and have used them to illustrate and analyze the problem. In [13], the values of inverse osmolalities are in fact normalized but the axis legend in Fig. 1 is labeled incorrectly as ‘‘1/Osm’’ (it should be unitless Miso/M). A similar label of the xaxis is seen in Fig. 1 in Salinos-Flores et al. [10]. However, it is not clear what osmolality value was considered as isosmotic because the saltwater Pacific oyster may have an iso-point which is much higher than the mammalian cell range (around 300 mOsm except for kidneys), so it is difficult to conclude whether ‘‘1’’ on the x-axis referred to 1 Osm1 or to the iso-point (1; 1): the normalized volume at that point was close to 1. In paper by Zhang et al. [14], it is even more confusing: not only is it not clear what values (normalized or non-normalized) were used in Fig. 1 (labeled as ‘‘1/osmolality’’), the BVH equation on p. 290 lacks biophysical meaning because of the unit discrepancy, which underlines the necessity of using normalized parameters. In [12], the x-axis is not normalized, and this is the only paper from those examined (in contrast to [10,13,14]), in which it is clearly seen that units Osm1 were used because the iso-point is at approximately (3.3; 1); however, marking the x-axis as ‘‘1/Osmol’’ could be confused with the number of osmoles (moles of osmotically active particles) in solute, which is not the unit of osmolal concentration Osm/kg H2O. Thus, it is not a surprise that such ‘‘diversity’’ in using and labeling of the inverse osmolality has led to the confusion in [13]: a ‘‘hybrid’’ of the correctly normalized values that were plotted on the erroneously labeled x-axis. An example where this issue was dealt with properly, i.e., the inverse osmolality was normalized, was the article by Gao and colleagues [2]: the x-axis on Fig. 3 was correctly labeled as ‘‘325 mOsm/kg/M’’, i.e., it was the same unitless normalized inverse osmolality as we use in formula (2). However, the sum of the slope (0.87) and the intercept (0.21) in that fitting equation deviates from 1 so we can conclude that the 2 independent parameter fitting mechanism was used, and thus, that equation given in the figure legend of Fig. 3 on p.46 and in the text on p. 44 in [2] does not describe a BVH plot given by Eq. (2) above and depicted in Fig. 1A (see detailed explanation in the following text). Note that in our consideration of isothermal conditions and absence of turgor pressure across the cell membrane, the terms ‘‘iso-tonic, hypertonic or hypotonic’’ can equally apply to isosmolal, hyper-osmolal or hypo-osmolal conditions respectively so we will use them interchangeably throughout the text. However, one must remember that it is the reverse osmolality x f-la (2) operates with so the normalization must be done to the osmolality (Osm/L) not to osmotic pressure (atm).

233

The improper use of non-normalized units in the right hand side of the BVH equation (2) is only one of the sources of errors and confusion in linearization and construction of BVH plots and the determination of vb, while the others will be discussed below. Note that for clarity and according to the mathematical tradition, we used CAPITAL letters for absolute values and small letters for the normalized unitless values, in contrast to [13] (Footnote 1). As we can now easily see from Eq. (2), the BVH plot must obey the following rules: (A) BOTH the slope (a) and the intercept (b) of the proper BVH line vBVH = aBVH x + bBVH are the functions of the osmotically inactive volume vb:

aBVH  1  v b

ð3Þ

bBVH  v b

ð4Þ

(B) The sum of the slope and the intercept, as it can clearly be seen by combining (3) and (4), must be exactly equal to 1 at any vb:

aBVH þ bBVH  1

ð5Þ

(C) The line must go exactly through the iso-point (1; 1); this conclusion comes from (12) and definition of the normalized values: viso  1 and xiso  1. (D) There is only one line that obeys the BVH law at given vb. The family of BVH plots at different vb’s is given in Fig. 1. (E) Thus, any other line that does not obey conditions (3)–(5) is not a BVH plot and can not be considered as far as the BVH law is concerned and can NOT be used to compute vb.

Fig. 1. BVH lines at different osmotically inactive volume vb. Note that all lines converge to the iso-point (1; 1). The sum of the slope (1  vb) and intercept (vb) of all lines equals 1 (formula (5) in the text).

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2

A

B

y = 0.5795x + 0.2608

1.8

2

R = 0.9348 1.6 1.4 2.15; 1.33 1.51; 1.26 1.70; 1.23

V/Vi

1.2

1.31; 1.07

1

1.00; 1.00

0.8 0.67; 0.73

0.6 0.43; 0.51

0.4

0.35; 0.41 0.29; 0.35 0.22; 0.30

0.2 0 0

0.5

1

1.5

2

2.5

M i /M

C

Fig. 2. Correct and incorrect linearization of equilibrium BVH data. (Panel A) Original data on MII Bovine oocytes in Fig. 2B [13] provided by Dr. Wang. This is NOT a correct BVH line (cf. with Fig. 1). The line does not go through the iso-point and the sum of intercept and slope (0.8404) deviates from exact 1. The actual data are given on the plot as a check point for Excel programs. (Panel B) Correct BVH line (Eq. (2) in the text) plotted using the data provided by Wang (Panel A) and using correct formula for calculation of vb (7) is in red solid line, the line that is plotted using the assumption of independent intercept and slope (8) (as it is done by Wang et al. [13]) is in black dotted. Blue circles P P are the actual experimental data. (Panel C) The unique case where statistical regression line and BVH plot coincide Ni¼1 xi ¼ Ni¼1 v i ¼ N: see explanation in the text and Appendix 2. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article.)

An example of incorrect linearization and plotting of BVH data and calculation of vb and the major source of wrong plotting. correct formula for calculation of vb However, despite the previous publications by Pegg and colleagues [7] and Armitage [1], and regardless of my recent effort

to thoroughly consider this case and offer the only proper calculations [3], cryobiologists and practical scientists continue to use plots which do not obey the rules described above, and thus, those lines can not be referred to as BVH plots, and vb’s in all such papers were calculated incorrectly. As an example, Fig. 2A depicts the original numerical data on osmotic reaction of MII bovine oocytes

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shown in Fig. 2B in [13]. These data, kindly provided by Dr. Wang, are averaged for each osmolality. The linearization regression line that goes through the average points used in this work is:

v Wang ¼ 0:58x þ 0:26

ð6Þ

Clearly, it is not a proper BVH plot, the sum of the slope and the intercept equals 0.84, the deviation error from 1 is 16%. If we analyze other publications, we can conclude that practically all have used the same and, thus, mathematically wrong approach. As a result, the plotted lines were not BVH plots and, thus, vb’s were calculated incorrectly. Let us now consider the major source of errors in linearization and construction of BVH plots, and as the result, incorrect calculation of vb. As we can see from Eq. (2) the BVH plot depends only on one parameter vb so the LSM method must apply to that sole parameter and the sum of the squared residuals of Eq. (2) P fv i  ½ð1  v b Þxi þ v b g2 must be differentiated in respect to vb only as carried out in formulas 58 and 59 in [3] and in Appendix 1. The osmotically inactive volume (and, thus, the intercept of the BVH line) can be then determined as follows:

vb ¼

PN

i¼1 ½ðxi  1Þðxi  PN 2 i¼1 ðxi  1Þ

v i Þ

ð7Þ

where subscript i is the ith osmolality tested (i = 1, . . ., N), xi the ith normalized inverse osmolality, and v i is the average normalized cell volume at a given ith osmolality (see more details on proper normalization in the section ‘‘An example of incorrect linearization and plotting of BVH data and calculation of vb and the major source of wrong plotting. correct formula for calculation of vb’’). Derivation of this formula is given in Appendix 1. The major problem rises from the fact that scientists use statistical packages for linearization. All the statistical packages and built-in functions (plotting, linear trend lines, Excel functions SLOPE and INTERCEPT, etc.), in contrast, presume that the slope a and the intercept b are statistically independent from each other. In this scenario, the least squares method (LSM) is applied in a way in which the P sum of the squared deviations (residuals) ½v i  ðaxi þ bÞ2 is differentiated in respect to a and b, and a system of 2 equations is solved (given by formulas 16 and 17 in our paper [3]) and in Appendix 2). In the case of the data relating to [13], vb determined from (2) is 0.49 so the proper BVH line would be:

v BVH ¼ 0:51x þ 0:49

ð8Þ

Both lines (6) and (8) and the original experimental data from [13] are plotted together in Fig. 2B. The slopes of the two curves (6) and (8) are practically the same (0.58 vs. 0.59). The osmotically inactive volume of the BVH line (0.41), however, almost doubles the value obtained using the 2-independent parameter statistics used by Wang et al. in [13] (0.26). The original line in [13] provides a better approximation of the experimental data: the sum of the PN 2 square deviations (residuals) of the theoretical values i¼1 d which corresponds to the lines from the respective experimental values is 0.0965 vs. 0.3509, i.e., the line in [13] and in f-la (6)) gives 3.6-times better approximation than the formula (8); however, it does not make it a valid BVH line! We can find an infinite number of other curves which would approximate the experimental data even better; moreover, a polynomial curve can be drawn exactly through all of the averaged points so there will be no residuals at all. But neither those curves, nor the straight lines described in Fig. 2 in [13] and (presumably) in all the other examples are proper BVH lines because they do not obey the BVH law and its consequences summarized by formulas (3)–(5). It may be appropriate to cite a saying ‘‘Once you have started with Copernicus, don’t finish with Ptolemy’’; in other words, if one claims the BVH relationship of

an ideal osmometer at the beginning, one should use only the BVH curve (Fig. 1) at the end! One can argue (as my anonymous reviewers on this paper and the previous paper did) that ‘‘in the real World’’, there is no exact BVH inside the cells, they are not ideal osmometers, and solutes also behave non-ideally at high extracellular concentrations and in an intracellular milieu so other models such as [9] have been proposed. But this is a completely different story; that point is very valid but we are discussing the mathematical correctness of the proper use of statistics (7) for calculation of the proper BVH equation (2). The other, ‘‘non-BVH’’ cases, while biophysically valid, are out of the scope of this paper: we ‘‘do not mix oranges and apples’’. As another example of this approach, see our critique and biophysical inapplicability of Kedem–Katchalsky formalism [5] in [4]. For the Math lover, comparison of formulas for the statistical two independent parameters (a and b) and BVH vs. one parameter to fit (vb), a least squares method is given in Appendix 3. Note that there is only one peculiar case when two lines completely coincide, P P namely when both Ni¼1 xi AND Ni¼1 v i equal exactly N, the number points (osmolalities) on the line. For the sake of completeness, this case (using synthetic data for N = 5) is depicted in Fig. 2C. Proper normalization of volume The other case, as we have already mentioned in the (Footnote 1), is the problem of proper normalization of the data. It is easy for reverse osmolality, it must be normalized and expressed as a unitless variable xi  Miso/Mi for each ith osmolality Mi, and it is straightforward. The situation with the normalized volume is more complicated because several ‘‘successful’’ (a statistical term proposed by my anonymous reviewer) volumetric measurements are usually made at a given ith-osmolality. There are two basic approaches to the normalization, which depend largely on a given situation so as the final decision as to what iso-tonic value(s) the volume data should be normalized to. The first way which can be called ‘standard’, and the most widely used method in the literature is that the iso-tonic value is averaged throughout all experiments:

V iso ¼

Siso X V iso;j Siso j¼1

ð9Þ

where subscript j represent the jth repetition of the measurement in an iso-tonic solution, Viso,j is the value of volume (absolute, not normalized!), and Siso is the amount of ‘‘successful’’ iso-measurements. All data are then normalized to this average iso-volume and the average of the normalized values for volume points at different osmolalities in formula (7) can be now found using the following:

PSi

vi ¼

V i;j j¼1 V iso

Ni

PSi

V i;j j¼1 Ni iso V iso;j j¼1 N iso

¼ PS

ð10Þ

where subscript Vi,j is the cell volume at the jth replicate of measurement at ith osmolality and Ni is the number of valid replicates (‘‘successful measurements’’) at that osmolality. It is easy to see that the average normalized iso-tonic value equates to 1:

PSiso

v iso ¼

V iso;j j¼1 V iso

Niso

PSiso

V iso;j j¼1 Siso iso V iso;j j¼1 N iso

¼ PS

1

ð11Þ

In this case, the iso-point (1; 1) has error bars because the values V of the normalized volumes Viso;j are in general not equal to 1, only iso the average value is. Despite the formula (10) appearing to be ‘‘intimidating’’ this is exactly what the majority of the researchers do regularly. It is a

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perfectly legitimate method if one considers that the ‘‘true’’ value of Viso was not known before the experiments but calculated from a formula (9) and can be used in all further experiments assuming that all the other aniso-osmotic experiments have the same conditions (for example, there is no shift in Coulter or microscopic image software setting or calibration beads are used ALL the time). There are however, situations when the normalization of anisotonic volumes to one average iso-value V iso can produce a substantial error. Let us consider for example that several replicates (either of the whole set of osmolalities, or several ones) are made at different days and systematic shifts in a Coulter counter, or image microscopy, or other volumetric devise used settings occur. That is not an uncommon scenario despite claims to the contrary in many papers, we suggest that calibration beads are not used at the beginning of every cycle or in every anisotonic solution. In this case, the absolute values of aniso-volumes may shift, and as a result, substantially skew the BVH-line and values of vb if those aniso-volumes are normalized to the previously known V iso . In this case, we recommend the normalization of each value of anisoosmotic measurement to each own iso-volume obtained for this particular replicate j. In this scenario, the formula for calculation of the average normalized volume at the given osmolality can be transferred to the following:

PS i

vi ¼

V i;j j¼1 V iso;j

ð12Þ

Ni

where Viso,j is the iso-osmotic volume obtained at the particular jth replicate. And again, as in formula (11), the average of the iso-tonic normalized volumes equates to a unit:

PSiso

v iso ¼

V iso;j j¼1 V iso;j

Niso

¼

Niso 1 Niso

ð13Þ

In this case, however, the iso-point (1; 1) has NO error bars because each iso-volume will be normalized to each own value Viso,j so that it will always equal to 1 at the iso-point at any jth replicate/ experiment. The formulas (10) and (12) give different values of the average PSi V i;j PSi V i;j – j¼1 V (unless all Viso,j are normalized volumes because j¼1 V iso

iso;j

equal), and as a result, different values of vb and, in general, different resulting BVH lines (though those two lines might accidentally be identical even if all Viso,j are not equal). Actual experiments by the author on a species of animal sperm are given in Fig. 3, the actual raw absolute volumes are given in Fig. 3A while Fig. 3B represents the first method of normalization (all data are normalized to the average = 54.4 lm3) while Fig. 3C represents the second method (the values are normalized to the iso-value at the same day run of a Coulter counter and the same set of donors). The values of the osmotically inactive volume calculated using formula (7) differ substantially (vb = 61.0% for Method 1 vs. 67.1% for Method 2, p < 0.05). Because different donors were used in each repetition, we use the second way of normalization (data will be published elsewhere). It looks like the same approach was used by Gao in [2] for human hematopoietic stem cells. Note, however, that both methods become identical if only one measurement of the iso-tonic volume is performed, the normalized iso-tonic volume will also be equal to 1 with no error bars. In regard to paper [13], Fig. 2A and B has error bars at the iso-point, which means that the first method (A) was used. At the same time, in Materials and Methods it is stated (though not clearly) that each oocyte was normalized to its own iso-tonic volume, so method (A) may in fact may not have been used. Regardless, which of the methods is used, together with conditions (5)–(7), the equality:

v iso  1

ð14Þ

is a necessary criterion to ensure that the normalization operations in Excel or other calculation application have been performed appropriately. For the sake of complicity, one may argue that ‘‘strictly speaking’’, the exact value of osmolality had not been known a priori and has also been obtained from experimental measurements, thus, the value of reverse osmolality (X-axis) should also have error bars. In reality though, researchers use the assumption that the independent variable is known and x has no error bars. However, if one wanted to introduce this further complication, the simplest way would be to use the average values of the variable xi in formula (28). And again, as in formula, (14) xiso  1. Analysis and possible causes of deviation from the BVH line Once the proper BVH line is plotted, it is useful to analyze how the experimental data fit the BVH curve. For example, taking into account the considerations which the Wang data described above have led us to, there is still an unease with the fact that the ‘‘independent a and b’’ equation such as Wang and colleagues used may better describe the experimental set, and this usually occurs as soon as two parameters instead of one are used in fitting the line. Moreover, this regression line is specifically designed in a way such that the sum of the residuals is always equal to zero, i.e., PN i¼1 ½v i  ðaxi þ bÞ ¼ 0 (see Appendix 4). Thus, this statistical line always goes between the experimental points. In contrast, for example, in the case of the correct BVH line for the data in [13] PN i¼1 fv i  ½ð1  v b Þxi þ v b g ¼ 1:5923. i.e., the experimental data lie largely above the true BVH line. NOTE, because of the linear relationships described in formulas, (6) and (8) that the linear correlation coefficients between the two theoretical lines (6) and (8) and the experimental data for the normalized volumes are always EQUAL to each other since they equal the correlation coefficient between the osmolalities and the average experimental values of the volume. In the Wang case, for all 3 correlation pairs R2 = 0.9348 (see Fig. 2A). This emphasizes that R2 describes not the ‘‘fitness’’ of the experimental data to the given curve (a common misconception) but the linearity of the data only. Even though, as we pointed above, the sum of the squared residual for the regression line in [13] (6) is 3.6 times smaller than for the correct BVH line (8), one shall not fall into the ‘‘fitting trap’’ as many investigators often do. An empirical line may be better at describing the current experimental results but if it does not follow the proper physical law, it is useless, or at best it can be called a ‘‘engineering’’ or ‘‘empirical’’ formula. And it cannot bear the name of the law it was initially designed to describe. Moreover, the statistical regression line vWang = ax + b, which often lies nicely between all experimental points, may actually mask very important cryobiological consequences (see below). At the same time, one should be aware if the experimental data deviate from the true BVH line vBVH = (1  vb)x + vb and the value of vb found by formula (7) is very different from the value of the intercept found by fitting two independent a and b, as it is in the case of the data by Wang et al. [13] (vb = 41%, b = 21%). A possible mathematical explanation comes from the fact that the hypotonic part of the experimental data provided by Dr. Wang in regard to the ideal osmometer is quite different from the hyperosmotic data. The panels A and B of Fig. 4 depict two of those subsets separately. As we can see, the statistical regression curves (calculated in presumption of independent slope and intercept) are now quite close to the BVH curves and almost coincide in both cases: 7.4% vs. 10.6% for hypertonic range, and 77% vs. 68% for hypotonicity. However, those lines are strikingly different for the hypertonic and hypotonic parts: the hypotonic lines are now flattened and shifted up so the

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B

A

C

Fig. 3. Two methods of normalization of volume in case of repetitions. (Panel A) The ‘‘raw experimental data’’: the x-axis is non-normalized osmolality, y-axis is the total cell volume in lm3. The experimental results are filled markers; the pink empty circles are the average of three experiments ± STD. The average iso-tonic volume V iso is marked by the larger filled red circle. Note that ‘‘Osm’’ is in fact osmolality Osm/kg H2O. (Panel B) Cell volumes are normalized to V iso . The results show that they do NOT equal exactly 1 at the iso-point and the average of the iso-point has error bars. (Panel C) Cell volumes are normalized to each own experimental Viso,j and as a result, ALL normalized volumes are merged at the iso-point (1; 1), STDV  0, and the average of the iso-point has no error bars. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

intercept of the regression line b is 10.5-times higher and vb is 6.4times higher than for the hyperosmotic range. Clearly, the oocytes did not behave as ideal osmometers as far as the whole range of osmolalities was concerned. One of the biophysical explanations for such increased values of vb at hypotonic conditions could be the osmotic regulation by active water transport known as the ‘‘cell volume decrease’’, which

was observed, for example, for granulocytes by Toupin et al. [11], in early mouse embryos by Pogorelov and colleagues [8], and recently reported and analyzed for the COS-7 cell line by Peckys and Mazur [6]. In this scenario, the apparent underestimated hypotonic cell volume which might be observed in relatively long experiments would flatten the hypo-osmotic part of the BVH line towards a lower slope and, thus, a higher intercept.

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B

A

Fig. 4. BVH for hypo vs. hyper-osmolality ranges. The Wang data plotted on the whole range of osmolalities as depicted in Fig. 2B. (Panel A) The lines are plotted for the hyperosmotic part of the Wang data. Note the low values of vb and b. (Panel B) The lines are plotted for the hypo-osmotic part of the Wang data. Note the much higher values of vb and b than for the hyperosmotic part depicted in Panel B. The cells do NOT behave as ideal osmometers in the whole range of data: e.g., the ‘‘volume decrease’’ due to active water anti-transport may play a role in hypo-osmotic solutions [6,8,11].

One may argue that had the original data been analyzed using the commonly used method of linearization, the values of the intercept b for hypo- and hyper-ranges would also have shown a 10-fold difference for the regression line (b = 7.4% vs. 77%). In our opinion, this did not happen, as in many other cases, exactly because the regression line on Fig. 2A ‘‘fitted nicely’’ the experimental data and did not concern the investigators so they missed what some may call a ‘‘curvature’’ of the regression line, which in fact may well be just a superposition of two BVH straight lines at hypo and hyper-ranges. On the other hand, had they, as many others previously, used the correctly calculated value of vb given by formula (7) and plotted the correct BVH line described by formula (8), we are sure that a similar situation with substantial deviation of the experimental data from the BVH line would have risen a red flag for many researchers. This illustrates the power of proper normalization and correct use of mathematical formula and statistics not only for the sake of mathematical purity, but for strictly practical purposes (e.g., checking the active water transport, particularly, for hypo-osmotic values). In general, we may consider the following criterion for the level of deviation from the BVH line in regard to the ideal osmometer considered over the whole range of the data:

PN  ðv v VBH Þ i¼1 i Abs N Absðdi Þ a ¼ PN v ðv i ÞN i¼1 i N P  N Abs i¼1 fv i  ½ð1  v b Þx þ v b g ¼ P 5% PN i¼1 v i

PN experimental volumes i¼1 v i ¼ 8:188 so the a-criterion would give 19%, which would definitely raise a red flag. Note that for the regression line, this criterion is not applicable because a is equal to zero (Appendix 4) and the line always ‘‘fits nicely’’ between the data points, so not only has this approach given the wrong values of vb, but this important biophysical information may have been overlooked in hundreds of papers where the regression line calculations (Appendix 2) were used instead of formula (7). Overall conclusion: the correct BVH plot In conclusion, if the normalization is done properly (by one of the two ways described above) and formula (7) is used for calculation of the osmotically inactive volume, the BVH line will have an intercept which is equal to vb, the slope will be equal to 1  vb, and the line will go exactly through the iso-point (1; 1) as shown in Fig. 1. The experimental average of the normalized iso-volume is also equal to the exact unit so the BVH line and experimental value both cross (1; 1) as shown in Fig. 2B. In contrast, the use of the statistical functions which operate with two independent parameters, such as in Excel and other applications, does NOT give correct values of vb and the slope, and the regression line do NOT cross the point (1; 1) with one exception depicted in Fig. 2C where two lines become identical. Practical recommendations (standard operating procedure) for calculating the correct vb and building the proper BVH plot

ð15Þ

where subscript ðv i ÞN is the volume that is now averaged over all the experimental data of all N osmolalities together, which of course is different from v i , the average of replicates at the given ith osmolality. P10 For the data of [13], the sums of residuals i¼1 ðv i  v VBH Þ for N = 10 osmolalities would be = 1.5923, the sum of the

In conclusion, we feel that it would be useful to outline a standard operating procedure that we follow for finding vb and building the BVH plot: (i) Normalize all data to x  Miso/M and v  V/Viso unitless variables. Remember: as soon as you have the simple BVH equation (2), the unit 1/M (often erroneously referred as the reversed number of osmotically active mole of particles ‘‘1/ Osm’’) is useless!

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I.I. Katkov / Cryobiology 62 (2011) 232–241

(ii) This normalization can be done be either normalizing to (1) the average iso-volume V iso or (2) its own iso-volume at the given replicates Viso,j when it is possible and preferable (see discussion in the section ‘‘An example of incorrect linearization and plotting of BVH data and calculation of vb and the major source of wrong plotting. correct formula for calculation of vb’’). Checkpoint 1: in both cases, the average of the values at iso-tonicity must be equal to 1; however, the isopoint will have errors bars in the former case and no error bars in the latter case. (iii) Plot the BVH line (2) with the CORRECTLY calculated vb using formula (7) for the intercept and a  1  vb for the slope. Checkpoint 2: the line must cross the experimental data at iso-point (1; 1) (Fig. 2B). Checkpoint 3: for the first time, use experimental data depicted in Fig. 2A, for which vb = 0.4062 and the correct line is depicted in Fig. 2B. We would be happy to provide an Excel template and input instructions for calculations. At the same time, we would highly discourage using the simulation or numeration software or similar imbedded LSM functions: the formula (7) is sufficient, simple and precise. (iv) Calculate the square of the correlation coefficient (R2) between the BVH plot and the experimental data. (v) Calculate the a-criterion of deviation between the BVH plot and the experimental data (formula (15)). If a P 5%, then (vi) Check the hyper- vs. hypotonic range: if vb’s for the two osmotic ranges are substantially different from each other, the cells do NOT behave as ideal osmometers in the whole range no matter how high the value of R2 is. One possible explanation (active water counter-transport) is described above, while other explanations may be true such as the presence of subpopulations and different developmental stage of the cell. (vii) Check (when possible) vb’s at different temperatures: a large difference in the vb value (at low individual variations among the cells at given T) may indicate issues with the osmoregulation as well as other factors such as an increase in intracellular viscosity and binding a part of the osmotically active water at one T to macromolecules at a different T, etc. (viii) We would be happy to provide Excel templates for calculations and input instructions upon request. Conflict of interest None declared Acknowledgments We are grateful to Dr. X. Wang for providing numerical data for Fig. 2B in [13] and for providing copies of papers [2,10,12,14], which we selected as appropriate examples that cover the majority of the problems associated with publications on BVH plotting. Appendix 1. Derivation of formula (7) and an example of calculation of vb

d2i ¼

i¼1

N X

fv i  ½ð1  v b Þxi þ v b g2 ¼ min

ð16Þ

We now apply the least square method (LSM) that states that the sum of the squared residuals of the N osmolalities used in the BVH plot must be minimal:

ð17Þ

i¼1

This sum of squares depends only on ONE parameter vb. Differentiation in regard to vb must result in 0:

@

P N

2 i¼1 di

 ¼ 2

@v b

N X ðxi  1Þfv i  ½ð1  v b Þxi þ v b g ¼ 0

ð18Þ

i¼1

Eq. (18) after rearrangements and dividing by 2 gives: N X ðxi  1Þ½ðxi  1Þv b  ðxi  v i Þ ¼ 0

ð19Þ

i¼1

Regrouping the known part and unknown part at the different sides of the equation gives us the following:

" # N N X X 2 ðxi  1Þ v b ¼ ðxi  1Þðxi  v i Þ i¼1

ð20Þ

i¼1

Solution of Eq. (20) gives us formula (7) for the calculation of vb: N P

½ðxi  1Þðxi  v i Þ

v b ¼ i¼1 PN

i¼1 ðxi

 1Þ2

We can now use the experimental data from Wang’s paper (Fig. 2A) and calculate the correct value of the osmotically inactive volume (we omitted the third decimal value depicted in Fig. 2A for the sake of a concise line):

vb ¼

ð0:22  1Þð0:22  0:30Þ þ ð0:29  1Þð0:29  0:35Þ þ ð2:15  1Þð2:15  1:33Þ ð0:22  1Þ2 þ ð0:29  1Þ2 þ ð2:15  1Þ2

1:6808 ¼ ¼ 0:4062 4:1375

ð21Þ

Appendix 2. Derivation of formulas for the 2-independent parameter regression line used by Wang et al. (the routinely used formulas) In contrast, the built-in statistical formulas and software packages of linearization use the assumption that the slope and the intercept of the regression line are independent, the same formula was also used by Wang (inexplicitly, ‘‘fitted to a straight line by the linear least square method’’, p. 59 in [13]), as well as in the overwhelming majority of other papers on linearization of BVH plots. We will continue to use terminology such as the ‘‘Wang line’’ keeping in mind that it is actually the ‘‘routinely used line’’:

v Wang ¼ ax þ b

ð22Þ

where a and b are now two independent parameters. Similarly, we will introduce the residual as was carried out above:

di  v i  v Wang ðxi Þ ¼ v i  ðaxi þ bÞ

To obtain the best fit vb for the BVH line (7), we first introduce the residual, the difference between the experiment and the theoretical line at a given i-th osmolality:

di  v i  v VBH ðxi Þ ¼ v i  ½ð1  v b Þxi þ v b 

N X

ð23Þ

and we indicate that the sum of the squared residuals must be minimized: N X i¼1

d2i 

N X

½v i  ðaxi þ bÞ2 ¼ min

ð24Þ

i¼1

In this case, there are two independent parameters so Eq. (24) must be differentiated in regard to both a and b TOGETHER, which gives a system of two equations:

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@

@

I.I. Katkov / Cryobiology 62 (2011) 232–241

P N

2 i¼1 di

@a P N

2 i¼1 di

 ¼ 2 

@b

Similarly, the intercept of the regression line will be equal (only in this very particular case!) to vb.

N X ðxi Þ½v i  ðaxi þ bÞ ¼ 0 i¼1

ð25Þ

N X ¼ 2 ð1Þ½v i  ðaxi þ bÞ ¼ 0

vb ¼

i¼1

which after simplification and taking to the account that PN i¼1 b ¼ Nb, where N is the number of osmolalities (points in the line) used in experiments, gives us the following system of two linear equations:

"

# " # N N N X X X  2 xi a þ ðxi Þ b ¼ ðxi v i Þ

"

# N N X X ðxi Þ a þ Nb ¼ ðv i Þ

i¼1

i¼1

i¼1

i¼1

ð26Þ

i¼1

The system (26) gives the following solution for independent a and b:

N

PN

i¼1 ðxi

a¼

v iÞ 

hP

i

N i¼1 ðxi Þ



hP

N i¼1 ð i Þ

v

b¼

10  10:275  9:63  8:188

N i¼1 ð i Þ

b¼

v

i



hP

N 2 i¼1 ðxi Þ

N X

N X

di 

½v i  ðaxi þ bÞ ¼

PN

¼

N X

N

vi 

N X

axi 

i¼1

hP

i

N i¼1 ðxi Þ



N X

N X

PN ðxi v i Þ  N  N þ N i¼1 ðxi v i Þ  N aBVH ¼ Pi¼1 ¼ PN N 2 ðx2 Þ  N i¼1 ðxi Þ  2  N þ N hP i hP i¼1 i i PN N N P N i¼1 ðxi v i Þ  i¼1 ðxi Þ  i¼1 ðv i Þ N Ni¼1 ðxi v i Þ  N  N a¼ ¼ P h i 2 PN P N Ni¼1 ðx2i Þ  N2 ðx Þ N N ðx2 Þ  ¼

i

i¼1

N

N i¼1 ðxi

io

viÞ

ð31Þ

i

v

i¼1 ðxi i Þ  N PN 2 i¼1 ðxi Þ  N

nhP

ð32Þ

N i¼1 ðxi

viÞ

PN

2 i¼1 ðxi Þ

i





N P

hP

2 ðxi Þ

i¼1



hP

N i¼1 ð i Þ

v

i

i2

N i¼1 ðxi Þ

ð33Þ

N 2 i¼1 ðxi Þ

i



hP

i

hP i hP io N N  i¼1 ðxi Þ  i¼1 ðxi v i Þ hP i2 P N N Ni¼1 ðx2i Þ  i¼1 ðxi Þ N i¼1 ð i Þ

v

N  Sx2  Sv  N  Sx  Sxv d

vi ¼

v

i¼1 d i

d

i¼1



hP

PN

N i¼1 ð i Þ

v

¼

N  Sx 2  Sv  d

N S2x

i

ð34Þ

( 

PN

N

PN

2 i¼1 ðxi Þ

2 i¼1 ðxi Þ



hP



N P

2 ) ðxi Þ

i¼1

i2 N i¼1 ðxi Þ

 Sv

ð35Þ

Substituting (33)–(35) into (32) gives the following: N X i¼1

di ¼

N  Sx2  Sv  S2x  Sv N  Sx  Sxv  S2x  Sv  d d 

ð30Þ

b0

N  Sx  Sxv  S2x  Sv d

b¼

ð29Þ

PN

i¼1

hP

where d is the denominator in formulas (26)

 þN

N X

hP

N

i¼1

Comparing this expression with the formula for the slope in (27) we can see that the BVH and regression lines become identical P P if both Ni¼1 xi and Ni¼1 v i equal exact N:

PN



i¼1

axi ¼



v i Þ

v

i

N i¼1 ðxi Þ

First, we will find the three sums as following:

v

v

nhP

i¼1

i¼1

¼ 0:2608

i¼1 ½ðxi  1Þð i  1Þ PN 2 i¼1 ðxi  1Þ PN PN PN i¼1 ðxi i Þ  i¼1 ðxi Þ  i¼1 ð i Þ P PN N 2 ðx Þ  2  ðx i¼1 i i¼1 i Þ þ N



N X ½v i  axi  b

i¼1 N X

¼

ð28Þ

There is however a very specific case when the results of LSM for models of linearization are exactly the same. To consider that, we first express the slope of BVH line in terms of sums:

¼

io

We need to prove that for any regression line, which coefficients are given by formula (26) we have an identical equality:

Appendix 3. Specific case when BVH is identical to the regression line

i¼1 ½ðxi  1Þðxi  PN 2 i¼1 ðxi  1Þ

v

This case for N = 5 is depicted in Fig. 2C, where the slope and intercept are 0.70 and 0.30 respectively.

i¼1

aBVH  1  v b ¼

v

ð27Þ

which is the regression line seen in Wang’s paper [13] depicted in Fig. 2A. As we can see, in general, the BVH and the regression (‘‘Wang’’) line with two independent parameters are different.

PN

PN

hP i2 P N N Ni¼1 ðx2i Þ  i¼1 ðxi Þ PN PN N i¼1 ðx2i Þ  N  i¼1 ðxi v i Þ ¼ PN N i¼1 ðx2i Þ  N 2 PN PN ðx2 Þ  i¼1 ðxi v i Þ ¼ i¼1PNi 2 i¼1 ðxi Þ  N

i¼1

10  13:398  9:632

 PN

v

nhP

¼ 0:5795

10  13:398  9:632 8:188  13:398  9:63  10:275

PN PN i¼1 ðxi i Þ  i¼1 ðxi Þ þ i¼1 ð i Þ P N 2 i¼1 ðxi Þ  2  i¼1 ðxi Þ þ N

2 i¼1 ðxi Þ

¼

Appendix 4. Proof that the sum of residuals of a regression line is equal to 0

PN

For the Wang data, it gives

a¼

PN

v

i¼1 ½ðxi  1Þðxi  i Þ PN 2 i¼1 ðxi  1Þ PN PN 2 i¼1 ðxi Þ  i¼1 ðxi i Þ PN 2 i¼1 ðxi Þ  N

i

hP i2 N 2 i¼1 ðxi Þ  i¼1 ðxi Þ hP i hP i hP i hP i N N N N 2 i¼1 ðv i Þ  i¼1 ðxi Þ  i¼1 ðxi Þ  i¼1 ðxi v i Þ b¼ hP i2 P N N Ni¼1 ðx2i Þ  i¼1 ðxi Þ N

¼

PN

N  Sx2  Sv  N  Sx  Sxv 0 d

This is the major feature of any linear regression line.

ð36Þ

I.I. Katkov / Cryobiology 62 (2011) 232–241

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