Hidden Markov model analysis to fluorescence blinking of fluorescently labeled DNA

In the present study, we analyze the fluorescence fluctuation data previously measured for charge-separation and charge-recombination processes in DNA [11, 12]. Figures 1 (c) and (d) are schematic chemical structures of the fluorescently labeled DNA (double helix structure) to be analyzed, where A, G, T, and C are the bases that make up DNA, X is the fluorescent molecule ATTO655 as a photosensitizer, and ^ZG is a deazaguanine as a hole trap to observe a charge-separation and charge recombination process in DNA at the single-molecule level. The panel (c) represents a schematic chemical structure for an emitting ON state, where ATTO655 repeats photon-absorption and emission. The panel (d) represents a schematic chemical structure of a not-emitting OFF state; during the charge-separated state, ATTO655 is in the radical anion form and cannot emit, therefore the duration of the OFF state corresponds to the lifetime of the charge-separated state.

Briefly, for the experimental setup, biotin group was introduced at the end of the duplex and was anchored on a glass surface through well-established biotinylated-BSA, streptavidin, biotinylated-DNA chemistry. The single molecule measurement was performed on a custom-made confocal fluorescence microscope consisting of an inverted optical microscope (IX73, Olympus). The output of the laser (OBIS 637 LX, Coherent, 637 nm, 4.6 $\mu$W) was focused in the sample by an objective (UPlanXApo, x60, oil, NA 1.42, Olympus), and the detection volume (confocal volume) was regulated by a pinhole (diameter 25 $\mu$m, Thorlabs MPH16). The scattered light was blocked by a band-pass filter (FF01-697/58-25, Semrock), and the emitted photons from ATTO655 were detected by an avalanche photodiode (SPCM-AQRH-14, Perkin-Elmer). A counting board (SPC-130EMN, Becker & Hickl GmbH) was used to count the output, and real-time monitoring of fluorescence intensity fluctuations has been achieved using the Spcm64 system. (Becker & Hickl GmbH). The time resolution of this device on the photon counting is 80 nsec.

We next briefly describe our calculation method of HMM [15]. In the HMM, a probability density function combined with various stochastic variables is introduced. The joint distribution of the HMM employed in the present study is the following form as

	$\displaystyle p({\bm{I}},\mathbf{S},\mathbf{\Theta},\bm{\pi},\mathbf{A})=p({\bm{I}}|\mathbf{S},\mathbf{\Theta})p(\mathbf{S}|\bm{\pi},\mathbf{A})p(\mathbf{\Theta})p(\bm{\pi})p(\mathbf{A}).$
			(1)

Here,

$\displaystyle{\bm{I}}=(I_{1},I_{2},\cdots,I_{N})$

(2)

is observed data corresponding to a fluorescence trajectory to be analyzed, and $N$ is the total numbers of the time grids. $\mathbf{S}$ is a time series of hidden variables as

$\displaystyle\mathbf{S}=({\bf s}_{1},{\bf s}_{2},\cdots,{\bf s}_{N}),$

(3)

which describes the internal state of each time in the fluorescence trajectory, and ${\bf s}_{i}$ is a $K$-dimensional vector in the time step $i$. To identify an ON or OFF state, we set $K$ = 2 [15]. $\mathbf{\Theta}$ describes the parameter characterizing a distribution function of observed data ${\bm{I}}$. In the present study, we assume that the distribution of each state follows the Gaussian distribution with a mean $\mu$ and a precision $\lambda$, where $\lambda^{-1}$ represents a variance. Then, we write $\mathbf{\Theta}$ as $\{{\bm{\mu}},{\bm{\lambda}}\}$ with ${\bf\mu}=(\mu_{{\rm ON}},\mu_{{\rm OFF}})$ and ${\bf\lambda}=(\lambda_{{\rm ON}},\lambda_{{\rm OFF}})$. $\bm{\pi}$ represents the probability of the initial-step hidden variable, and $\bm{\pi}=(\pi_{{\rm ON}},\pi_{{\rm OFF}})$. $\mathbf{A}$ describes a 2$\times$2 transition matrix for the time evolution of the hidden variables. $p(\mathbf{S}|\bm{\pi},\mathbf{A})$ in Eq. (1) describes the conditional probability distribution of $\mathbf{S}$ with $\mathbf{A}$ and $\bm{\pi}$ fixed, and similarly, $p(\bm{I}|\mathbf{S},\mathbf{\Theta})$ is the conditional probability distribution of $\bm{I}$ after $\mathbf{S}$ and $\mathbf{\Theta}$ were determined.

The purpose of the HMM simulation is to infer the hidden-variable time series $\mathbf{S}$. For this purpose, we need to calculate a posterior distribution which is a conditional distribution function under the observed trajectory $\bm{I}$ as

$\displaystyle p(\mathbf{S},{\bm{\mu}},{\bm{\lambda}},\bm{\pi},\mathbf{A}|\bm{I})=\frac{p(\bm{I},\mathbf{S},\bm{\mu},\bm{\lambda},\bm{\pi},\mathbf{A})}{p(\bm{I})},$

(4)

where $p(\bm{I})$ is the marginal distribution written as

$\displaystyle p(\bm{I})=\sum_{\mathbf{S}}\int p(\bm{I},\mathbf{S},\bm{\mu},\bm{\lambda},\bm{\pi},\mathbf{A})d\bm{\mu}d\bm{\lambda}d\bm{\pi}d\mathbf{A}.$

(5)

To calculate the posterior distribution in Eq. (4) approximately, we use a blocking Gibbs sampling based on Bayesian inference. Optimization details can be found in Ref. 15. By evaluation of the posterior distribution, the hidden-variable time series $\mathbf{S}$ in Eq. (3) can be obtained as a simulation result. On the calculation condition, the total number of the iteration steps of the Gibbs sampling $N_{itr}=1000$, and the hyperparameters of each distribution are set to the same as Ref. 15.

Next, we describe how to analyze the hidden-variable time series $\mathbf{S}$ in Eq. (3) obtained from the HMM simulation. For this purpose, we consider a probability density of duration $\tau$ as

$\displaystyle p(\tau)=\frac{1}{N_{e}}\sum_{\alpha=1}^{N_{e}}\delta(\tau-\tau_{\alpha}),$

(6)

where $p(\tau)$ is normalized as $\int p(\tau)d\tau=1$. We consider $p(\tau_{{\rm ON}})$ and $p(\tau_{{\rm OFF}})$, and $\tau_{\alpha}$ in Eq. (6) is the $\alpha$-th ON or OFF event duration. The {$\tau_{\alpha}$} data are collected from the hidden-variable time series $\mathbf{S}$ [15], and $N_{e}$ is the total number of the {$\tau_{\alpha}$} data. In the practical calculation, $p(\tau)$ is evaluated discretely as

$\displaystyle p(\tau_{k})=\frac{1}{C}\sum_{\alpha=1}^{N_{e}}\delta(\tau_{k}-\tau_{\alpha}),$

(7)

where $p(\tau_{k})$ is proportional to the total number of the events with the duration from $\tau_{k-1}$ to $\tau_{k}$, and $\tau_{k}$ is given as $k\Delta_{\tau}$ with $\Delta_{\tau}$ being a grid spacing of duration. $C$ is a normalization constant given as

$\displaystyle C=\sum_{k=1}^{N_{\tau}}\sum_{\alpha=1}^{N_{e}}\delta(\tau_{k}-\tau_{\alpha})\Delta\tau,$

(8)

where $N_{\tau}$ is the total number of the duration grids.

The calculation condition is as follows: We analyze the fluorescence trajectories of 40 molecules. The length of the fluorescence trajectories is ranging from 0.46 sec to 9.27 sec. For the time bin $\Delta$ of the trajectory, we consider the three size; 62.5 $\mu$sec, 125 $\mu$sec, and 250 $\mu$sec. The total number of the time grids of each trajectory, $N$ in Eqs. (2) and (3), depends on the trajectory length and $\Delta$. The total number of the {$\tau_{\alpha}$} data, $N_{e}$ in Eq. (6) or (7), is 2,078 for the ON case and 2,067 for the OFF case with the $\Delta$=250-$\mu$sec case. Similarly, for the $\Delta$=125-$\mu$sec case, $N_{e}$ is 2,341 (ON) and 2,334 (OFF), and for the $\Delta$=62.5-$\mu$sec case, $N_{e}$ is 2,697 (ON) and 2,678 (OFF). The grid spacing of duration $\Delta\tau$ is 500 $\mu$sec, and the total number of the duration grids $N_{\tau}$ is 462 for the ON case and 122 for the OFF case with the $\Delta$=250-$\mu$sec case. Similarly, for the $\Delta$=125-$\mu$sec case, $N_{\tau}$ is 383 (ON) and 122 (OFF), and for the $\Delta$=62.5-$\mu$sec case, $N_{\tau}$ is 322 (ON) and 122 (OFF).

Figure 2 shows three experimental fluorescence trajectories corresponding to Eq. (2), where photon counts are evaluated in the time bin $\Delta$ = (a) 250 $\mu$sec, (b) 125 $\mu$sec, and (c) 62.5 $\mu$sec. These data are all the same trajectory, but $\Delta$ is different, so the integrated values for each bin are different. It can clearly be seen that the noise becomes more appreciable as the $\Delta$ becomes smaller.

Figure 3 superimposes the hidden-variable time series $\mathbf{S}$ in Eq. (3) obtained from the HMM simulations, denoted by the green dashed lines, onto the original fluorescent trajectories (red solid lines). A black-solid line of 0.5 is the line to distinguish an ON state (above 0.5) and an OFF state (below 0.5) for the hidden variable time series. In the practical calculation, this threshold is slightly adjusted depending on the data within 0.4-0.6. As can be seen from the figure, the hidden-variable time series well capture the switching of the ON$\to$OFF and OFF$\to$ON process, from which we properly evaluate the ON and OFF duration, $\tau_{{\rm ON}}$ and $\tau_{{\rm OFF}}$. Examples of $\tau_{{\rm ON}}$ and $\tau_{{\rm OFF}}$ are illustrated in the Fig. 3 (a). Since the extremely noisy data affects the state identification, short-duration events might slightly be overestimated in the analysis of time series with smaller $\Delta$. We note that the moving-average method [39] does not work for the ON/OFF identification from the present fluorescence trajectories. This is because the data is noisy, so there is a limitation for data smoothing. As a result, a large number of artificial short-duration events are generated, compared to the HMM. In this sense, the HMM is highly effective for the noise removing.

Figure 4 displays our calculated probability density for the ON [(a), (c), and (e)] and OFF [(b), (d), and (f)] duration, $p(\tau_{k})$ in Eq. (7). The upper (a) and (b) describe the results for fluorescence trajectories with $\Delta$ = 250 $\mu$sec. The middle (c) and (d) and the lower (e) and (f) panels describe the results for $\Delta$ = 125 $\mu$sec and $\Delta$ = 62.5 $\mu$sec, respectively. Note that these data are collected from the fluorescence trajectories of the 40 molecules.

To analyze the trends of these data, we performed fittings of model probability density functions. For the ON distribution, we use an exponential distribution as

$\displaystyle P(\tau_{{\rm ON}})=\frac{1}{s}\exp\biggl{(}-\frac{\tau_{{\rm ON}}}{s}\biggr{)},$

(9)

where $s$ is a parameter for the exponential function. For the OFF distribution, we consider a log-normal distribution as

$\displaystyle P(\tau_{{\rm OFF}})=\frac{1}{\sqrt{2\pi\sigma^{2}}\tau_{{\rm OFF}}}\exp\biggl{(}-\frac{(\ln\tau_{{\rm OFF}}-\mu)^{2}}{2\sigma^{2}}\biggr{)},$

(10)

where $\mu$ and $\sigma$ represent mean and standard deviation of the distribution, respectively. To evaluate the fitting accuracy, we calculate the following measures; root mean squared error (RMSE) as

$\displaystyle{\rm RMSE}=\sqrt{\frac{1}{N_{D}}\sum_{k=1}^{N_{D}}\bigl{(}p(\tau_{k})-P(\tau_{k})\bigr{)}^{2}},$

(11)

and coefficient of determination ${\rm R}^{2}$ as

$\displaystyle{\rm R}^{2}=1-\frac{\sum_{k=1}^{N_{D}}\bigl{(}p(\tau_{k})-P(\tau_{k})\bigr{)}^{2}}{\sum_{k=1}^{N_{D}}\bigl{(}p(\tau_{k})-\bar{p}\bigr{)}^{2}}$

(12)

with the $\bar{p}$ in Eq. (12) being the mean of probability density as

$\displaystyle\bar{p}=\frac{1}{N_{D}}\sum_{k=1}^{N_{D}}p(\tau_{k}).$

(13)

Here, $\tau_{k}$ represents the $k$-th duration grid, introduced in Eq. (7), but the calculations of Eqs.(11), (12), and (13) exclude the data of $p(\tau_{k})=0$. Thus, the total number of the data points $N_{D}$ is 231 for the ON case and 88 for the OFF case with the $\Delta$=250-$\mu$sec case. Similarly, for the $\Delta$=125-$\mu$sec case, $N_{D}$ is 218 (ON) and 85 (OFF), and for the $\Delta$=62.5-$\mu$sec case, $N_{D}$ is 209 (ON) and 87 (OFF). The $P(\tau_{k})$ is the value of the fitting function of the $k$-th duration grid.

The model parameters in Eqs. (9) and (10) are determined as follows: We first perform least-squared fitting. Then, we set the resulting parameters to the initial guess, and perform the Kolmogorov-Smirnov (KS) test [40]. In this test, we tune the model parameters to maximize $p$-value for measuring goodness-of-fit, where the $p$-value is defined as

$\displaystyle p\mathchar 45\relax{\rm value}=p(z)=2\sum_{j=1}^{\infty}(-1)^{j-1}\exp(-2j^{2}z^{2})$

(14)

with

$\displaystyle z=\sqrt{N_{e}}\max_{\tau}\bigl{|}S_{N_{e}}(\tau)-F(\tau)\bigr{|}$

(15)

and $N_{e}$ being the total number of the {$\tau_{\alpha}$} data [Eq. (6)]. $S_{N_{e}}(\tau)$ is an empirical distribution function as

$\displaystyle S_{N_{e}}(\tau)=\frac{1}{N_{e}}\sum_{\alpha=1}^{N_{e}}{\bf 1}_{\tau_{\alpha}<\tau},$

(16)

where ${\bf 1}_{A}$ is an indicator function for the condition $A$, and $F(\tau)$ is a cumulative distribution function as

$\displaystyle F(\tau)=\int_{0}^{\tau}P(\tau^{\prime})d\tau^{\prime},$

(17)

where $P(\tau)$ is a probability density function to be tested. The conventional significance level for the $p$-value is 0.05, and if the obtained $p$-value is above 0.05, the test model function is considered to have validity.

Figure 4 shows the resulting fitted curves with blue-solid lines. We see from the figure that the fittings are satisfactorily. Optimized parameters and fitting accuracy are given in Table 1, where the first and second columns contain the results for $P(\tau_{{\rm ON}})$ and $P(\tau_{{\rm OFF}})$, respectively. From the table, the RMSE and ${\rm R}^{2}$ support good accuracy; very low RMSE and ${\rm R}^{2}$ sufficiently close to 1. The resulting $p$-values are also listed, where they are 0.2-0.6 for $P(\tau_{{\rm ON}})$ and nearly 0.3 for $P(\tau_{{\rm OFF}})$. These $p$-values are larger than the conventional significance level of 0.05. Thus, the null hypothesis (the data distribution differs from the assumed distribution function) is rejected.

Next, we discuss the characteristic relaxation time in terms of the resulting model parameters. For the relaxation time of the ON$\to$OFF process, it is estimated as 20-25 msec (see $s$ in Table 1). In addition, on the characteristic relaxation time $e^{\mu}$ of the OFF$\to$ON process, we obtain $e^{1.95}=7.03$ msec for the $\Delta=250$-$\mu$sec case, $e^{1.82}=6.17$ msec for the $\Delta=125$-$\mu$sec case, and $e^{1.68}=5.37$ msec for the $\Delta=62.5$-$\mu$sec case. The estimate based on the auto-correlation function analysis [12, 11] is nearly 5 msec, and thus, the present HMM estimates are consistent with the estimates of the correlation analyses. We note that the both relaxation times of the ON and OFF states become short as the time-bin size $\Delta$ of the fluorescence trajectory becomes small. This may reflect that the longer duration event is split into the shorter duration events with considering the smaller $\Delta$.

The ON distribution is basically a monotonically decreasing, so the validity of the exponential function is identified visibly. On the other hand, the OFF distribution has an asymmetric structure, so it is necessary to carefully verify the validity of the log-normal function. To find the best function for probability density of the OFF duration, we investigate two other representative asymmetric distributions, the Weibull and Gamma distributions. The Weibull distribution $P_{W}(\tau)$ is written as

$\displaystyle P_{W}(\tau)=\frac{m}{\eta}\biggl{(}\frac{\tau}{\eta}\biggr{)}^{m-1}\exp\biggl{\{}-\biggl{(}\frac{\tau}{\eta}\biggr{)}^{m}\biggr{\}},$

(18)

where $m$ and $\eta$ are the shape and scale parameters of the distribution, respectively. Also, the Gamma distribution $P_{G}(\tau)$ is written as

$\displaystyle P_{G}(\tau)=\frac{b^{-a}}{\Gamma(a)}\tau^{a-1}\exp\biggl{(}-\frac{\tau}{b}\biggr{)},$

(19)

where $a$ and $b$ are the shape and scale parameters of the distribution, respectively. $\Gamma(a)$ is the Gamma function.

Figure 5 compares the fittings to the OFF probability density: The panels (a), (d), and (g) are the fitting results of the Weibull distribution $P_{W}(\tau_{{\rm OFF}})$ in Eq. (18), based on the least-squared fitting plus the KS-$p$ value maximization, and (b), (e), and (h) are the fitting results of the Gamma distribution $P_{G}(\tau_{{\rm OFF}})$ in Eq. (19). The panels (c), (f), and (i) are the log-normal distribution $P(\tau_{{\rm OFF}})$ in Eq. (10). Also, (a), (b), and (c) describe the results for the time bin $\Delta=250$ $\mu$sec, (d), (e), and (f) describe the results for $\Delta=125$ $\mu$sec, and (g), (h), and (i) are the results for $\Delta=62.5$ $\mu$sec. The fitted parameters are summarized in Table 2. From the figure, we see that the log-normal function gives a good fit, while the Weibull and Gamma functions seem to underestimate the peak height around $\tau_{{\rm OFF}}\sim$ 5 msec. In the Weibull and Gamma functions, the tail behavior in the low duration side is not described properly. The resulting $p$-values for the Weibull and Gamma distributions are appreciably small compared to those of the log-normal distribution, and well below the significant level of 0.05.

The fact that $\tau_{{\rm OFF}}$ follows the log-normal distribution means that $\tau_{{\rm OFF}}$ is based on the multiplicative stochastic process as [41, 42, 43]

$\displaystyle\tau_{{\rm OFF}}=\prod_{i=1}^{M}n_{i},$

(20)

where $n_{i}$ is the $i$-th factor affecting $\tau_{{\rm OFF}}$ and $M$ is the total numbers of the factors. Now, taking the logarithm of Eq. (20), we obtain

$\displaystyle\ln\tau_{{\rm OFF}}=\sum_{i=1}^{M}\ln n_{i}$

(21)

If $n_{i}$ is independent with each other and $M$ is sufficiently large, $\ln\tau_{{\rm OFF}}$ follows the normal distribution based on the central limit theorem, or equivalently, $\tau_{{\rm OFF}}$ follows the log-normal distribution. Now, an important question is what is the factor $n_{i}$ affecting the stability of the OFF state, leaving to be explored.

Using the above determined model distribution functions, we discuss aspects of the ON$\to$OFF and OFF$\to$ON transition processes. The ON$\to$OFF transition corresponds to a process in which the photon-absorption and emission cycle changes to a charge-separated state. On the other hand, the OFF$\to$ON transition corresponds to a process in which the charge-separated state returns to the ground state. In addition, it was confirmed that the lifetime distribution of the ON state decays exponentially, and that of the OFF state exhibits the log-normal distribution.

Based on these results, we will discuss aspects of the transition processes in terms of the failure analysis in the semiconductor device [44]. In this analysis, for example, the ON$\to$OFF process is discussed as follows: First, an ON state is defined as a non-faulty state, and an OFF state is defined as a faulty state. The failure analysis tells us the nature of the failures; we can classify whether the ON$\to$OFF process is based on an accidental event or a wear-out event, as well as the failure type of the semiconductor product.

We first consider a failure probability density $f(\tau)$ representing the probability density that a failure will occur between $\tau$ and $\tau+d\tau$. A failure rate $\lambda(\tau)$ is defined in terms of $f(\tau)$ as

$\displaystyle\lambda(\tau)=\frac{f(\tau)}{R(\tau)},$

(22)

where $R(\tau)$ represents survival function defined as

$\displaystyle R(\tau)=\int^{\infty}_{\tau}f(t)dt=1-F(\tau)$

(23)

with $F(\tau)$ being the failure distribution function as

$\displaystyle F(\tau)=\int^{\tau}_{0}f(t)dt.$

(24)

The failure rate $\lambda(\tau)$ describes the frequency of breaking down after the usage time $\tau$. Based on $\lambda(\tau)$, the nature of the failures in semiconductor products can properly classified: Figure 6 shows typical behavior of $\lambda(\tau)$, called bathtub-like curve; initially, the $\lambda(\tau)$ decreases because the initial defective products with a certain extent continue to fail. This period is called the early failure period. Then, the $\lambda(\tau)$ becomes constant. This period is called the random failure period. Finally, the $\lambda(\tau)$ rapidly increases as the product deteriorates due to wear. This is wear-out period.

In Sec. III.2, we confirmed that the probability density of the ON duration follows the exponential function of $P(\tau_{{\rm ON}})$ in Eq. (9) and that of the OFF duration follows the log-normal function $P(\tau_{{\rm OFF}})$ in Eq. (10). Now, these $P(\tau_{{\rm ON}})$ and $P(\tau_{{\rm OFF}})$ correspond to $f(\tau)$ in Eq. (22): Thus, the $\lambda(\tau_{{\rm ON}})=P(\tau_{{\rm ON}})\big{/}R(\tau_{{\rm ON}})$ describes the failure rate where the ON state breaks to the OFF state after duration $\tau_{{\rm ON}}$. Similarly, the $\lambda(\tau_{{\rm OFF}})$ represents the failure rate where the OFF state changes to the ON state after duration $\tau_{{\rm OFF}}$. By comparing the $\lambda(\tau_{{\rm ON}})$ and $\lambda(\tau_{{\rm OFF}})$ with the bathtub-like curve in Fig. 6, we can discuss the nature of the transition processes of the ON$\to$OFF and OFF$\to$ON.

Figure 7 shows our calculated failure rate $\lambda(\tau)$ in Eq. (22), denoted by red-dashed curve. Survival function $R(\tau)$ in Eq. (23) is also plotted with green-dotted curve, and blue-solid curve is the model probability density $P(\tau)$ given in Fig. 4. The upper (a) and (b), middle (c) and (d), the lower (e) and (f) panels are the results for the time series with $\Delta$ = 250 $\mu$sec, $\Delta$ = 125 $\mu$sec, and $\Delta$ = 62.5 $\mu$sec, respectively. Also, the left (a), (c), and (e) panels are results for the ON$\to$OFF process, while the right (b), (d), and (f) panels correspond to the results for the OFF$\to$ON process. We see that the $\lambda(\tau_{{\rm ON}})$ is constant, which is a consequence of the exponential decay of $P(\tau_{{\rm ON}})$. The behavior of $\lambda(\tau_{{\rm ON}})$ is the same as the random failure period in the bathtub-like curve of Fig. 6, and then the ON$\to$OFF process would be recognized as an accidental event; the light-emitting ON state switches to the not-emitting OFF state accidentally.

On the other hand, the OFF$\to$ON process would not be so simple; the $\lambda(\tau_{{\rm OFF}})$ first increases, reaches a maximum, then decreases. The initial increasing trend of the $\lambda(\tau_{{\rm OFF}})$ seems to correspond to the wear-out period in Fig. 6. However, after that, it decreases, which is not observed in the bathtub curve of Fig. 6. In the case of practical semiconductor products, once the wear-out failures begin, all the products would break down within the wear-out period. Therefore, this decreasing trend may not be observed in the semiconductor-product case. In the present case, the situation might be a little different; at $\tau_{{\rm OFF}}=0$, $P(\tau_{{\rm OFF}})=0$ and the OFF state does not react in the very beginning. Then, the OFF state begins to break down. Around the inflection point of $R(\tau_{{\rm OFF}})$, $\lambda(\tau_{{\rm OFF}})$ takes a peak, and then decreases. The peak position is at around 5 msec, and most of the OFF states return to the ON states in about 30 msec. In any case, when it comes to wear-out failure, various factors, such as the details of the molecular structure and the environment, can be relevant to the failure.