In this study, we tested the i.i.d. assumption commonly implied in single-molecule biomechanical experiments. This assumption enables individual measurements acquired by repeated sequential tests to be treated as independent realizations of the same random variable, thereby allowing the measurements to be used for statistical analysis of this random variable. Should this assumption be violated, measurements from different positions in a test sequence would be realizations of different random variables, which would require the data to be segregated into subgroups for separate analyses.
A binary adhesion score is the simplest measurement of single-molecule experiments because it requires only a single probability value for its description. By comparison, measurement of a single-bond rupture force or bond lifetime has to be described by a probability function because it spans a continuous domain. However, it seems reasonable that memory in adhesion scores would accompany memory in rupture forces and lifetimes. Many of the ideas developed here for the former may be applicable to the latter. For example, the i.i.d. assumption is violated if the rupture force or lifetime distribution depends on where in a test sequence the rupture forces or lifetimes are measured. Another example is the unfolding of protein domains, which involves disruption of similar kinds of noncovalent interactions within one molecule as opposed to between two molecules (6, 7). Because refolding may take longer than the intermission between two consecutive tests, incomplete refolding or misfolding may violate the i.i.d. assumption.
We introduced a memory index 
p to quantify the deviation from i.i.d. because correlation among different tests in a time sequence represents the impact of the past on the future. This includes at least three aspects: (i) the magnitude (i.e., to what extent the past memory impacts the future), (ii) the direction (i.e., whether the impact is positive or negative), and (iii) the duration (i.e., how long the memory lasts). To quantify the duration, we can vary the time between two consecutive tests, which was 0.5 s in the experiments analyzed herein. It seems reasonable to suspect that the memory may fade if this time is prolonged. Another question is how long ago a previous test will still have an impact. The present study treats the simplest scenario, in which only the immediate past test is assumed to influence the next test. Relaxing this assumption can include more general scenarios to allow influences from tests further upstream, which would require multistep memories.
Our analysis identified three distinct behaviors (no memory, positive memory, and negative memory), which were exhibited by three molecular systems. These behaviors have been demonstrated by visual observations of different distributions of adhesion clusters (Fig. 2), by two types of model analyses (Figs. 3 and 4), and by extensive data with rigorous statistical analysis (Fig. 5). These different behaviors reflect specific properties of the molecular and cellular systems. The differences are not due to different experimental setups, because the same micropipette adhesion frequency assay was used in all experiments, performed using the same equipment in the same laboratory.
At the level of molecular interactions, adhesion memory likely reflects kinetic processes triggered by the measured binding events. The mathematical model for the adhesion frequency assay predicts that the average number of bonds is 
1 when Pa has midrange values (4). The data in Fig. 5 thus suggest that a single TCR/pMHC bond, engaged for memory. It has been shown that calcium responses (17) or cytotoxic activities (18) in T cells can be triggered by engagement of TCR with even a single pMHC. An intriguing question is whether these two forms of exquisite sensitivity are related or are two manifestations of the same mechanism.
Like the TCR/pMHC interaction, the homotypic interaction between C-cadherins mediates both adhesion and signaling. Contrary to the TCR/pMHC interaction, engagement of C-cadherin in the previous test down-regulates the probability of adhesion in the next test, which is also intriguing. The damping of receptor responsiveness could reflect a physiological feedback mechanism that protects against both acute and chronic receptor overstimulation (19).
Similar abilities to remember a once-presented stimulus on the level of an individual receptor, and to quickly respond at the system level, have been observed for visual and olfactory systems, both mediated by G protein-coupled receptors (19, 20). For example, light adaptation occurs within several seconds and begins at intensities so low that most photoreceptors receive only a few photons (20). The duration of the adaptation is in turn determined by the lifetimes of the chemical processes underlying the adaptive response.
Enzymatic cascades usually include both positive and negative feedback loops to allow precise control of the outcome of the incoming signal. The present work suggests that similar feedback control mechanisms may exist in adhesion cascades as well.
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