Fig. 1[and supporting information (SI) Movie 1] illustrate a simple experiment: the micropipette adhesion frequency assay (4). A human red blood cell (RBC) pressurized by micropipette aspiration is used as an adhesion sensor to test interactions between ligands coated on the RBC membrane and receptors expressed on a second cell (Fig. 1A). The receptor-expressing cell is put into contact with the RBC for a given duration (Fig. 1B) and then retracted. If adhesion results, retraction will stretch the RBC (Fig. 1C), otherwise the RBC will smoothly return to its initial shape (Fig. 1D). When adhesion does occur, additional quantities can be measured by using the RBC picoforce transducer or any other ultrasensitive force technique, including rupture force (3), adhesion lifetime (2), molecular elasticity (5), protein unfolding (6), and protein refolding (7).

Single-molecule biomechanical measurements are inherently stochastic because molecular events (e.g., unfolding of a protein domain or unbinding of a receptor–ligand bond) are determined not only by the weak, noncovalent interactions (within a single molecule or between two interacting molecules) but also by thermal excitations from the environment. In a given adhesion test, both positive (adhesion, scored 1) and negative (no adhesion, scored 0) outcomes are possible. When adhesion occurs, its rupture force or lifetime can be any positive value. Estimation of the adhesion probability requires averaging a large number of adhesion scores (4), and estimation of the probability distribution of single-bond rupture forces or lifetimes requires histogram analysis of a large number of measurements (2, 3). Experimentally, these data are obtained by sequentially repeating the measurement many times, yielding a sequence of random numbers (e.g., random sequences of 0s and 1s from the micropipette adhesion frequency assay).

A crucial assumption that allows probability to be calculated by using measurements from sequentially repeated tests is that all measurements are identical yet independent from each other, i.e., the “independent and identically distributed” (i.i.d.) assumption. However, no analysis had been done to test this assumption in single-molecule biomechanical experiments.

Various statistical tests could be used to test the i.i.d. assumption. Probability plots can be employed to visually determine whether data are from Bernoulli sequences, an approach that can be subjective (8). Another widely used procedure employs a χ² statistic of empirical transitional probabilities to test serial independence (9). Here we develop a model for size distribution of the consecutive adhesion events expected for a one-step Markov process. Fitting the model to experimental data allows us to quantify the magnitude and direction of deviation from the i.i.d. assumption in terms of a “memory” index. Here, memory represents the ability of the system to remember the result of the previous test, as evidenced by a change in the likelihood of the outcome in the subsequent test. We found that nature has provided examples for all three theoretically possible scenarios: no memory, positive memory, and negative memory.

Adhesion between K562 cells transfected with lymphocyte function-associated antigen 1 (LFA-1) and RBCs reconstituted with intercellular adhesion molecule 1 (ICAM-1) (10) exhibited the behavior describable by Bernoulli sequences, with no memory. LFA-1/ICAM-1 interaction mediates the adhesion and migration of leukocytes during an inflammatory reaction (11), as well as their formation of immunological synapses with other cells (12).

Markov sequences with positive feedback (adhesion probability increased by adhesion in the immediate past) were observed in adhesion between T lymphocytes expressing T cell receptor (TCR) and RBCs coated with an antigen peptide bound to a major histocompatibility molecule (pMHC). TCR/pMHC interaction is of central importance to adaptive immunity because it determines how T cells discriminate between different pMHC ligands and transduce distinct signals for various downstream effector functions (13).

Markov sequences with negative feedback (adhesion probability decreased by adhesion in the immediate past) were observed in homotypic adhesions between CHO cells transfected with C-cadherin and RBCs coated with C-cadherin. C-cadherin mediates adhesion between Xenopus laevis blastomeres and plays an essential role in the maintenance of embryo integrity (14) and in morphogenetic cell movements (15).

The result of such n repeated tests is a random sequence whose value X_i at the ith position is either 0 or 1. In a previous analysis (4), the running adhesion frequency, defined as F_i = (X₁ + X₂ + … + X_i)/i (1 ≤ i ≤ n), was plotted vs. i, the test cycle index (Fig. 2A–C). F_i fluctuates when i is small because of the small number of statistics, but it should approach an asymptotic curve as i approaches n, unless the sequence is too short. For sequences of sufficient length, F_n is the average adhesion frequency, which is the best estimator for the probability of adhesion in each test if the i.i.d. assumption holds, i.e., the sequence is Bernoullian. A gradual decline in running adhesion frequency, observed for some receptor–ligand interactions (4), could result from receptor extraction from the cell membrane when the two cells are forced to detach from each other. For the three molecular systems studied here, F_i approaches a plateau equal to the averaged adhesion probability P_a.

Another way to visualize the sequences in Fig. 2 A–C is to plot the nonzero X_iF_n vs. i (Fig. 2 D–F). The zero scores for no-adhesion events have been omitted for clarity. Three molecular systems are shown: LFA-1/ICAM-1 interaction (Fig. 2D), TCR/pMHC interaction (Fig. 2E), and homotypic interaction between C-cadherins (Fig. 2F). For each interaction, three sequences were obtained by using three pairs of cells tested under the same conditions. Symbols in Fig. 2 D–F match those in Fig. 2 A–C. The variations in the horizontal levels among the three sequences (i.e., F_n values) reflect cell-to-cell variations in heterogeneous cell populations that expressed lognormally distributed receptor–ligand densities. To compare different receptor–ligand interactions, we chose data sets having similar mean adhesion levels (solid lines, Fig. 2 D–F).

Three distinct behaviors seem apparent, even with a brief glance at the adhesion score sequences. Compared with those for the LFA-1/ICAM-1 interaction (Fig. 2D), the sequences for the TCR/pMHC interaction (Fig. 2E) appear more “clustered,” whereas those for the C-cadherin interaction (Fig. 2F) are less “clustered.” Here, “cluster” refers to consecutive adhesion events uninterrupted by no-adhesion events. In Fig. 2 A–C, clustering manifests as uninterrupted ascending segments in the running adhesion frequency curves.

Fig. 2 suggests that the likelihood of an adhesion in the future test may be influenced by the outcomes of past tests, depending on the biological systems. To analyze this possibility quantitatively, we assume that the adhesion score sequence is Markovian and stationary. The one-step transitional probabilities are independent of the test cycle index i and could be defined in terms of the conditional probabilities: where n_ij is the number of i →j transitions directly calculated from the adhesion score sequence; p₁₀ + p₁₁ = 1 and p₀₀ + p₀₁ = 1. By definition, p₀₁ is the probability of having adhesion (i.e., X_i+1 = 1) under the condition that the previous adhesion test is not successful (i.e., X_i = 0). It is also given a short notation p by definition. p₁₁ is the probability of having adhesion in the next test if the previous test also results in adhesion. It is also given a short notation p + Δp by definition. Δp represents an increment (positive or negative) in the probability of adhesion in the (i + 1)th test due to the occurrence of adhesion in the ith test, which can be thought of as a memory index. If the i.i.d. assumption holds, p₀₁ = p₁₁ = p and Δp = 0, and we recover the Bernoulli sequence. p can be estimated from the average adhesion frequency because it also equals P_a. If the i.i.d. assumption is violated (p₀₁ ≠ p₁₁ and Δp ≠ 0), p will not be equal to P_a (see Eq. 5 below).

Direct calculations of p₀₁ and p₁₁ for the data in Fig. 2 D–F show that their values are close to each other and to the averaged adhesion probability, P_a, for LFA-1/ICAM-1 interaction, providing preliminary validation for the i.i.d. assumption. By comparison, transition 1 → 1 is more favorable than transition 0 → 1 for the TCR/pMHC interaction, whereas the opposite seems true for the C-cadherin interaction, indicating that the i.i.d. assumption might be violated for these two cases.

Closer inspection of the scaled adhesion scores in Fig. 2 D–F reveals that they are clustered at different sizes. It seems intuitive that, for a given “cluster size” m (i.e., m consecutive adhesions), the number of times it appears in an adhesion score sequence contains statistical information about that sequence. Our intuition is supported by the data in Fig. 3, which shows the cluster size distribution enumerated from the adhesion score sequences in Fig. 2. Compared with the distribution for the LFA-1/ICAM-1 interaction (Fig. 3A), the distribution for the TCR/pMHC interaction (Fig. 3B) has more clusters of large size, whereas the distribution for the C-cadherin interaction (Fig. 3C) has more single adhesion events surrounded by no-adhesion events.

To quantify the differences among the three cases in Fig. 3, we derived a formula to express the number, M_B, of clusters of size m expected in a Bernoulli sequence of length n and probability p for the positive outcome in each test: The first term in the upper branch on the right-hand side of Eq. 2 represents summation over the probabilities of having a cluster of size m in all possible positions, i.e., for clusters starting at i = 2 to i = n − (m − 1). Clusters starting from X₁ or ending with X_n are accounted for by the second term in the upper branch. The lower branch of our formula accounts for the sequence of all 1s. It becomes apparent from the above derivation that Eq. 2 assumes equal probability for the cluster to take any position in the sequence. This can be thought of as a stationary assumption.

The total number of positive adhesion scores in the entire sequence can be calculated by multiplying Eq. 2 by m and then summing over m from 1 to n. It can be shown by direct calculation that ΣmM_B(m, n, p) = np. Here, np is the expected total number of adhesion events. This outcome is predicted and shows that Eq. 2 is self-consistent.

M_B is plotted vs. P_a (= p) in Fig. 4A for n = 50 and for clusters of sizes m = 1–4. The actual cluster size distributions enumerated from measured adhesion score sequences (e.g., Fig. 3A) should be realizations of the underlying stochastic process. We used computer simulations to characterize the statistical properties of this stochastic process. The mean ± SEM of the number of clusters of size 1 from 3 (open triangles, mimic experiments where three to five pairs of cells were usually tested) and 50 (filled triangles, a good approximation to Eq. 2) simulated Bernoulli sequences for several P_a values are shown in Fig. 4A, which evidently agree well with M_B(1, 50, P_a) given by Eq. 2.

We next extend Eq. 2 to the case of a Markov sequence by including a single-step memory. The four conditional probabilities defined in Eq. 1 form a one-step transition probability matrix [P] of a stationary Markov sequence. Using Bayes' theorem for total probability, the unconditional probabilities for the (i + 1)th test are related to those for the ith test by [P]: By applying the Chapman–Kolmogorov equation (8), the n-step transition matrix can be obtained as [P⁽ⁿ⁾] = [P]ⁿ. With the initial condition of P(X₁ = 1) = p and P(X₁ = 0) = 1 − p, it can be shown by mathematical induction that P(X_i = 1) = p(1 − Δpⁱ)/(1 − Δp). The formula for the expected cluster size distribution can now be extended as follows: Setting Δp = 0 reduces Eq. 4 to Eq. 2, as required. Eq. 4 has another special case at Δp = 1 − p when it simplifies to M_M(m, n, p, 1 − p) = p(1 − p)^n−m, which describes the situation where the first adhesion event in the sequence would increase the probability for adhesion in the next test to 1, leading to continuous adhesion for all tests until the end of the experiment.

Experimentally, the adhesion probability can be estimated from the adhesion frequency F_n. The expected value of F_n can be calculated as follows: If Δp = 0 (i.e., a Bernoulli sequence), F_n approaches p as n becomes large. However, if Δp ≠ 0 (i.e., a Markov sequence), then F_n approaches p/(1 − Δp).

M_M is plotted vs. P_a (related to p and Δp by Eq. 5) in Fig. 4B for n = 50, m = 1 (i.e., solitary 1s bound by 0s from both ends) for Δp ranging from −0.3 to 0.5 in increments of 0.1. The case of Δp = 0 (i.e., Bernoulli sequence) is plotted as a solid curve. Cases of Δp > 0 (i.e., memory with positive feedback) are plotted as dotted curves. Cases of Δp < 0 (i.e., memory with negative feedback) are plotted as dashed curves. Increasing Δp shifts the curve downward toward a smaller number of isolated 1s in an adhesion score sequence.

It can be shown by direct calculation that Σ_m=1ⁿ mM_M(m, n, p, Δp) = p[n − Δp(1 − Δpⁿ)/(1 − Δp)]/(1 − Δp). Here, the left-hand side sums adhesion events distributed in various clusters of different sizes, and the right-hand side is the expected number of adhesion events in n repeated tests, nE(F_n) (cf. Eq. 5). This predicted result confirms that Eq. 4 is self-consistent.

In Fig. 3, Eq. 4 was fit (curves) to the measured cluster size distributions (bars) (see Materials and Methods for procedure detail). Three different types of behaviors can be clearly discerned. A memory index Δp ≈ 0 was returned from fitting the LFA-1/ICAM-1 data in Fig. 3A. Because of the limited amount of data, small fluctuations of Δp from zero could be observed even for Bernoulli sequences, as seen from computer simulations of a small number of cell pairs. Fitting the TCR/pMHC data (Fig. 3B) returned a positive Δp, indicating the presence of memory with positive feedback, whereas fitting the C-cadherin data (Fig. 3C) returned a negative Δp, indicating the presence of memory with negative feedback. These results support our preliminary conclusions based on the observations in Fig. 2 and preliminary analysis using the p₀₁, p₁₁, and P_a comparison.

The above fittings used the distribution of clusters of all sizes (i.e., all m values) for a given P_a to evaluate Δp. However, differences in the distribution of cluster sizes are dominated by the difference in the expected number of clusters of size 1 (Fig. 3, comparing the first bar in each panel). Thus, the number of clusters of size 1 in experimental adhesion score sequences can be used as a simple indicator to evaluate the memory effect.

Analysis so far has used the raw data shown in Fig. 2, which have similar P_a values. To obtain further support with ≈10× more data, we varied contact times and/or receptor–ligand densities to obtain different average adhesion frequencies, which also allowed us to examine the potential effects of molecular densities on Δp through P_a. For each system, the experimental number of clusters of size 1, M_exp(1), for each P_a value was plotted in Fig. 4B to compare with the theoretical curves, M_M[1, 50, p(50, P_a, Δp), Δp]. It can be seen that the LFA-1/ICAM-1 data are scattered evenly from both sides of the solid curve corresponding to Δp = 0. By comparison, almost all of the TCR/pMHC data are below the theoretical curve for Δp = 0, and most of the C-cadherin data are above that curve. These results further support our conclusions regarding three types of behaviors.

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 Δp 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 and4),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 P_a has midrange values (4). The data in Fig. 5 thus suggest that a single TCR/pMHC bond, engaged for <1 s, is sufficient to generate significant 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.

K562 cells expressing LFA-1 were a gift from T. A. Springer (Harvard Medical School, Boston, MA). Purified mouse glycosyl phosphatidylinositol (GPI)-anchored ICAM-1 was a gift from P. Selvaraj (Emory University School of Medicine, Atlanta, GA). T cells from OTI transgenic mice expressing H-2K^b MHC-restricted OTI TCR specific for an OVA peptide were a gift from B. D. Evavold (Emory University School of Medicine). Mouse H-2K^b MHC bound with an ovalbumin-derived peptide OVA (SIINFEKL, amino acids 257–264) was produced by the National Institutes of Health Tetramer Facility at Emory University. To isolate TCR binding, a chimeric MHC molecule that replaced the mouse H-2K^b α3 domain with the human HLA-A2 α3 domain was used to eliminate CD8 binding and was a gift from J. Altman (Emory University School of Medicine). CHO cells expressing full-length C-cadherin were generated with a plasmid provided by B. Gumbiner (University of Virginia School of Medicine, Charlottesville, VA), who also provided stably transfected CHO cells that secreted Fc-tagged C-cadherin, which was purified as described previously (22).

Human RBCs were isolated from whole peripheral blood of healthy donors, in accordance with a protocol approved by the Institutional Review Board of the Georgia Institute of Technology. GPI-ICAM-1 was reconstituted in RBC membrane by a 2.5-hr incubation. Biotin–streptavidin coupling was used to coat biotinylated pMHC monomers onto the RBC surface. Chromium chloride coupling was used to coat an anti-human IgG Fc antibody (Sigma–Aldrich, St. Louis, MO) on the RBC surface, with subsequent incubation of the RBCs with Fc-tagged C-cadherin.

Site densities of the receptors and ligands on cell membranes were measured by flow cytometry. The specificity of measured adhesion was confirmed by using blocking monoclonal antibodies directed against the receptors and/or ligands involved, by not coating the ligands on the RBCs, and by using EDTA to chelate the divalent cations required for LFA-I/CAM-1 binding and for C-cadherin binding. All treatments substantially reduced average adhesion frequencies.

Three methods were used to evaluate the memory index. Direct calculation uses the transition probabilities defined by Eq. 1 to express Δp:

The theoretical prediction M_M(m, n, p, Δp) from Eq. 4 was fit either to the experimental distribution of adhesion clusters, M_exp(m), shown in Fig. 3, or to the measured number of clusters of size 1, M_exp (1), shown in Fig. 4B by the least squares method. p in Eq. 4 as a function of P_a, Δp, and n is solved from Eq. 5 as follows: