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- Memory in receptor–ligand-mediated cell adhesion

**Results**

The micropipette adhesion frequency assay repeats, sequentially,^{}*n* tests with a single pair of cells (4). Each test is performed^{}by using computer-automated and piezoelectric translator-driven^{}micromanipulation to control the contact time and area, ensuring^{}it to be as nearly identical to any other tests in the same^{}sequence as possible. Each test generates a random binary adhesion^{}score. The probability of adhesion depends on the kinetic rates^{}of receptor–ligand interaction, surface densities of interacting^{}molecules, and contact time and area.^{}

The result of such *n* repeated tests is a random sequence whose^{}value *X** _{i}* at the

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Another way to visualize the sequences in Fig. 2 *A–C* is^{}to plot the nonzero *X*_{i}*F** _{n}* vs.

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. 2*D*), the sequences for the TCR/pMHC^{}interaction (Fig. 2*E*) appear more "clustered," whereas those^{}for the C-cadherin interaction (Fig. 2*F*) 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

Direct calculations of *p*_{01} and *p*_{11} 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. 3*A*), the distribution for the^{}TCR/pMHC interaction (Fig. 3*B*) has more clusters of large size,^{}whereas the distribution for the C-cadherin interaction (Fig. 3*C*)^{}has more single adhesion events surrounded by no-adhesion events.^{}

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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 *X*_{1} 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^{}*mM** _{B}*(

*M*_{B} is plotted vs. *P*_{a} (= *p*) in Fig. 4*A* 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. 3*A*)^{}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. 4*A*, 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 *i*th test by [*P*]:

By applying the Chapman–Kolmogorov equation^{}(8), the *n*-step transition matrix can be obtained as [*P*^{(n)}]^{}= [** P**]

Setting *p* = 0 reduces Eq. **4** to Eq. **2**, as required. Eq. **4** has^{}another special case at *p* = 1 – *p* when it simplifies to^{}*M*_{M}(*m*, *n*, *p*, 1 – *p*) = *p*(1 – *p*)^{n}* ^{–m}*, which describes

Experimentally, the adhesion probability can be estimated from^{}the adhesion frequency *F** _{n}*. The expected value of

If *p* = 0 (i.e., a Bernoulli sequence), *F** _{n}* approaches

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

It can be shown by direct calculation that *mM*_{M}(*m*, *n*, *p*, *p*) = *p*[*n* – *p*(1 – *p ^{n}*)/(1

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 *p* 0 was returned from fitting the LFA-1/ICAM-1^{}data in Fig. 3*A*. Because of the limited amount of data, small^{}fluctuations of *p* 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. 3*B*) returned^{}a positive *p*, indicating the presence of memory with positive^{}feedback, whereas fitting the C-cadherin data (Fig. 3*C*) returned^{}a negative *p*, 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 *p*_{01}, *p*_{11}, 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 *p*. 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 10x 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 *p* 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. 4*B* to compare with the theoretical curves, *M*_{M}[1, 50, *p*(50,^{}*P*_{a}, *p*), *p*]. It can be seen that the LFA-1/ICAM-1 data are scattered^{}evenly from both sides of the solid curve corresponding to *p*^{}= 0. By comparison, almost all of the TCR/pMHC data are below^{}the theoretical curve for *p* = 0, and most of the C-cadherin^{}data are above that curve. These results further support our^{}conclusions regarding three types of behaviors.^{}

The memory index *p* was obtained from fitting experimental cluster^{}size distribution with Eq. **4** and is plotted in Fig. 5(solid^{}bars) along with *p* estimated from direct calculation (using^{}Eq. **6** below) (open bars). Comparable results were obtained by^{}both approaches for all *P*_{a} values tested for all three systems^{}exhibiting qualitatively distinct behaviors. *p* values for the^{}LFA-1/ICAM-1 system are not statistically significantly different^{}from zero (*P* value 0.23) except in one instance (*P* value =^{}0.06). By comparison, a vast majority of the *p* values for the^{}TCR/pMHC system are statistically significantly greater than^{}zero, whereas half of the *p* values for the C-cadherin system^{}are statistically significantly less than zero and are marked^{}with asterisks on the top of corresponding solid bars to indicate^{}*P* value 0.05.^{}

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