Single-molecule force spectroscopy (SMFS) techniques play a pivotal role in unraveling the mechanics and conformational transitions of biological macromolecules under external forces. Among these techniques, multiplexed magnetic tweezers (MT) are particularly well suited to probe very small forces, ≤1 pN, critical for studying noncovalent interactions and regulatory conformational changes at the single-molecule level. However, to apply and measure such small forces, a reliable and accurate force-calibration procedure is crucial. Here, we introduce a new approach to calibrate MT based on thermal motion using the Hadamard variance (HV). To test our method, we perform bead-tether Brownian dynamics simulations that mimic our experimental system and compare the performance of the HV method against two established techniques: power spectral density (PSD) and Allan variance (AV) analyses. Our analysis includes an assessment of each method’s ability to mitigate common sources of additive noise, such as white and pink noise, as well as drift, which often complicate experimental data analysis. We find that the HV method exhibits overall similar or higher precision and accuracy, yielding lower force estimation errors across a wide range of signal-to-noise ratios (SNRs) and drift speeds compared with the PSD and AV methods. Notably, the HV method remains robust against drift, maintaining consistent uncertainty levels across the entire studied SNR and drift speed spectrum. We also explore the HV method using experimental MT data, where we find overall smaller force estimation errors compared with PSD and AV approaches. Overall, the HV method offers a robust method for achieving sub-pN resolution and precision in multiplexed MT measurements. Its potential extends to other SMFS techniques, presenting exciting opportunities for advancing our understanding of mechanosensitivity and force generation in biological systems. To make our methods widely accessible to the research community, we provide a well-documented Python implementation of the HV method as an extension to the Tweezepy package.

中文翻译：

---

使用 Hadamard 方差进行精确的漂移不变单分子力校准


单分子力谱 （SMFS） 技术在解开生物大分子在外力作用下的力学和构象转变中起着关键作用。在这些技术中，多路复用磁镊 （MT） 特别适合探测非常小的力 ≤1 pN，这对于研究单分子水平的非共价相互作用和调节构象变化至关重要。然而，要施加和测量如此小的力，可靠和准确的力校准程序至关重要。在这里，我们介绍了一种使用 Hadamard 方差 （HV） 基于热运动校准 MT 的新方法。为了测试我们的方法，我们执行了珠系布朗动力学模拟，模拟我们的实验系统，并将 HV 方法的性能与两种已建立的技术进行比较：功率谱密度 （PSD） 和艾伦方差 （AV） 分析。我们的分析包括评估每种方法减轻常见加性噪声源（如白噪声和粉红噪声）以及漂移的能力，这通常使实验数据分析复杂化。我们发现 HV 方法总体上表现出相似或更高的精度和准确度，与 PSD 和 AV 方法相比，在很宽的信噪比 （SNR） 和漂移速度范围内产生较低的力估计误差。值得注意的是，HV 方法对漂移保持稳健性，在整个研究的 SNR 和漂移速度谱中保持一致的不确定性水平。我们还使用实验 MT 数据探索了 HV 方法，与 PSD 和 AV 方法相比，我们发现总体上力估计误差较小。总体而言，HV 方法为在多路复用 MT 测量中实现亚 pN 分辨率和精度提供了一种可靠的方法。 它的潜力延伸到其他 SMFS 技术，为促进我们对生物系统中机械敏感性和力产生的理解提供了令人兴奋的机会。为了使我们的方法被研究社区广泛使用，我们提供了 HV 方法的有据可查的 Python 实现作为 Tweezepy 包的扩展。